Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 381806 12 pageshttpdxdoiorg1011552013381806

Research ArticleUnsteady Hydromagnetic Heat and Mass Transfer Flow of a HeatRadiating and Chemically Reactive Fluid Past a Flat PorousPlate with Ramped Wall Temperature

R Nandkeolyar1 M Das2 and P Sibanda1

1 School of Mathematics Statistics and Computer Science University of KwaZulu-Natal Private Bag X01Scottsville Pietermaritzburg 3209 South Africa

2Department of Mathematics School of Applied Sciences KIIT University Bhubaneswar 751024 India

Correspondence should be addressed to P Sibanda sibandapukznacza

Received 8 March 2013 Revised 19 July 2013 Accepted 19 July 2013

Academic Editor Oluwole Daniel Makinde

Copyright copy 2013 R Nandkeolyar et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Unsteady hydromagnetic free convective flow of a viscous incompressible electrically conducting and heat radiating fluid pasta flat plate with ramped wall temperature and suctionblowing is studied The governing equations are first subjected to Laplacetransformation and then inverted numerically using INVLAP routine of Matlab The numerical solutions of the fluid properties arepresented graphically while the skin friction and heat and mass transfer coefficients are presented in tabular form The results areverified by a careful comparison with results in the literature for certain parameter values

1 Introduction

Magnetohydrodynamic (MHD) free convective flow hasattracted many researchers due to its applications in manyfluid-engineering problems such as in MHD generators andpumps plasma studies nuclear reactors boundary layer flowcontrol and geothermal energy extraction Mention maybe made of studies by Gupta [1 2] Cramer [3] Pop [4]Kuiken [5] Wilks [6] Hossain [7] Aldoss et al [8] Helmy[9] Kim [10] Takhar et al [11] and Ahmed et al [12] Inall these studies the effects of thermal radiation are nottaken into account However in industrial applications suchas glass production furnace design thermonuclear fusioncasting and levitation and in space technology applicationssuch as cosmical flights propulsion systems plasma physicsand space reentry aerodynamics which operate at highertemperature radiation effect becomes significant Hossainand Takhar [13] considered the effects of radiation on mixedconvection along a vertical plate with uniform temperatureBakier and Gorla [14] studied the effects of radiation onmixed convection flow over a horizontal surface embeddedin a saturated porous medium Takhar et al [15] considered

the effects of radiation on MHD free convection flow of agas past a semi-infinite vertical plate Chamkha [16] studiedthermal radiation and buoyancy effects on MHD flow overan accelerating permeable surface with a heat source or sinkAzzam [17] analyzed the effects of radiation onMHD free andforced convection flow past a semi-infinite moving verticalplate with high temperature differences Israel-Cookey et al[18] studied the effects of viscous dissipation and radiation onunsteady MHD free convection flow past an infinite heatedvertical plate in a porous medium with time dependentsuction Mahmoud [19] studied the thermal radiation effecton unsteady MHD free convection flow of an electricallyconducting fluid past an infinite vertical porous plate takingviscous dissipation into account Bestman and Adjepong[20] investigated unsteady MHD free convection flow ofan incompressible optically thick fluid with radiative heattransfer near a moving plate in a rotating medium

Chamkha et al [21] investigated radiation effects on freeconvection flow past a semi-infinite vertical plate in thepresence of mass transfer Thermal radiation effects on non-Darcy free convection flow with lateral mass transfer were

2 Mathematical Problems in Engineering

studied by El-Hakiem and El-Amin [22] They presented aboundary layer analysis of the influence of thermal radiationand lateral mass flux on non-Darcy natural convection overa vertical flat plate in a fluid saturated porous mediumPrasad et al [23] investigated the effects of radiation andmasstransfer on two-dimensional flow past an impulsively startedinfinite vertical plate Makinde and Ogulu [24] consideredthe heat and mass transfer flows of a variable viscosityfluid past a vertical porous plate permeated by a transversemagnetic field with thermal radiation effects The effectsof thermal radiation and variable fluid viscosity on freeconvective flow and heat transfer past a porous stretchingsurface are investigated by Mukhopadhyay and Layek [25]Pal [26] studied heat and mass transfer in a stagnation pointflow towards a stretching surface in the presence of buoyancyforce and thermal radiation

Combined heat and mass transfer flow in the presenceof chemically reactive species concentration with or withoutapplied magnetic field has bearing on many transport pro-cesses present in nature and also in science and engineer-ing applications In processes such as drying evaporationenergy transfer in a cooling tower and the flow in a desertcooler heat and mass transfer occur simultaneously Freeconvection processes of involving the combined mechanismare also encountered inmany natural processes and industrialapplications such as in the curing of plastics the cleaningand chemical processing of materials and the manufacturingof pulp and insulated cables Chamkha [27] studied MHDflow over a uniformly stretched vertical permeable surfacesubject to a chemical reaction Afify [28] analyzed the MHDfree convective flow and mass transfer over a stretching sheetwith a homogeneous chemical reaction of order 119899 where 119899was taken to be 0 1 2 or 3 The influence of a chemicalreaction on heat and mass transfer from vertical surfaces inporous media subject to Soret and Dufour effects was studiedby Postelnicu [29] He showed that the thickness of theconcentration boundary layer decreases as the Lewis numberincreases a phenomenon also evident when a chemicalreaction is absent Kandasamy and Palanimani [30] studiedthe effects of a chemical reaction on heat andmass transfer ona magnetohydrodynamic boundary layer flow over a wedgewith ohmic heating and viscous dissipation in a porousmedium Pal and Mondal [31] studied the effects of SoretDufour chemical reaction and thermal radiation on MHDnon-Darcy unsteadymixed convective heat andmass transferover a stretching sheet

In all the above investigations the analytical or numericalsolution is obtained assuming that the temperature at theinterface was continuous and well defined However thereexist several problems of physical interest which may requirenonuniform or arbitrary wall conditions Several researchers[32ndash35] have investigated free convection from a verticalplate with stepdiscontinuities in the surface temperatureChandran et al [36] considered unsteady natural convectionflow of a viscous incompressible fluid near a vertical platewith ramped wall temperature Seth et al [37] studiedunsteady natural convection flow of a viscous incompressibleelectrically conducting fluid past an impulsively moving ver-tical plate in a porousmediumwith ramped wall temperature

taking into account the effects of thermal radiation Theycompared the results of natural convection near a rampedtemperature plate with those of natural convection near anisothermal plate Recently Seth et al [38] extended thisproblem to include the effects of rotation

Suction or blowing through the wall plays importantrole in boundary layer formation An increase in suctionvelocity may lead to a delay in the formation of a boundarylayer and an increase in blowing may lead to a decrease inthe skin friction at the plate which in turn may lead to adecrease in the rate of heat transfer at the plate Makinde [39]studied the effects of wall porosity on free convection flowwith thermal radiation and mass transfer Mbeledogu andOgulu [40] studied the heat andmass transfer rates in rotatingflow past a vertical porous flat plate Afify [41] discussedthe MHD free convective heat and mass transfer flow overa stretching sheet in the presence of suctioninjection withthermal diffusion and diffusion thermoeffects

The aim of the present study is to investigate the effectsof suctionblowing on unsteady hydromagnetic free con-vective flow and mass transfer of a viscous incompressibleelectrically conducting and heat radiating fluid past a flatplate with ramped wall temperature in the presence ofchemical reaction Such a fluid flow finds many engineeringapplications such as those in MHD devices and in severalnatural phenomena occurring subject to radiation in thepresence of chemically reactive species

2 Problem Formulation

Consider the unsteady free convective heat andmass transferflow of a viscous incompressible electrically conductingand heat radiating fluid past an impulsively moving verticalporous flat plate We choose the coordinate system in such away that the 119909-axis is along the plate in the upward directionthe119910-axis normal to the plate and the 119911-axis perpendicular tothe 119909119910-plane The fluid is permeated by a uniform transversemagnetic field 119861

0applied parallel to 119910-axis Initially at time

1199051015840le 0 the fluid and plate are at rest at a uniform temperature

1198791015840

infinand species concentration 1198621015840

infin At time 1199051015840 gt 0 the plate

begins to move in the 119909-direction with uniform velocity 1198800

Instantaneously the plate temperature is raised or lowered to1198791015840

infin+ (1198791015840

119908minus 1198791015840

infin)11990510158401199050when 1199051015840 lt 119905

0 and for 1199051015840 gt 119905

0the plate

is maintained at constant temperature 1198791015840119908 A constant species

concentration 1198621015840119908is maintained at the plate for 1199051015840 gt 0 Since

the plate is of infinite extent in the 119909- and 119911-directions and iselectrically nonconducting all physical quantities except thepressure are functions of 119910 and 1199051015840 only

The fluid is a metallic liquid whose magnetic Reynoldsnumber is small and hence the induced magnetic fieldproduced by the fluid motion is negligible in comparison tothe applied one [42] so that the magnetic field B = (0 119861

0 0)

Also no external electric field is applied so the effect ofpolarization of magnetic field is negligible [43] that is E =(0 0 0)The fluid flow is induced by the impulsive movementof the plate in the 119909-direction and there is constant flow offluid in 119910-direction due to the pores in the plate so that the

Mathematical Problems in Engineering 3

velocity vector is q = (119906 minusV0 0) With these assumptions the

governing model equations are given by

1205971199061015840

1205971199051015840minus V0

1205971199061015840

120597119910

= ]12059721199061015840

1205971199102minus

1205901198612

0

120588

1199061015840

+ 119892120573119879(1198791015840minus 1198791015840

infin) + 119892120573

119862(1198621015840minus 1198621015840

infin)

(1)

1205971198791015840

1205971199051015840minus V0

1205971198791015840

120597119910

=

119896

120588119888119901

12059721198791015840

1205971199102minus

1

120588119888119901

1205971199021015840

119903

1205971199101015840 (2)

1205971198621015840

1205971199051015840minus V0

1205971198621015840

120597119910

= 119863

12059721198621015840

12059711991010158402minus 119896119903(1198621015840minus 1198621015840

infin) (3)

where 1199061015840 V0 120588 119892 120573

119879 120573119862 1198791015840 1198621015840 119888

119901 119896 119902119903 ] 120590 119863 and 119896

119903are

respectively fluid velocity in 119909-direction suctioninjectionvelocity in 119910 direction fluid density acceleration due to grav-ity volumetric coefficient of thermal expansion volumetriccoefficient of expansion or contraction temperature of thefluid near the plate species concentration specific heat atconstant pressure thermal conductivity radiative flux kine-matic coefficient of viscosity electrical conductivity chemicalmolecular diffusivity and chemical reaction coefficient

Assuming that there is no slip between the plate and thefluid the initial and boundary conditions for the fluid flowproblem are

1199061015840= 0 119879

1015840= 1198791015840

infin 1198621015840= 1198621015840

infinfor 119910 ge 0 1199051015840 le 0 (4a)

1199061015840= 1198800

at 119910 = 0 for 1199051015840 gt 0 (4b)

1198791015840= 1198791015840

infin+

(1198791015840

119908minus 1198791015840

infin) 1199051015840

1199050

at 119910 = 0 for 0 lt 1199051015840 le 1199050 (4c)

1198791015840= 1198791015840

119908at 119910 = 0 for 1199051015840 gt 119905

0 (4d)

1198621015840= 1198621015840

119908at 119910 = 0 for 1199051015840 gt 0 (4e)

1199061015840997888rarr 0 119879

1015840997888rarr 119879

1015840

infin 1198621015840997888rarr 119862

1015840

infin

as 119910 997888rarr infin for 1199051015840 gt 0(4f)

For an optically thick fluid in addition to emission thereis also self-absorption and usually the absorption coefficientis wavelength dependent and large so that we can adopt theRosseland approximation for radiative flux vector 1199021015840

119903 The

radiative flux vector 1199021015840119903under the Rosseland approximation

is

1199021015840

119903= minus

4120590lowast

3119896lowast

12059711987910158404

120597119910

(5)

where 119896lowast is the mean absorption coefficient and 120590lowast is theStefan-Boltzmann constant Assuming a small temperaturedifference between the fluid temperature 1198791015840 and the freestream temperature 1198791015840

infin 11987910158404 is expanded in a Taylor series

about the free stream temperature1198791015840infin Neglecting second and

higher order terms in (1198791015840 minus 1198791015840infin) we obtain

11987910158404cong 411987910158403

infin1198791015840minus 311987910158404

infin (6)

Using (5) and (6) in (2) we obtain

1205971198791015840

1205971199051015840minus V0

1205971198791015840

120597119910

=

119896

120588119888119901

12059721198791015840

1205971199102+

1

120588119888119901

16120590lowast11987910158403

infin

3119896lowast

12059721198791015840

120597y2 (7)

Introducing the following dimensionless variables

120578 =

1199101015840

11988001199050

119906 =

1199061015840

1198800

119905 =

1199051015840

1199050

119879 =

1198791015840minus 1198791015840

infin

1198791015840

119908minus 1198791015840

infin

119862 =

1198621015840minus 1198621015840

infin

1198621015840

119908minus 1198621015840

infin

(8)

the governing equations (1) (3) and (7) in dimensionlessform become

120597119906

120597119905

minus 119878

120597119906

120597120578

=

1205972119906

1205971205782+ 119879 + Gm119862 minus119872119906

120597119879

120597119905

minus 119878

120597119879

120597120578

=

(1 + 119873)

Pr1205972119879

1205971205782

120597119862

120597119905

minus 119878

120597119862

120597120578

=

1

119878119888

1205972119862

1205971205782minus 119870119903119862

(9)

where 119878 = V01198800is the suction or blowing parameter Gm =

119892120573119862](1198621015840119908minus 1198621015840

infin)1198803

0is the mass Grashof number 119872 =

1205901198612

0]12058811988020is the magnetic parameter 119873 = 16120590

lowast11987910158403

infin3119896119896lowast

is the thermal radiation parameter Pr = 120588]119888119901119896 is the

Prandtl number Sc = ]119863 is the Schmidt number and119870119903= 119896119903]21198631198802

0is the chemical reaction parameter Here

119878 gt 0 corresponds to suction whereas 119878 lt 0 correspondsto blowing The characteristic time 119905

0and the characteristic

velocity 1198800are defined as

1199050=

]

1198802

0

1198800= [119892120573

119879] (1198791015840119908minus 1198791015840

infin)]

13

(10)

The initial and boundary conditions (4a)ndash(4f) in nondi-mensional form are

119906 = 0 119879 = 0 119862 = 0 for 120578 ge 0 119905 le 0 (11a)

119906 = 1 119862 = 1 at 120578 = 0 for 119905 gt 0 (11b)

119879 = 119905 at 120578 = 0 for 0 lt 119905 le 1 (11c)

119879 = 1 at 120578 = 0 for 119905 gt 1 (11d)

119906 997888rarr 0 119879 997888rarr 0 119862 997888rarr 0 as 120578 997888rarr infin for 119905 gt 0(11e)

The system of partial differential equations (9) subject tothe initial and boundary conditions (11a)ndash(11e) represents themodel for unsteady hydromagnetic free convective flow ofa viscous incompressible electrically conducting and heatradiating fluid past an infinite porous flat plate with rampedwall temperature in the presence of a chemically reactivespecies

4 Mathematical Problems in Engineering

Table 1 A comparison of minus120575119879120575120578|120578=0

when 119878 = 0

119905 119873 Seth et al [37] Present results

02

05 03472 03471791 03007 03006665 01736 017359010 01282 0128204

04

05 04910 04909851 04252 04252065 02455 024549310 01813 0181308

06

05 06013 06013321 05208 05207695 03007 030066610 02221 0222057

08

05 06944 06943581 06013 06013325 03472 034717910 02564 0256409

3 Analytic Solutions

Using the Laplace transform technique on the system ofequations (9) subject to the initial and boundary conditions(11a)ndash(11e) we obtain

(120578 119904) = minus [

Gm1199041198631

1198901205821120578+

(1 minus 119890minus119904)

11990421198632

1198901205822120578]

+

1

119904

+

(1 minus 119890minus119904)

11990421198631

+

Gm1199041198632

1198901205823120578

(12)

119879 (120578 119904) =

(1 minus 119890minus119904)

1199042

1198901205822120578 (13)

119862 (120578 119904) =

1198901205821120578

119904

(14)

where 119862(120578 119904) 119879(120578 119904) (120578 119904) are respectively the Laplacetransforms of 119862(120578 119905) 119879(120578 119905) and 119906(120578 119905) 119904 gt 0 is the Laplacetransform parameter and

1205821= 05 minus119878Sc minus radic1198782Sc2 + 4Sc (119870

119903+ 119904) (15a)

1205822= 05

minus119878(

Pr1 + 119873

) minusradic1198782(

Pr1 + 119873

)

2

+ 4 (

Pr1 + 119873

) 119904

(15b)

1205823= 05 minus119878 minus radic119878

2+ 4 (119872 + 119904) (15c)

1198631= 1205822

1+ 1198781205821minus (119872 + 119904) (15d)

1198632= 1205822

2+ 1198781205822minus (119872 + 119904) (15e)

An exact Laplace transform inversion of (12) can beobtained when there is no suctionblowing that is when

119878 = 0 However for a nonzero 119878 the inversion of (12) isnot possible Thus the inversion of (12)ndash(14) was obtainednumerically using INVLAP routine in Matlab However tocompare the results which are obtained using INVLAP rou-tine the exact inversion of (13) and (14) gave the solutions

119879 (120578 119905) = 119890minus119887120578[119875 (120578 119905) minus 119867 (119905 minus 1) 119875 (120578 119905 minus 1)] (16)

119862 (120578 119905) =

1

2

[119890minus120578(119886minusradic119889) erfc (119905

1) + 119890minus120578(119886+radic119889) erfc (119905

2)] (17)

where erfc is the complementary error function 119867 is theHeaviside unit step function and

119875 (120578 119905) = (

119905

2

+

119898120578

4radic119899

) 119890120578radic119899 erfc (119905

3)

+ (

119905

2

minus

119898120578

4radic119899

) 119890minus120578radic119899 erfc (119905

4)

1199051 1199052= plusmnradic

119889119905

119888

+

120578

2

radic

119888

119905

1199053 1199054= plusmnradic

119899119905

119898

+

120578

2

radic

119898

119905

119886 =

119878Sc2

119887 =

Pr1198782 (1 + 119873)

119888 = Sc

119889 =

1198782Sc2

4

+ Sc119870119903 119898 =

Pr1 + 119873

119899 =

Pr21198782

4(1 + 119873)2

(18)

In the absence of suctionblowing and the thermal radi-ation effect (119878 = 0 and 119873 = 0) the solution 119879(120578 119905) in (16) isin agreement with the solution obtained by Chandran et al[36]

31 Skin Friction Nusselt Number and Sherwood NumberThe nondimensional quantities of engineering interest arethe skin friction 120591 which is a measure of shear stress at theplate theNusselt numberNu whichmeasures the rate of heattransfer and the Sherwood number Sh which measures therate of mass transfer at the plateThe Nusselt number Nu andthe Sherwood number Sh have the exact values

NuRe= minus (1 + 119873)(

120597119879

120597120578

)

120578=0

= minus [119891 (119905) minus 119867 (119905 minus 1) 119891 (119905 minus 1)]

ShRe= minus(

120597119862

120597120578

)

120578=0

= (119886 + radic119889) + radic

119888

120587119905

119890minus119889119905119888

minus radic119889 erfc(radic119889119905119888

)

(19)

Mathematical Problems in Engineering 5

Table 2 Comparison of exact and numerical values of NuRe and ShRe for different values of 119905 and 119878 when119870119903= 05 Sc = 1 Pr = 071 and

119873 = 1

119905 119878

NuRe NuRe ShRe ShReExact Numerical Exact Numerical

02minus1 053388346 0533883 094620854 09462090 060133195 0601332 138566159 13856621 067588346 0675883 194620854 1946209

04minus1 071843936 0718439 064706152 06470620 085041184 0850412 106475703 10647571 100243936 1002439 164706152 1647062

06minus1 084692696 0846927 053359416 05335940 104153758 1041537 093657181 09365721 127292696 1272927 153359416 1533594

08minus1 094692675 0946927 047550301 04755030 120266405 1202664 086753072 08675311 151492675 1514927 147550301 1475503

0 1 2 3 4 50

01

02

03

04

05

06

07

08

Exact solution Numerical solution

S = minus1 0 1

T(120578

120578

t)

(a)

Exact solution Numerical solution

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

S = minus1 0 1

C(120578

120578

t)

(b)

Figure 1 Comparison of numerical and exact solutions of (a) 119879(120578 119905) and (b) 119862(120578 119905) for different values of 119878 when Pr = 071 Sc = 1119873 = 1119870119903= 05 and 119905 = 05

where Re = 1198800119871] is the Reynolds number 119871 is some

characteristic length and

119891 (119905) = (1 + 119873) [(

119898

2radic119899

+ radic119899119905) erfc(radic119899119905119898

)

minus(

119898

2radic119899

+ 119887119905 + radic119899119905) minus radic119898119905

120587

119890minus119899119905119898

]

(20)

The values of skin friction 120591 can be obtained numericallyusing the Matlab INVLAP routine

4 Validation of Numerical Results

In order to validate the numerical results obtained using theMatlab INVLAP routine the values of NuRe are comparedwith the exact values obtained by Seth et al [37] in Table 1Further the numerical values of NuRe and ShRe arecomparedwith the exact analytical values in Table 2Thefluidtemperature and species concentration profiles are plottedusing both the numerical and exact values and are compared

6 Mathematical Problems in Engineering

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

M = 3 5 7

Gm = 1

Suction (S = 1) Blowing (S = minus1)

u(120578t)

120578

(a)

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

M = 3 5 7

Gm = minus1

Suction (S = 1) Blowing (S = minus1)

u(120578t)

120578

(b)

Figure 2 Effect of119872 on 119906(120578 119905) when Sc = 1 119870119903= 05119873 = 1 Pr = 071 and 119905 = 07

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

1

18 2 220

005

01

015

N = 05 10 15

Gm = 1

Suction (S = 1) Blowing (S = minus1)

u(120578t)

u(120578t)

120578

120578

(a)

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

1

N = 05 10 15

Gm = minus1

08 1 12

015

02

025

Suction (S = 1) Blowing (S = minus1)

u(120578t)

u(120578t)

120578

120578

(b)

Figure 3 Effect of119873 on 119906(120578 119905) when Sc = 1 119870119903= 05119872 = 3 Pr = 071 and 119905 = 07

in Figure 1 An excellent agreement between the values isobserved which validates the accuracy of the INVLAP results

5 Results and Discussion

The unsteady hydromagnetic free convective flow of a vis-cous incompressible electrically conducting and heat radi-ating fluid past an infinite vertical flat plate in the presence ofsuctionblowing and chemically reactive species concentra-tion has been studiednumerically using the Matlab INVLAP

routine The fluid velocity 119906(120578 119905) temperature 119879(120578 119905) andspecies concentration 119862(120578 119905) profiles are given in Figures 23 4 5 6 7 8 9 10 and 11 whereas the numerical valuesof the skin friction heat and mass transfer coefficients arepresented in Tables 3 and 4 when 119905 = 07 Gm = 1 (whichcorresponds to assisting buoyancy) and Gm = minus1 (whichcorresponds to opposing buoyancy)

Figures 2 to 7 show the effects of the magnetic fieldthermal radiation suctionblowing chemical reaction massdiffusion and thermal diffusion on the fluid velocity Withassisting mass buoyancy (ie Gm = 1) the fluid velocity

Mathematical Problems in Engineering 7

0 1 2 3 4 5 60

01

02

03

04

05

06

07

08

09

1

S = 1 2 3

S = minus1 minus2 minus3

Gm = 1

u(120578t)

120578

(a)

0 1 2 3 4 5 60

01

02

03

04

05

06

07

08

09

1

S = minus1 minus2 minus3

Gm = minus1

u(120578t)

120578

S = 1 2 3

(b)

Figure 4 Effect of 119878 on 119906(120578 119905) when Sc = 1 119870119903= 05119873 = 1 Pr = 071119872 = 3 and 119905 = 07

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

08 1 12 14 16

015

02

025

03

Gm = 1

u(120578t)

120578

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

Kr = 05 10 15

(a)

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

1

12 14 16

005

01

015

Gm = minus1

u(120578t)

u(120578t)

120578

120578

Suction (S = 1) Blowing (S = minus1)

Kr = 0 10 15

(b)

Figure 5 Effect of119870119903on 119906(120578 119905) when Sc = 1119872 = 3119873 = 1 Pr = 071 and 119905 = 07

decreases with an increase in the magnetic parameter 119872chemical reaction parameter 119870

119903 Schmidt number Sc and

Prandtl number Pr while it increases with the thermalradiation parameter 119873 Since the Schmidt number Sc is theratio of viscous to mass diffusivity an increase in Sc impliesa decrease in the mass diffusion rate Hence the magneticfield and the chemical reaction rate tend to reduce the fluidvelocity whereas the mass diffusivity thermal diffusivity andthermal radiation tend to increase the fluid velocity Withopposing mass buoyancy (ie for Gm = minus1) the behaviourof the fluid velocity with respect to all governing parametersremains the same except with respect to Sc when the fluidvelocity decreases with an increase in mass diffusivity

Figure 4 shows that the fluid velocity decreases with anincrease in suction velocity whereas it increases with anincrease in blowing for both assisting and opposing massbuoyancy This result may be useful in engineering applica-tions where the formation of boundary layer is to be delayedThis can be done by increasing the suction velocity We alsoobserve that the thickness of the momentum boundary layeris less in the presence of opposing mass buoyancy than in thepresence of assisting mass buoyancy

Figures 8 9 and 10 show the effects of thermal radi-ation thermal diffusion and suctionblowing on the fluidtemperature We observe from Figures 8 and 9 that the fluidtemperature increases with119873 while it decreases with Pr The

8 Mathematical Problems in Engineering

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

Sc = 05 10 15

Gm = 1

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

(a)

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

Sc = 05 10 15

Gm = minus1

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

(b)

Figure 6 Effect of Sc on 119906(120578 119905) when119872 = 3 119870119903= 05119873 = 1 Pr = 071 and 119905 = 07

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

1

Pr = 071 1 7

Gm = 1

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

(a)

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

1

Pr = 071 1 7

Gm = minus1

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

(b)

Figure 7 Effect of Pr on 119906(120578 119905) when Sc = 1 119870119903= 05119873 = 1119872 = 3 and 119905 = 07

Prandtl number is the ratio of viscosity to thermal diffusivityand an increase in Pr implies a decrease in thermal diffusivityThermal radiation and diffusivity tend to increase the fluidtemperature Figure 10 shows that an increase in suctionvelocity causes a reduction in the fluid temperature whereasan increase in blowing increases the fluid temperature

Figures 11 12 and 13 show the effects of the chemicalreaction mass diffusion and suction or blowing on thespecies concentration We observe from Figures 11 and 12that the species concentration decreases with an increase inthe chemical reaction parameter or the Schmidt numberThus the chemical reaction rate tends to reduce the species

concentration whereas mass diffusivity increases the speciesconcentration It is clear from Figure 13 that the suctionvelocity reduces the species concentration while blowingcauses an increase in the species concentration We observefrom Figures 11 and 12 that the species concentration is higherin the case of blowing than in the case of suction We alsoobserve from Figures 2ndash11 that the momentum thermal andconcentration boundary layers are thicker in the case ofblowing than in the case of suction

Table 3 shows the effects of the magnetic field thermalradiation suctionblowing chemical reactionmass diffusionand thermal diffusion on the skin friction in the presence

Mathematical Problems in Engineering 9

Table 3 Effect of fluid and physical parameters on the skin friction minus120591 when 119905 = 07

119872 119873 119878 119870119903

Sc Pr minus120591 (119878 gt 0Gm = 1)

minus120591 (119878 lt 0Gm = 1)

minus120591 (119878 gt 0Gm = minus1)

minus120591 (119878 lt 0Gm = minus1)

3 1 mdash 05 1 071 1747428 0788854 2397344 14387705 mdash mdash mdash mdash mdash 2291296 1325411 2865812 18999277 mdash mdash mdash mdash mdash 2737296 1766244 3258082 2287031mdash 05 mdash mdash mdash mdash 1763388 0794076 2413303 1443992mdash 10 mdash mdash mdash mdash 1747428 0788854 2397344 1438770mdash 15 mdash mdash mdash mdash 1735676 0784937 2385592 1434853mdash mdash 1 mdash mdash mdash 1747428 mdash 2397344 mdashmdash mdash 2 mdash mdash mdash 2463830 mdash 3043645 mdashmdash mdash 3 mdash mdash mdash 3291595 mdash 3786658 mdashmdash mdash minus1 mdash mdash mdash mdash 0788854 mdash 1438770mdash mdash minus2 mdash mdash mdash mdash 0540588 mdash 1120403mdash mdash minus3 mdash mdash mdash mdash 0393845 mdash 0888908mdash mdash mdash 05 mdash mdash 1747428 0788854 2397344 1438770mdash mdash mdash 10 mdash mdash 1761672 0803098 2383100 1424526mdash mdash mdash 15 mdash mdash 1774216 0815642 2370555 1411981mdash mdash mdash mdash 05 mdash 1666559 0771682 2478212 1455942mdash mdash mdash mdash 10 mdash 1747428 0788854 2397344 1438770mdash mdash mdash mdash 15 mdash 1796602 0798486 2348170 1429138mdash mdash mdash mdash mdash 071 1747428 0788854 2397344 1438770mdash mdash mdash mdash mdash 10 1766522 0795089 2416438 1445005mdash mdash mdash mdash mdash 70 1884422 0830142 2534338 1480058

0 1 2 3 4 50

01

02

03

04

05

06

07

N = 05 10 15

Suction (S = 1) Blowing (S = minus1)

T(120578

120578

t)

Figure 8 Effect of119873 on 119879(120578 119905) when Pr = 071 and 119905 = 07

of assisting and opposing mass buoyancy In the case ofassisting mass buoyancy an increase in 119872 119870

119903 Sc and Pr

tends to increase the skin friction whereas an increase in119873 reduces the skin friction for both suction and blowingcases Thus the skin friction increases with an increase inthe magnetic field parameter and the chemical reaction rate

0 1 2 3 4 50

01

02

03

04

05

06

07

Pr = 071 1 7

Suction (S = 1) Blowing (S = minus1)

T(120578

120578

t)

Figure 9 Effect of Pr on 119879(120578 119905) when119873 = 1 and 119905 = 07

while it decreaseswith an increase inmass diffusivity thermaldiffusivity and thermal radiation in the case of assisting massbuoyancy However for opposing mass buoyancy the skinfriction increases with mass diffusivity while decreasing withan increase in the rate of the chemical reaction

10 Mathematical Problems in Engineering

Table 4 Values of NuRe and ShRe for different values of governing parameters when 119905 = 07

119873 119878 Pr 119870119903

Sc NuRe NuRe ShRe ShRe(Suction) (Blowing) (Suction) (Blowing)

05 mdash 071 mdash mdash 1249449 0752449 mdash mdash10 mdash 071 mdash mdash 1396642 0899642 mdash mdash15 mdash 071 mdash mdash 1527010 1030010 mdash mdash1 1 071 05 10 1396642 mdash 1500355 mdash1 2 071 05 10 1712939 mdash 2280316 mdash1 3 071 05 10 2069304 mdash 3174187 mdash1 minus1 071 05 10 mdash 0899642 mdash 05003551 minus2 071 05 10 mdash 0718939 mdash 02803161 minus3 071 05 10 mdash 0578304 mdash 01741871 mdash 071 mdash mdash 1396642 0899642 mdash mdash1 mdash 10 mdash mdash 1723721 1023721 mdash mdash1 mdash 70 mdash mdash 6662968 1762968 mdash mdashmdash mdash mdash 05 1 mdash mdash 1500355 0500355mdash mdash mdash 10 1 mdash mdash 1691322 0691322mdash mdash mdash 15 1 mdash mdash 1865496 0865496mdash mdash mdash 05 05 mdash mdash 0921462 0421462mdash mdash mdash 05 10 mdash mdash 1500355 0500355mdash mdash mdash 05 15 mdash mdash 2035801 0535801

0 1 2 3 4 7650

01

02

03

04

05

06

07

S = minus1 minus2 minus3

S = 1 2 3

T(120578

120578

t)

Figure 10 Effect of 119878 on 119879(120578 119905) when Pr = 071119873 = 1 and 119905 = 07

Table 4 shows the effects of thermal radiation suc-tionblowing and thermal diffusion on the heat transfer ratefrom the plate It also shows the effects of suctionblowingchemical reaction rate andmass diffusion on the rate of masstransfer at the plate We observe that the Nusselt numberincreases with 119873 and Pr Thus the thermal radiation tendsto accelerate heat transfer at the plate while thermal diffusionhas reverse effect Table 4 also shows that the rate of heattransfer at the plate increases with an increase in suctionvelocity and decreases with blowing The Sherwood numberincreases with 119870

119903and Sc for both suction and blowing An

increase in the suction velocity causes an increase in therate of mass transfer whereas an increase in blowing has theopposite effect

0 05 1 15 2 25 3 35 4

Suction (S = 1) Blowing (S = minus1)

0

01

02

03

04

05

06

07

08

09

1

C(120578

120578

t)

Kr = 0 05 10

Figure 11 Effect of119872 on 119862(120578 119905) when Sc = 1 and 119905 = 07

6 Conclusions

The unsteady hydromagnetic free convection flow of aviscous incompressible and heat-radiating fluid past animpulsively started vertical porous flat plate with rampedtemperature in the presence of a chemically reactive speciesconcentration was studied For certain parameter valuesexact solutions of the model equations were found usingLaplace transforms The Matlab INVLAP routine was used tofind the numerical results in the general case The numericalresults were found to be in excellent agreement with the

Mathematical Problems in Engineering 11

Suction (S = 1) Blowing (S = minus1)

Sc = 05 10 15

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

C(120578

120578

t)

Figure 12 Effect of Sc on 119862(120578 119905) when 119870119903= 05 and 119905 = 07

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

S = 1 2 3

S = minus1 minus2 minus3

C(120578

120578

t)

Figure 13 Effect of 119878 on 119862(120578 119905) when 119870119903= 05 Sc = 1 and 119905 = 07

results of Seth et al [37] and with the exact results Theimportant findings of the present work may be summarizedas follows

(i) In the case of assisting mass buoyancy force the fluidvelocity is reduced by an increase in themagnetic fieldand the chemical reaction rate

(ii) Blowing causes an acceleration in the fluid velocitywhile suction has reverse effect on it

(iii) An increase in the chemical reaction rate tends toreduce the species concentration whereas the massdiffusivity parameter has reverse effect

(iv) Thermal radiation thermal diffusivity and blowingtend to increase the fluid temperature whereas suc-tion has the opposite effect

(v) In the case of an assisting mass buoyancy force themagnetic field and chemical reaction parameter tendto increase the skin friction while themass diffusivitythermal diffusivity and thermal radiation have thereverse effect

(vi) Thermal diffusivity and blowing tend to reduce therate of heat transfer at the plate whereas thermalradiation and suction have the opposite effect

Acknowledgment

The authors would like to thank the referee for his valuableand constructive suggestions

References

[1] A S Gupta ldquoSteady and transient free convection of anelectrically conducting fluid froma vertical plate in the presenceof a magnetic fieldrdquo Applied Scientific Research vol 9 no 1 pp319ndash333 1960

[2] A S Gupta ldquoLaminar free convection flow of an electricallyconducting fluid from a vertical plate with uniform surface heatflux and variable wall temperature in the presence of amagneticfieldrdquoZeitschrift fur AngewandteMathematik undPhysik vol 13pp 324ndash333 1962

[3] K R Cramer ldquoSeveral magnetohydrodynamic free convectionsolutionsrdquo Journal of Heat Transfer vol 85 pp 35ndash40 1963

[4] I Pop ldquoOn the unsteady hydromagnetic free convection flowpast a vertical infinite flat platerdquo Indian Journal of Physics vol43 pp 196ndash200 1969

[5] H K Kuiken ldquoMagnetohydrodynamic free convection in astrong cross fieldrdquo Journal of Fluid Mechanics vol 40 no 1 pp21ndash38 1970

[6] G Wilks ldquoMagnetohydrodynamic free convection about asemi-infinite vertical plate in a strong cross fieldrdquo Zeitschrift furAngewandte Mathematik und Physik ZAMP vol 27 no 5 pp621ndash631 1976

[7] M A Hossain ldquoEffect of hall current on unsteady hydromag-netic free convection flow near an infinite vertical porous platerdquoJournal of the Physical Society of Japan vol 55 no 7 pp 2183ndash2190 1986

[8] T K Aldoss M A Al-Nimr M A Jarrah and B J Al-SharsquoerldquoMagnetohydrodynamicmixed convection from a vertical plateembedded in a porous mediumrdquo Numerical Heat Transfer PartA vol 28 no 5 pp 635ndash645 1995

[9] K A Helmy ldquoMHD unsteady free convection flow past avertical porous platerdquo Zeitschrift fur Angewandte Mathematikund Mechanik vol 78 no 4 pp 255ndash270 1998

[10] Y J Kim ldquoUnsteadyMHD convective heat transfer past a semi-infinite vertical porous moving plate with variable suctionrdquoInternational Journal of Engineering Science vol 38 no 8 pp833ndash845 2000

[11] H S Takhar S Roy and G Nath ldquoUnsteady free convectionflow over an infinite vertical porous plate due to the combinedeffects of thermal and mass diffusion magnetic field and Hallcurrentsrdquo Heat and Mass Transfer vol 39 no 10 pp 825ndash8342003

[12] N Ahmed H K Sarmah and D Kalita ldquoThermal diffusioneffect on a three-dimensional MHD free convection with masstransfer flow from a porous vertical platerdquo Latin AmericanApplied Research vol 41 pp 165ndash176 2011

12 Mathematical Problems in Engineering

[13] M A Hossain and H S Takhar ldquoRadiation effect on mixedconvection along a vertical plate with uniform surface temper-aturerdquoHeat and Mass Transfer vol 31 no 4 pp 243ndash248 1996

[14] A Y Bakier and R S R Gorla ldquoThermal radiation effect onmixed convection from horizontal surfaces in saturated porousmediardquo Transport in Porous Media vol 23 no 3 pp 357ndash3631996

[15] H S Takhar R S R Gorla and VM Soundalgekar ldquoRadiationeffects onMHD free convection flowof a gas past a semi-infinitevertical platerdquo International Journal of Numerical Methods forHeat and Fluid Flow vol 6 no 2 pp 77ndash83 1996

[16] A J Chamkha ldquoThermal radiation and buoyancy effects onhydromagnetic flow over an accelerating permeable surfacewith heat source or sinkrdquo International Journal of EngineeringScience vol 38 no 15 pp 1699ndash1712 2000

[17] G E-D A Azzam ldquoRadiation effects on the MHDmixed free-forced convective flow past a semi-infinitemoving vertical platefor high temperature differencesrdquo Physica Scripta vol 66 no 1pp 71ndash76 2002

[18] C Israel-Cookey A Ogulu and V B Omubo-Pepple ldquoInflu-ence of viscous dissipation and radiation on unsteady MHDfree-convection flow past an infinite heated vertical plate ina porous medium with time-dependent suctionrdquo InternationalJournal of Heat andMass Transfer vol 46 no 13 pp 2305ndash23112003

[19] M A A Mahmoud ldquoThermal radiation effect on unsteadyMHD free convection flow past a vertical plate with tem-perature-dependent viscosityrdquo Canadian Journal of ChemicalEngineering vol 87 no 1 pp 47ndash52 2009

[20] A R Bestman and S K Adjepong ldquoUnsteady hydromagneticfree-convection flow with radiative heat transfer in a rotatingfluidmdashI Incompressible optically thin fluidrdquo Astrophysics andSpace Science vol 143 no 1 pp 73ndash80 1988

[21] A J Chamkha H S Takhar and V M Soundalgekar ldquoRadia-tion effects on free convection flow past a semi-infinite verticalplate with mass transferrdquo Chemical Engineering Journal vol 84no 3 pp 335ndash342 2001

[22] M A El-Hakiem and M F El-Amin ldquoThermal radiation effecton non-Darcy natural convection with lateral mass transferrdquoHeat and Mass Transfer vol 37 no 2-3 pp 161ndash165 2001

[23] V R Prasad N B Reddy and R Muthucumaraswamy ldquoRadia-tion and mass transfer effects on two-dimensional flow past animpulsively started infinite vertical platerdquo International Journalof Thermal Sciences vol 46 no 12 pp 1251ndash1258 2007

[24] O D Makinde and A Ogulu ldquoThe effect of thermal radiationon the heat and mass transfer flow of a variable viscosity fluidpast a vertical porous plate permeated by a transverse magneticfieldrdquo Chemical Engineering Communications vol 195 no 12pp 1575ndash1584 2008

[25] SMukhopadhyay andG C Layek ldquoEffects of thermal radiationand variable fluid viscosity on free convective flow and heattransfer past a porous stretching surfacerdquo International Journalof Heat andMass Transfer vol 51 no 9-10 pp 2167ndash2178 2008

[26] D Pal ldquoHeat andmass transfer in stagnation-point flow towardsa stretching surface in the presence of buoyancy force andthermal radiationrdquoMeccanica vol 44 no 2 pp 145ndash158 2009

[27] A J Chamkha ldquoMHDflowof a uniformly streched vertical per-meable surface in the presence of heat generationabsorptionand a chemical reactionrdquo International Communications inHeatand Mass Transfer vol 30 no 3 pp 413ndash422 2003

[28] A A Afify ldquoMHD free convective flow and mass transferover a stretching sheet with chemical reactionrdquo Heat and MassTransfer vol 40 no 6-7 pp 495ndash500 2004

[29] A Postelnicu ldquoInfluence of chemical reaction on heat andmasstransfer by natural convection from vertical surfaces in porousmedia considering Soret and Dufour effectsrdquo Heat and MassTransfer vol 43 no 6 pp 595ndash602 2007

[30] R Kandasamy and P G Palanimani ldquoEffects of chemical reac-tions heat and mass transfer on nonlinear magnetohydrody-namic boundary layer flow over a wedge with a porous mediumin the presence of ohmic heating and viscous dissipationrdquoJournal of Porous Media vol 10 no 5 pp 489ndash501 2007

[31] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[32] A A Hayday D A Bowlus and R A McGraw ldquoFree convec-tion from a vertical plate with step discontinuities in surfacetemperaturerdquo ASME Journal of Heat Transfer vol 89 pp 244ndash250 1967

[33] M Kelleher ldquoFree convection from a vertical plate with dis-continuous wall temperaturerdquo Journal of Heat Transfer Trans-actions ASME vol 93 no 4 pp 349ndash356 1971

[34] T-T Kao ldquoLaminar free convective heat transfer response alonga vertical flat plate with step jump in surface temperaturerdquoLetters inHeat andMass Transfer vol 2 no 5 pp 419ndash428 1975

[35] S Lee and M M Yovanovich ldquoLaminar natural convectionfrom a vertical plate with a step change in wall temperaturerdquoJournal of Heat Transfer vol 113 no 2 pp 501ndash504 1991

[36] P Chandran N C Sacheti and A K Singh ldquoNatural convec-tion near a vertical plate with ramped wall temperaturerdquo Heatand Mass Transfer vol 41 no 5 pp 459ndash464 2005

[37] G S Seth M S Ansari and R Nandkeolyar ldquoMHD naturalconvection flow with radiative heat transfer past an impulsivelymoving plate with ramped wall temperaturerdquo Heat and MassTransfer vol 47 no 5 pp 551ndash561 2011

[38] G S Seth R Nandkeolyar and M S Ansari ldquoEffects ofthermal radiation and rotation on unsteady hydromagnetic freeconvection flow past an impulsively moving vertical plate withramped temperature in a porous mediumrdquo Journal of AppliedFluid Mechanics vol 6 no 1 pp 27ndash38 2013

[39] O D Makinde ldquoFree convection flow with thermal radiationand mass transfer past a moving vertical porous platerdquo Interna-tional Communications in Heat and Mass Transfer vol 32 no10 pp 1411ndash1419 2005

[40] I U Mbeledogu and A Ogulu ldquoHeat and mass transfer of anunsteady MHD natural convection flow of a rotating fluid pasta vertical porous flat plate in the presence of radiative heattransferrdquo International Journal of Heat and Mass Transfer vol50 no 9-10 pp 1902ndash1908 2007

[41] A A Afify ldquoSimilarity solution in MHD effects of thermaldiffusion and diffusion thermo on free convective heat andmass transfer over a stretching surface considering suction orinjectionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2202ndash2214 2009

[42] K Cramer and S PaiMagnetofluid Dynamics for Engineers andApplied Physicists McGraw Hill New York NY USA 1973

[43] R C Meyer ldquoOn reducing aerodynamic heat-transfer ratesby magnetohydrodynamic techniquesrdquo Journal of Aero SpaceScience vol 25 pp 561ndash572 1958

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

studied by El-Hakiem and El-Amin [22] They presented aboundary layer analysis of the influence of thermal radiationand lateral mass flux on non-Darcy natural convection overa vertical flat plate in a fluid saturated porous mediumPrasad et al [23] investigated the effects of radiation andmasstransfer on two-dimensional flow past an impulsively startedinfinite vertical plate Makinde and Ogulu [24] consideredthe heat and mass transfer flows of a variable viscosityfluid past a vertical porous plate permeated by a transversemagnetic field with thermal radiation effects The effectsof thermal radiation and variable fluid viscosity on freeconvective flow and heat transfer past a porous stretchingsurface are investigated by Mukhopadhyay and Layek [25]Pal [26] studied heat and mass transfer in a stagnation pointflow towards a stretching surface in the presence of buoyancyforce and thermal radiation

Combined heat and mass transfer flow in the presenceof chemically reactive species concentration with or withoutapplied magnetic field has bearing on many transport pro-cesses present in nature and also in science and engineer-ing applications In processes such as drying evaporationenergy transfer in a cooling tower and the flow in a desertcooler heat and mass transfer occur simultaneously Freeconvection processes of involving the combined mechanismare also encountered inmany natural processes and industrialapplications such as in the curing of plastics the cleaningand chemical processing of materials and the manufacturingof pulp and insulated cables Chamkha [27] studied MHDflow over a uniformly stretched vertical permeable surfacesubject to a chemical reaction Afify [28] analyzed the MHDfree convective flow and mass transfer over a stretching sheetwith a homogeneous chemical reaction of order 119899 where 119899was taken to be 0 1 2 or 3 The influence of a chemicalreaction on heat and mass transfer from vertical surfaces inporous media subject to Soret and Dufour effects was studiedby Postelnicu [29] He showed that the thickness of theconcentration boundary layer decreases as the Lewis numberincreases a phenomenon also evident when a chemicalreaction is absent Kandasamy and Palanimani [30] studiedthe effects of a chemical reaction on heat andmass transfer ona magnetohydrodynamic boundary layer flow over a wedgewith ohmic heating and viscous dissipation in a porousmedium Pal and Mondal [31] studied the effects of SoretDufour chemical reaction and thermal radiation on MHDnon-Darcy unsteadymixed convective heat andmass transferover a stretching sheet

In all the above investigations the analytical or numericalsolution is obtained assuming that the temperature at theinterface was continuous and well defined However thereexist several problems of physical interest which may requirenonuniform or arbitrary wall conditions Several researchers[32ndash35] have investigated free convection from a verticalplate with stepdiscontinuities in the surface temperatureChandran et al [36] considered unsteady natural convectionflow of a viscous incompressible fluid near a vertical platewith ramped wall temperature Seth et al [37] studiedunsteady natural convection flow of a viscous incompressibleelectrically conducting fluid past an impulsively moving ver-tical plate in a porousmediumwith ramped wall temperature

taking into account the effects of thermal radiation Theycompared the results of natural convection near a rampedtemperature plate with those of natural convection near anisothermal plate Recently Seth et al [38] extended thisproblem to include the effects of rotation

Suction or blowing through the wall plays importantrole in boundary layer formation An increase in suctionvelocity may lead to a delay in the formation of a boundarylayer and an increase in blowing may lead to a decrease inthe skin friction at the plate which in turn may lead to adecrease in the rate of heat transfer at the plate Makinde [39]studied the effects of wall porosity on free convection flowwith thermal radiation and mass transfer Mbeledogu andOgulu [40] studied the heat andmass transfer rates in rotatingflow past a vertical porous flat plate Afify [41] discussedthe MHD free convective heat and mass transfer flow overa stretching sheet in the presence of suctioninjection withthermal diffusion and diffusion thermoeffects

The aim of the present study is to investigate the effectsof suctionblowing on unsteady hydromagnetic free con-vective flow and mass transfer of a viscous incompressibleelectrically conducting and heat radiating fluid past a flatplate with ramped wall temperature in the presence ofchemical reaction Such a fluid flow finds many engineeringapplications such as those in MHD devices and in severalnatural phenomena occurring subject to radiation in thepresence of chemically reactive species

2 Problem Formulation

Consider the unsteady free convective heat andmass transferflow of a viscous incompressible electrically conductingand heat radiating fluid past an impulsively moving verticalporous flat plate We choose the coordinate system in such away that the 119909-axis is along the plate in the upward directionthe119910-axis normal to the plate and the 119911-axis perpendicular tothe 119909119910-plane The fluid is permeated by a uniform transversemagnetic field 119861

0applied parallel to 119910-axis Initially at time

1199051015840le 0 the fluid and plate are at rest at a uniform temperature

1198791015840

infinand species concentration 1198621015840

infin At time 1199051015840 gt 0 the plate

begins to move in the 119909-direction with uniform velocity 1198800

Instantaneously the plate temperature is raised or lowered to1198791015840

infin+ (1198791015840

119908minus 1198791015840

infin)11990510158401199050when 1199051015840 lt 119905

0 and for 1199051015840 gt 119905

0the plate

is maintained at constant temperature 1198791015840119908 A constant species

concentration 1198621015840119908is maintained at the plate for 1199051015840 gt 0 Since

the plate is of infinite extent in the 119909- and 119911-directions and iselectrically nonconducting all physical quantities except thepressure are functions of 119910 and 1199051015840 only

The fluid is a metallic liquid whose magnetic Reynoldsnumber is small and hence the induced magnetic fieldproduced by the fluid motion is negligible in comparison tothe applied one [42] so that the magnetic field B = (0 119861

0 0)

Also no external electric field is applied so the effect ofpolarization of magnetic field is negligible [43] that is E =(0 0 0)The fluid flow is induced by the impulsive movementof the plate in the 119909-direction and there is constant flow offluid in 119910-direction due to the pores in the plate so that the

Mathematical Problems in Engineering 3

velocity vector is q = (119906 minusV0 0) With these assumptions the

governing model equations are given by

1205971199061015840

1205971199051015840minus V0

1205971199061015840

120597119910

= ]12059721199061015840

1205971199102minus

1205901198612

0

120588

1199061015840

+ 119892120573119879(1198791015840minus 1198791015840

infin) + 119892120573

119862(1198621015840minus 1198621015840

infin)

(1)

1205971198791015840

1205971199051015840minus V0

1205971198791015840

120597119910

=

119896

120588119888119901

12059721198791015840

1205971199102minus

1

120588119888119901

1205971199021015840

119903

1205971199101015840 (2)

1205971198621015840

1205971199051015840minus V0

1205971198621015840

120597119910

= 119863

12059721198621015840

12059711991010158402minus 119896119903(1198621015840minus 1198621015840

infin) (3)

where 1199061015840 V0 120588 119892 120573

119879 120573119862 1198791015840 1198621015840 119888

119901 119896 119902119903 ] 120590 119863 and 119896

119903are

respectively fluid velocity in 119909-direction suctioninjectionvelocity in 119910 direction fluid density acceleration due to grav-ity volumetric coefficient of thermal expansion volumetriccoefficient of expansion or contraction temperature of thefluid near the plate species concentration specific heat atconstant pressure thermal conductivity radiative flux kine-matic coefficient of viscosity electrical conductivity chemicalmolecular diffusivity and chemical reaction coefficient

Assuming that there is no slip between the plate and thefluid the initial and boundary conditions for the fluid flowproblem are

1199061015840= 0 119879

1015840= 1198791015840

infin 1198621015840= 1198621015840

infinfor 119910 ge 0 1199051015840 le 0 (4a)

1199061015840= 1198800

at 119910 = 0 for 1199051015840 gt 0 (4b)

1198791015840= 1198791015840

infin+

(1198791015840

119908minus 1198791015840

infin) 1199051015840

1199050

at 119910 = 0 for 0 lt 1199051015840 le 1199050 (4c)

1198791015840= 1198791015840

119908at 119910 = 0 for 1199051015840 gt 119905

0 (4d)

1198621015840= 1198621015840

119908at 119910 = 0 for 1199051015840 gt 0 (4e)

1199061015840997888rarr 0 119879

1015840997888rarr 119879

1015840

infin 1198621015840997888rarr 119862

1015840

infin

as 119910 997888rarr infin for 1199051015840 gt 0(4f)

For an optically thick fluid in addition to emission thereis also self-absorption and usually the absorption coefficientis wavelength dependent and large so that we can adopt theRosseland approximation for radiative flux vector 1199021015840

119903 The

radiative flux vector 1199021015840119903under the Rosseland approximation

is

1199021015840

119903= minus

4120590lowast

3119896lowast

12059711987910158404

120597119910

(5)

where 119896lowast is the mean absorption coefficient and 120590lowast is theStefan-Boltzmann constant Assuming a small temperaturedifference between the fluid temperature 1198791015840 and the freestream temperature 1198791015840

infin 11987910158404 is expanded in a Taylor series

about the free stream temperature1198791015840infin Neglecting second and

higher order terms in (1198791015840 minus 1198791015840infin) we obtain

11987910158404cong 411987910158403

infin1198791015840minus 311987910158404

infin (6)

Using (5) and (6) in (2) we obtain

1205971198791015840

1205971199051015840minus V0

1205971198791015840

120597119910

=

119896

120588119888119901

12059721198791015840

1205971199102+

1

120588119888119901

16120590lowast11987910158403

infin

3119896lowast

12059721198791015840

120597y2 (7)

Introducing the following dimensionless variables

120578 =

1199101015840

11988001199050

119906 =

1199061015840

1198800

119905 =

1199051015840

1199050

119879 =

1198791015840minus 1198791015840

infin

1198791015840

119908minus 1198791015840

infin

119862 =

1198621015840minus 1198621015840

infin

1198621015840

119908minus 1198621015840

infin

(8)

the governing equations (1) (3) and (7) in dimensionlessform become

120597119906

120597119905

minus 119878

120597119906

120597120578

=

1205972119906

1205971205782+ 119879 + Gm119862 minus119872119906

120597119879

120597119905

minus 119878

120597119879

120597120578

=

(1 + 119873)

Pr1205972119879

1205971205782

120597119862

120597119905

minus 119878

120597119862

120597120578

=

1

119878119888

1205972119862

1205971205782minus 119870119903119862

(9)

where 119878 = V01198800is the suction or blowing parameter Gm =

119892120573119862](1198621015840119908minus 1198621015840

infin)1198803

0is the mass Grashof number 119872 =

1205901198612

0]12058811988020is the magnetic parameter 119873 = 16120590

lowast11987910158403

infin3119896119896lowast

is the thermal radiation parameter Pr = 120588]119888119901119896 is the

Prandtl number Sc = ]119863 is the Schmidt number and119870119903= 119896119903]21198631198802

0is the chemical reaction parameter Here

119878 gt 0 corresponds to suction whereas 119878 lt 0 correspondsto blowing The characteristic time 119905

0and the characteristic

velocity 1198800are defined as

1199050=

]

1198802

0

1198800= [119892120573

119879] (1198791015840119908minus 1198791015840

infin)]

13

(10)

The initial and boundary conditions (4a)ndash(4f) in nondi-mensional form are

119906 = 0 119879 = 0 119862 = 0 for 120578 ge 0 119905 le 0 (11a)

119906 = 1 119862 = 1 at 120578 = 0 for 119905 gt 0 (11b)

119879 = 119905 at 120578 = 0 for 0 lt 119905 le 1 (11c)

119879 = 1 at 120578 = 0 for 119905 gt 1 (11d)

119906 997888rarr 0 119879 997888rarr 0 119862 997888rarr 0 as 120578 997888rarr infin for 119905 gt 0(11e)

The system of partial differential equations (9) subject tothe initial and boundary conditions (11a)ndash(11e) represents themodel for unsteady hydromagnetic free convective flow ofa viscous incompressible electrically conducting and heatradiating fluid past an infinite porous flat plate with rampedwall temperature in the presence of a chemically reactivespecies

4 Mathematical Problems in Engineering

Table 1 A comparison of minus120575119879120575120578|120578=0

when 119878 = 0

119905 119873 Seth et al [37] Present results

02

05 03472 03471791 03007 03006665 01736 017359010 01282 0128204

04

05 04910 04909851 04252 04252065 02455 024549310 01813 0181308

06

05 06013 06013321 05208 05207695 03007 030066610 02221 0222057

08

05 06944 06943581 06013 06013325 03472 034717910 02564 0256409

3 Analytic Solutions

Using the Laplace transform technique on the system ofequations (9) subject to the initial and boundary conditions(11a)ndash(11e) we obtain

(120578 119904) = minus [

Gm1199041198631

1198901205821120578+

(1 minus 119890minus119904)

11990421198632

1198901205822120578]

+

1

119904

+

(1 minus 119890minus119904)

11990421198631

+

Gm1199041198632

1198901205823120578

(12)

119879 (120578 119904) =

(1 minus 119890minus119904)

1199042

1198901205822120578 (13)

119862 (120578 119904) =

1198901205821120578

119904

(14)

where 119862(120578 119904) 119879(120578 119904) (120578 119904) are respectively the Laplacetransforms of 119862(120578 119905) 119879(120578 119905) and 119906(120578 119905) 119904 gt 0 is the Laplacetransform parameter and

1205821= 05 minus119878Sc minus radic1198782Sc2 + 4Sc (119870

119903+ 119904) (15a)

1205822= 05

minus119878(

Pr1 + 119873

) minusradic1198782(

Pr1 + 119873

)

2

+ 4 (

Pr1 + 119873

) 119904

(15b)

1205823= 05 minus119878 minus radic119878

2+ 4 (119872 + 119904) (15c)

1198631= 1205822

1+ 1198781205821minus (119872 + 119904) (15d)

1198632= 1205822

2+ 1198781205822minus (119872 + 119904) (15e)

An exact Laplace transform inversion of (12) can beobtained when there is no suctionblowing that is when

119878 = 0 However for a nonzero 119878 the inversion of (12) isnot possible Thus the inversion of (12)ndash(14) was obtainednumerically using INVLAP routine in Matlab However tocompare the results which are obtained using INVLAP rou-tine the exact inversion of (13) and (14) gave the solutions

119879 (120578 119905) = 119890minus119887120578[119875 (120578 119905) minus 119867 (119905 minus 1) 119875 (120578 119905 minus 1)] (16)

119862 (120578 119905) =

1

2

[119890minus120578(119886minusradic119889) erfc (119905

1) + 119890minus120578(119886+radic119889) erfc (119905

2)] (17)

where erfc is the complementary error function 119867 is theHeaviside unit step function and

119875 (120578 119905) = (

119905

2

+

119898120578

4radic119899

) 119890120578radic119899 erfc (119905

3)

+ (

119905

2

minus

119898120578

4radic119899

) 119890minus120578radic119899 erfc (119905

4)

1199051 1199052= plusmnradic

119889119905

119888

+

120578

2

radic

119888

119905

1199053 1199054= plusmnradic

119899119905

119898

+

120578

2

radic

119898

119905

119886 =

119878Sc2

119887 =

Pr1198782 (1 + 119873)

119888 = Sc

119889 =

1198782Sc2

4

+ Sc119870119903 119898 =

Pr1 + 119873

119899 =

Pr21198782

4(1 + 119873)2

(18)

In the absence of suctionblowing and the thermal radi-ation effect (119878 = 0 and 119873 = 0) the solution 119879(120578 119905) in (16) isin agreement with the solution obtained by Chandran et al[36]

31 Skin Friction Nusselt Number and Sherwood NumberThe nondimensional quantities of engineering interest arethe skin friction 120591 which is a measure of shear stress at theplate theNusselt numberNu whichmeasures the rate of heattransfer and the Sherwood number Sh which measures therate of mass transfer at the plateThe Nusselt number Nu andthe Sherwood number Sh have the exact values

NuRe= minus (1 + 119873)(

120597119879

120597120578

)

120578=0

= minus [119891 (119905) minus 119867 (119905 minus 1) 119891 (119905 minus 1)]

ShRe= minus(

120597119862

120597120578

)

120578=0

= (119886 + radic119889) + radic

119888

120587119905

119890minus119889119905119888

minus radic119889 erfc(radic119889119905119888

)

(19)

Mathematical Problems in Engineering 5

Table 2 Comparison of exact and numerical values of NuRe and ShRe for different values of 119905 and 119878 when119870119903= 05 Sc = 1 Pr = 071 and

119873 = 1

119905 119878

NuRe NuRe ShRe ShReExact Numerical Exact Numerical

02minus1 053388346 0533883 094620854 09462090 060133195 0601332 138566159 13856621 067588346 0675883 194620854 1946209

04minus1 071843936 0718439 064706152 06470620 085041184 0850412 106475703 10647571 100243936 1002439 164706152 1647062

06minus1 084692696 0846927 053359416 05335940 104153758 1041537 093657181 09365721 127292696 1272927 153359416 1533594

08minus1 094692675 0946927 047550301 04755030 120266405 1202664 086753072 08675311 151492675 1514927 147550301 1475503

0 1 2 3 4 50

01

02

03

04

05

06

07

08

Exact solution Numerical solution

S = minus1 0 1

T(120578

120578

t)

(a)

Exact solution Numerical solution

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

S = minus1 0 1

C(120578

120578

t)

(b)

Figure 1 Comparison of numerical and exact solutions of (a) 119879(120578 119905) and (b) 119862(120578 119905) for different values of 119878 when Pr = 071 Sc = 1119873 = 1119870119903= 05 and 119905 = 05

where Re = 1198800119871] is the Reynolds number 119871 is some

characteristic length and

119891 (119905) = (1 + 119873) [(

119898

2radic119899

+ radic119899119905) erfc(radic119899119905119898

)

minus(

119898

2radic119899

+ 119887119905 + radic119899119905) minus radic119898119905

120587

119890minus119899119905119898

]

(20)

The values of skin friction 120591 can be obtained numericallyusing the Matlab INVLAP routine

4 Validation of Numerical Results

In order to validate the numerical results obtained using theMatlab INVLAP routine the values of NuRe are comparedwith the exact values obtained by Seth et al [37] in Table 1Further the numerical values of NuRe and ShRe arecomparedwith the exact analytical values in Table 2Thefluidtemperature and species concentration profiles are plottedusing both the numerical and exact values and are compared

6 Mathematical Problems in Engineering

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

M = 3 5 7

Gm = 1

Suction (S = 1) Blowing (S = minus1)

u(120578t)

120578

(a)

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

M = 3 5 7

Gm = minus1

Suction (S = 1) Blowing (S = minus1)

u(120578t)

120578

(b)

Figure 2 Effect of119872 on 119906(120578 119905) when Sc = 1 119870119903= 05119873 = 1 Pr = 071 and 119905 = 07

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

1

18 2 220

005

01

015

N = 05 10 15

Gm = 1

Suction (S = 1) Blowing (S = minus1)

u(120578t)

u(120578t)

120578

120578

(a)

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

1

N = 05 10 15

Gm = minus1

08 1 12

015

02

025

Suction (S = 1) Blowing (S = minus1)

u(120578t)

u(120578t)

120578

120578

(b)

Figure 3 Effect of119873 on 119906(120578 119905) when Sc = 1 119870119903= 05119872 = 3 Pr = 071 and 119905 = 07

in Figure 1 An excellent agreement between the values isobserved which validates the accuracy of the INVLAP results

5 Results and Discussion

The unsteady hydromagnetic free convective flow of a vis-cous incompressible electrically conducting and heat radi-ating fluid past an infinite vertical flat plate in the presence ofsuctionblowing and chemically reactive species concentra-tion has been studiednumerically using the Matlab INVLAP

routine The fluid velocity 119906(120578 119905) temperature 119879(120578 119905) andspecies concentration 119862(120578 119905) profiles are given in Figures 23 4 5 6 7 8 9 10 and 11 whereas the numerical valuesof the skin friction heat and mass transfer coefficients arepresented in Tables 3 and 4 when 119905 = 07 Gm = 1 (whichcorresponds to assisting buoyancy) and Gm = minus1 (whichcorresponds to opposing buoyancy)

Figures 2 to 7 show the effects of the magnetic fieldthermal radiation suctionblowing chemical reaction massdiffusion and thermal diffusion on the fluid velocity Withassisting mass buoyancy (ie Gm = 1) the fluid velocity

Mathematical Problems in Engineering 7

0 1 2 3 4 5 60

01

02

03

04

05

06

07

08

09

1

S = 1 2 3

S = minus1 minus2 minus3

Gm = 1

u(120578t)

120578

(a)

0 1 2 3 4 5 60

01

02

03

04

05

06

07

08

09

1

S = minus1 minus2 minus3

Gm = minus1

u(120578t)

120578

S = 1 2 3

(b)

Figure 4 Effect of 119878 on 119906(120578 119905) when Sc = 1 119870119903= 05119873 = 1 Pr = 071119872 = 3 and 119905 = 07

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

08 1 12 14 16

015

02

025

03

Gm = 1

u(120578t)

120578

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

Kr = 05 10 15

(a)

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

1

12 14 16

005

01

015

Gm = minus1

u(120578t)

u(120578t)

120578

120578

Suction (S = 1) Blowing (S = minus1)

Kr = 0 10 15

(b)

Figure 5 Effect of119870119903on 119906(120578 119905) when Sc = 1119872 = 3119873 = 1 Pr = 071 and 119905 = 07

decreases with an increase in the magnetic parameter 119872chemical reaction parameter 119870

119903 Schmidt number Sc and

Prandtl number Pr while it increases with the thermalradiation parameter 119873 Since the Schmidt number Sc is theratio of viscous to mass diffusivity an increase in Sc impliesa decrease in the mass diffusion rate Hence the magneticfield and the chemical reaction rate tend to reduce the fluidvelocity whereas the mass diffusivity thermal diffusivity andthermal radiation tend to increase the fluid velocity Withopposing mass buoyancy (ie for Gm = minus1) the behaviourof the fluid velocity with respect to all governing parametersremains the same except with respect to Sc when the fluidvelocity decreases with an increase in mass diffusivity

Figure 4 shows that the fluid velocity decreases with anincrease in suction velocity whereas it increases with anincrease in blowing for both assisting and opposing massbuoyancy This result may be useful in engineering applica-tions where the formation of boundary layer is to be delayedThis can be done by increasing the suction velocity We alsoobserve that the thickness of the momentum boundary layeris less in the presence of opposing mass buoyancy than in thepresence of assisting mass buoyancy

Figures 8 9 and 10 show the effects of thermal radi-ation thermal diffusion and suctionblowing on the fluidtemperature We observe from Figures 8 and 9 that the fluidtemperature increases with119873 while it decreases with Pr The

8 Mathematical Problems in Engineering

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

Sc = 05 10 15

Gm = 1

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

(a)

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

Sc = 05 10 15

Gm = minus1

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

(b)

Figure 6 Effect of Sc on 119906(120578 119905) when119872 = 3 119870119903= 05119873 = 1 Pr = 071 and 119905 = 07

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

1

Pr = 071 1 7

Gm = 1

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

(a)

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

1

Pr = 071 1 7

Gm = minus1

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

(b)

Figure 7 Effect of Pr on 119906(120578 119905) when Sc = 1 119870119903= 05119873 = 1119872 = 3 and 119905 = 07

Prandtl number is the ratio of viscosity to thermal diffusivityand an increase in Pr implies a decrease in thermal diffusivityThermal radiation and diffusivity tend to increase the fluidtemperature Figure 10 shows that an increase in suctionvelocity causes a reduction in the fluid temperature whereasan increase in blowing increases the fluid temperature

Figures 11 12 and 13 show the effects of the chemicalreaction mass diffusion and suction or blowing on thespecies concentration We observe from Figures 11 and 12that the species concentration decreases with an increase inthe chemical reaction parameter or the Schmidt numberThus the chemical reaction rate tends to reduce the species

concentration whereas mass diffusivity increases the speciesconcentration It is clear from Figure 13 that the suctionvelocity reduces the species concentration while blowingcauses an increase in the species concentration We observefrom Figures 11 and 12 that the species concentration is higherin the case of blowing than in the case of suction We alsoobserve from Figures 2ndash11 that the momentum thermal andconcentration boundary layers are thicker in the case ofblowing than in the case of suction

Table 3 shows the effects of the magnetic field thermalradiation suctionblowing chemical reactionmass diffusionand thermal diffusion on the skin friction in the presence

Mathematical Problems in Engineering 9

Table 3 Effect of fluid and physical parameters on the skin friction minus120591 when 119905 = 07

119872 119873 119878 119870119903

Sc Pr minus120591 (119878 gt 0Gm = 1)

minus120591 (119878 lt 0Gm = 1)

minus120591 (119878 gt 0Gm = minus1)

minus120591 (119878 lt 0Gm = minus1)

3 1 mdash 05 1 071 1747428 0788854 2397344 14387705 mdash mdash mdash mdash mdash 2291296 1325411 2865812 18999277 mdash mdash mdash mdash mdash 2737296 1766244 3258082 2287031mdash 05 mdash mdash mdash mdash 1763388 0794076 2413303 1443992mdash 10 mdash mdash mdash mdash 1747428 0788854 2397344 1438770mdash 15 mdash mdash mdash mdash 1735676 0784937 2385592 1434853mdash mdash 1 mdash mdash mdash 1747428 mdash 2397344 mdashmdash mdash 2 mdash mdash mdash 2463830 mdash 3043645 mdashmdash mdash 3 mdash mdash mdash 3291595 mdash 3786658 mdashmdash mdash minus1 mdash mdash mdash mdash 0788854 mdash 1438770mdash mdash minus2 mdash mdash mdash mdash 0540588 mdash 1120403mdash mdash minus3 mdash mdash mdash mdash 0393845 mdash 0888908mdash mdash mdash 05 mdash mdash 1747428 0788854 2397344 1438770mdash mdash mdash 10 mdash mdash 1761672 0803098 2383100 1424526mdash mdash mdash 15 mdash mdash 1774216 0815642 2370555 1411981mdash mdash mdash mdash 05 mdash 1666559 0771682 2478212 1455942mdash mdash mdash mdash 10 mdash 1747428 0788854 2397344 1438770mdash mdash mdash mdash 15 mdash 1796602 0798486 2348170 1429138mdash mdash mdash mdash mdash 071 1747428 0788854 2397344 1438770mdash mdash mdash mdash mdash 10 1766522 0795089 2416438 1445005mdash mdash mdash mdash mdash 70 1884422 0830142 2534338 1480058

0 1 2 3 4 50

01

02

03

04

05

06

07

N = 05 10 15

Suction (S = 1) Blowing (S = minus1)

T(120578

120578

t)

Figure 8 Effect of119873 on 119879(120578 119905) when Pr = 071 and 119905 = 07

of assisting and opposing mass buoyancy In the case ofassisting mass buoyancy an increase in 119872 119870

119903 Sc and Pr

tends to increase the skin friction whereas an increase in119873 reduces the skin friction for both suction and blowingcases Thus the skin friction increases with an increase inthe magnetic field parameter and the chemical reaction rate

0 1 2 3 4 50

01

02

03

04

05

06

07

Pr = 071 1 7

Suction (S = 1) Blowing (S = minus1)

T(120578

120578

t)

Figure 9 Effect of Pr on 119879(120578 119905) when119873 = 1 and 119905 = 07

while it decreaseswith an increase inmass diffusivity thermaldiffusivity and thermal radiation in the case of assisting massbuoyancy However for opposing mass buoyancy the skinfriction increases with mass diffusivity while decreasing withan increase in the rate of the chemical reaction

10 Mathematical Problems in Engineering

Table 4 Values of NuRe and ShRe for different values of governing parameters when 119905 = 07

119873 119878 Pr 119870119903

Sc NuRe NuRe ShRe ShRe(Suction) (Blowing) (Suction) (Blowing)

05 mdash 071 mdash mdash 1249449 0752449 mdash mdash10 mdash 071 mdash mdash 1396642 0899642 mdash mdash15 mdash 071 mdash mdash 1527010 1030010 mdash mdash1 1 071 05 10 1396642 mdash 1500355 mdash1 2 071 05 10 1712939 mdash 2280316 mdash1 3 071 05 10 2069304 mdash 3174187 mdash1 minus1 071 05 10 mdash 0899642 mdash 05003551 minus2 071 05 10 mdash 0718939 mdash 02803161 minus3 071 05 10 mdash 0578304 mdash 01741871 mdash 071 mdash mdash 1396642 0899642 mdash mdash1 mdash 10 mdash mdash 1723721 1023721 mdash mdash1 mdash 70 mdash mdash 6662968 1762968 mdash mdashmdash mdash mdash 05 1 mdash mdash 1500355 0500355mdash mdash mdash 10 1 mdash mdash 1691322 0691322mdash mdash mdash 15 1 mdash mdash 1865496 0865496mdash mdash mdash 05 05 mdash mdash 0921462 0421462mdash mdash mdash 05 10 mdash mdash 1500355 0500355mdash mdash mdash 05 15 mdash mdash 2035801 0535801

0 1 2 3 4 7650

01

02

03

04

05

06

07

S = minus1 minus2 minus3

S = 1 2 3

T(120578

120578

t)

Figure 10 Effect of 119878 on 119879(120578 119905) when Pr = 071119873 = 1 and 119905 = 07

Table 4 shows the effects of thermal radiation suc-tionblowing and thermal diffusion on the heat transfer ratefrom the plate It also shows the effects of suctionblowingchemical reaction rate andmass diffusion on the rate of masstransfer at the plate We observe that the Nusselt numberincreases with 119873 and Pr Thus the thermal radiation tendsto accelerate heat transfer at the plate while thermal diffusionhas reverse effect Table 4 also shows that the rate of heattransfer at the plate increases with an increase in suctionvelocity and decreases with blowing The Sherwood numberincreases with 119870

119903and Sc for both suction and blowing An

increase in the suction velocity causes an increase in therate of mass transfer whereas an increase in blowing has theopposite effect

0 05 1 15 2 25 3 35 4

Suction (S = 1) Blowing (S = minus1)

0

01

02

03

04

05

06

07

08

09

1

C(120578

120578

t)

Kr = 0 05 10

Figure 11 Effect of119872 on 119862(120578 119905) when Sc = 1 and 119905 = 07

6 Conclusions

The unsteady hydromagnetic free convection flow of aviscous incompressible and heat-radiating fluid past animpulsively started vertical porous flat plate with rampedtemperature in the presence of a chemically reactive speciesconcentration was studied For certain parameter valuesexact solutions of the model equations were found usingLaplace transforms The Matlab INVLAP routine was used tofind the numerical results in the general case The numericalresults were found to be in excellent agreement with the

Mathematical Problems in Engineering 11

Suction (S = 1) Blowing (S = minus1)

Sc = 05 10 15

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

C(120578

120578

t)

Figure 12 Effect of Sc on 119862(120578 119905) when 119870119903= 05 and 119905 = 07

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

S = 1 2 3

S = minus1 minus2 minus3

C(120578

120578

t)

Figure 13 Effect of 119878 on 119862(120578 119905) when 119870119903= 05 Sc = 1 and 119905 = 07

results of Seth et al [37] and with the exact results Theimportant findings of the present work may be summarizedas follows

(i) In the case of assisting mass buoyancy force the fluidvelocity is reduced by an increase in themagnetic fieldand the chemical reaction rate

(ii) Blowing causes an acceleration in the fluid velocitywhile suction has reverse effect on it

(iii) An increase in the chemical reaction rate tends toreduce the species concentration whereas the massdiffusivity parameter has reverse effect

(iv) Thermal radiation thermal diffusivity and blowingtend to increase the fluid temperature whereas suc-tion has the opposite effect

(v) In the case of an assisting mass buoyancy force themagnetic field and chemical reaction parameter tendto increase the skin friction while themass diffusivitythermal diffusivity and thermal radiation have thereverse effect

(vi) Thermal diffusivity and blowing tend to reduce therate of heat transfer at the plate whereas thermalradiation and suction have the opposite effect

Acknowledgment

The authors would like to thank the referee for his valuableand constructive suggestions

References

[1] A S Gupta ldquoSteady and transient free convection of anelectrically conducting fluid froma vertical plate in the presenceof a magnetic fieldrdquo Applied Scientific Research vol 9 no 1 pp319ndash333 1960

[2] A S Gupta ldquoLaminar free convection flow of an electricallyconducting fluid from a vertical plate with uniform surface heatflux and variable wall temperature in the presence of amagneticfieldrdquoZeitschrift fur AngewandteMathematik undPhysik vol 13pp 324ndash333 1962

[3] K R Cramer ldquoSeveral magnetohydrodynamic free convectionsolutionsrdquo Journal of Heat Transfer vol 85 pp 35ndash40 1963

[4] I Pop ldquoOn the unsteady hydromagnetic free convection flowpast a vertical infinite flat platerdquo Indian Journal of Physics vol43 pp 196ndash200 1969

[5] H K Kuiken ldquoMagnetohydrodynamic free convection in astrong cross fieldrdquo Journal of Fluid Mechanics vol 40 no 1 pp21ndash38 1970

[6] G Wilks ldquoMagnetohydrodynamic free convection about asemi-infinite vertical plate in a strong cross fieldrdquo Zeitschrift furAngewandte Mathematik und Physik ZAMP vol 27 no 5 pp621ndash631 1976

[7] M A Hossain ldquoEffect of hall current on unsteady hydromag-netic free convection flow near an infinite vertical porous platerdquoJournal of the Physical Society of Japan vol 55 no 7 pp 2183ndash2190 1986

[8] T K Aldoss M A Al-Nimr M A Jarrah and B J Al-SharsquoerldquoMagnetohydrodynamicmixed convection from a vertical plateembedded in a porous mediumrdquo Numerical Heat Transfer PartA vol 28 no 5 pp 635ndash645 1995

[9] K A Helmy ldquoMHD unsteady free convection flow past avertical porous platerdquo Zeitschrift fur Angewandte Mathematikund Mechanik vol 78 no 4 pp 255ndash270 1998

[10] Y J Kim ldquoUnsteadyMHD convective heat transfer past a semi-infinite vertical porous moving plate with variable suctionrdquoInternational Journal of Engineering Science vol 38 no 8 pp833ndash845 2000

[11] H S Takhar S Roy and G Nath ldquoUnsteady free convectionflow over an infinite vertical porous plate due to the combinedeffects of thermal and mass diffusion magnetic field and Hallcurrentsrdquo Heat and Mass Transfer vol 39 no 10 pp 825ndash8342003

[12] N Ahmed H K Sarmah and D Kalita ldquoThermal diffusioneffect on a three-dimensional MHD free convection with masstransfer flow from a porous vertical platerdquo Latin AmericanApplied Research vol 41 pp 165ndash176 2011

12 Mathematical Problems in Engineering

[13] M A Hossain and H S Takhar ldquoRadiation effect on mixedconvection along a vertical plate with uniform surface temper-aturerdquoHeat and Mass Transfer vol 31 no 4 pp 243ndash248 1996

[14] A Y Bakier and R S R Gorla ldquoThermal radiation effect onmixed convection from horizontal surfaces in saturated porousmediardquo Transport in Porous Media vol 23 no 3 pp 357ndash3631996

[15] H S Takhar R S R Gorla and VM Soundalgekar ldquoRadiationeffects onMHD free convection flowof a gas past a semi-infinitevertical platerdquo International Journal of Numerical Methods forHeat and Fluid Flow vol 6 no 2 pp 77ndash83 1996

[16] A J Chamkha ldquoThermal radiation and buoyancy effects onhydromagnetic flow over an accelerating permeable surfacewith heat source or sinkrdquo International Journal of EngineeringScience vol 38 no 15 pp 1699ndash1712 2000

[17] G E-D A Azzam ldquoRadiation effects on the MHDmixed free-forced convective flow past a semi-infinitemoving vertical platefor high temperature differencesrdquo Physica Scripta vol 66 no 1pp 71ndash76 2002

[18] C Israel-Cookey A Ogulu and V B Omubo-Pepple ldquoInflu-ence of viscous dissipation and radiation on unsteady MHDfree-convection flow past an infinite heated vertical plate ina porous medium with time-dependent suctionrdquo InternationalJournal of Heat andMass Transfer vol 46 no 13 pp 2305ndash23112003

[19] M A A Mahmoud ldquoThermal radiation effect on unsteadyMHD free convection flow past a vertical plate with tem-perature-dependent viscosityrdquo Canadian Journal of ChemicalEngineering vol 87 no 1 pp 47ndash52 2009

[20] A R Bestman and S K Adjepong ldquoUnsteady hydromagneticfree-convection flow with radiative heat transfer in a rotatingfluidmdashI Incompressible optically thin fluidrdquo Astrophysics andSpace Science vol 143 no 1 pp 73ndash80 1988

[21] A J Chamkha H S Takhar and V M Soundalgekar ldquoRadia-tion effects on free convection flow past a semi-infinite verticalplate with mass transferrdquo Chemical Engineering Journal vol 84no 3 pp 335ndash342 2001

[22] M A El-Hakiem and M F El-Amin ldquoThermal radiation effecton non-Darcy natural convection with lateral mass transferrdquoHeat and Mass Transfer vol 37 no 2-3 pp 161ndash165 2001

[23] V R Prasad N B Reddy and R Muthucumaraswamy ldquoRadia-tion and mass transfer effects on two-dimensional flow past animpulsively started infinite vertical platerdquo International Journalof Thermal Sciences vol 46 no 12 pp 1251ndash1258 2007

[24] O D Makinde and A Ogulu ldquoThe effect of thermal radiationon the heat and mass transfer flow of a variable viscosity fluidpast a vertical porous plate permeated by a transverse magneticfieldrdquo Chemical Engineering Communications vol 195 no 12pp 1575ndash1584 2008

[25] SMukhopadhyay andG C Layek ldquoEffects of thermal radiationand variable fluid viscosity on free convective flow and heattransfer past a porous stretching surfacerdquo International Journalof Heat andMass Transfer vol 51 no 9-10 pp 2167ndash2178 2008

[26] D Pal ldquoHeat andmass transfer in stagnation-point flow towardsa stretching surface in the presence of buoyancy force andthermal radiationrdquoMeccanica vol 44 no 2 pp 145ndash158 2009

[27] A J Chamkha ldquoMHDflowof a uniformly streched vertical per-meable surface in the presence of heat generationabsorptionand a chemical reactionrdquo International Communications inHeatand Mass Transfer vol 30 no 3 pp 413ndash422 2003

[28] A A Afify ldquoMHD free convective flow and mass transferover a stretching sheet with chemical reactionrdquo Heat and MassTransfer vol 40 no 6-7 pp 495ndash500 2004

[29] A Postelnicu ldquoInfluence of chemical reaction on heat andmasstransfer by natural convection from vertical surfaces in porousmedia considering Soret and Dufour effectsrdquo Heat and MassTransfer vol 43 no 6 pp 595ndash602 2007

[30] R Kandasamy and P G Palanimani ldquoEffects of chemical reac-tions heat and mass transfer on nonlinear magnetohydrody-namic boundary layer flow over a wedge with a porous mediumin the presence of ohmic heating and viscous dissipationrdquoJournal of Porous Media vol 10 no 5 pp 489ndash501 2007

[31] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[32] A A Hayday D A Bowlus and R A McGraw ldquoFree convec-tion from a vertical plate with step discontinuities in surfacetemperaturerdquo ASME Journal of Heat Transfer vol 89 pp 244ndash250 1967

[33] M Kelleher ldquoFree convection from a vertical plate with dis-continuous wall temperaturerdquo Journal of Heat Transfer Trans-actions ASME vol 93 no 4 pp 349ndash356 1971

[34] T-T Kao ldquoLaminar free convective heat transfer response alonga vertical flat plate with step jump in surface temperaturerdquoLetters inHeat andMass Transfer vol 2 no 5 pp 419ndash428 1975

[35] S Lee and M M Yovanovich ldquoLaminar natural convectionfrom a vertical plate with a step change in wall temperaturerdquoJournal of Heat Transfer vol 113 no 2 pp 501ndash504 1991

[36] P Chandran N C Sacheti and A K Singh ldquoNatural convec-tion near a vertical plate with ramped wall temperaturerdquo Heatand Mass Transfer vol 41 no 5 pp 459ndash464 2005

[37] G S Seth M S Ansari and R Nandkeolyar ldquoMHD naturalconvection flow with radiative heat transfer past an impulsivelymoving plate with ramped wall temperaturerdquo Heat and MassTransfer vol 47 no 5 pp 551ndash561 2011

[38] G S Seth R Nandkeolyar and M S Ansari ldquoEffects ofthermal radiation and rotation on unsteady hydromagnetic freeconvection flow past an impulsively moving vertical plate withramped temperature in a porous mediumrdquo Journal of AppliedFluid Mechanics vol 6 no 1 pp 27ndash38 2013

[39] O D Makinde ldquoFree convection flow with thermal radiationand mass transfer past a moving vertical porous platerdquo Interna-tional Communications in Heat and Mass Transfer vol 32 no10 pp 1411ndash1419 2005

[40] I U Mbeledogu and A Ogulu ldquoHeat and mass transfer of anunsteady MHD natural convection flow of a rotating fluid pasta vertical porous flat plate in the presence of radiative heattransferrdquo International Journal of Heat and Mass Transfer vol50 no 9-10 pp 1902ndash1908 2007

[41] A A Afify ldquoSimilarity solution in MHD effects of thermaldiffusion and diffusion thermo on free convective heat andmass transfer over a stretching surface considering suction orinjectionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2202ndash2214 2009

[42] K Cramer and S PaiMagnetofluid Dynamics for Engineers andApplied Physicists McGraw Hill New York NY USA 1973

[43] R C Meyer ldquoOn reducing aerodynamic heat-transfer ratesby magnetohydrodynamic techniquesrdquo Journal of Aero SpaceScience vol 25 pp 561ndash572 1958

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

velocity vector is q = (119906 minusV0 0) With these assumptions the

governing model equations are given by

1205971199061015840

1205971199051015840minus V0

1205971199061015840

120597119910

= ]12059721199061015840

1205971199102minus

1205901198612

0

120588

1199061015840

+ 119892120573119879(1198791015840minus 1198791015840

infin) + 119892120573

119862(1198621015840minus 1198621015840

infin)

(1)

1205971198791015840

1205971199051015840minus V0

1205971198791015840

120597119910

=

119896

120588119888119901

12059721198791015840

1205971199102minus

1

120588119888119901

1205971199021015840

119903

1205971199101015840 (2)

1205971198621015840

1205971199051015840minus V0

1205971198621015840

120597119910

= 119863

12059721198621015840

12059711991010158402minus 119896119903(1198621015840minus 1198621015840

infin) (3)

where 1199061015840 V0 120588 119892 120573

119879 120573119862 1198791015840 1198621015840 119888

119901 119896 119902119903 ] 120590 119863 and 119896

119903are

respectively fluid velocity in 119909-direction suctioninjectionvelocity in 119910 direction fluid density acceleration due to grav-ity volumetric coefficient of thermal expansion volumetriccoefficient of expansion or contraction temperature of thefluid near the plate species concentration specific heat atconstant pressure thermal conductivity radiative flux kine-matic coefficient of viscosity electrical conductivity chemicalmolecular diffusivity and chemical reaction coefficient

Assuming that there is no slip between the plate and thefluid the initial and boundary conditions for the fluid flowproblem are

1199061015840= 0 119879

1015840= 1198791015840

infin 1198621015840= 1198621015840

infinfor 119910 ge 0 1199051015840 le 0 (4a)

1199061015840= 1198800

at 119910 = 0 for 1199051015840 gt 0 (4b)

1198791015840= 1198791015840

infin+

(1198791015840

119908minus 1198791015840

infin) 1199051015840

1199050

at 119910 = 0 for 0 lt 1199051015840 le 1199050 (4c)

1198791015840= 1198791015840

119908at 119910 = 0 for 1199051015840 gt 119905

0 (4d)

1198621015840= 1198621015840

119908at 119910 = 0 for 1199051015840 gt 0 (4e)

1199061015840997888rarr 0 119879

1015840997888rarr 119879

1015840

infin 1198621015840997888rarr 119862

1015840

infin

as 119910 997888rarr infin for 1199051015840 gt 0(4f)

For an optically thick fluid in addition to emission thereis also self-absorption and usually the absorption coefficientis wavelength dependent and large so that we can adopt theRosseland approximation for radiative flux vector 1199021015840

119903 The

radiative flux vector 1199021015840119903under the Rosseland approximation

is

1199021015840

119903= minus

4120590lowast

3119896lowast

12059711987910158404

120597119910

(5)

where 119896lowast is the mean absorption coefficient and 120590lowast is theStefan-Boltzmann constant Assuming a small temperaturedifference between the fluid temperature 1198791015840 and the freestream temperature 1198791015840

infin 11987910158404 is expanded in a Taylor series

about the free stream temperature1198791015840infin Neglecting second and

higher order terms in (1198791015840 minus 1198791015840infin) we obtain

11987910158404cong 411987910158403

infin1198791015840minus 311987910158404

infin (6)

Using (5) and (6) in (2) we obtain

1205971198791015840

1205971199051015840minus V0

1205971198791015840

120597119910

=

119896

120588119888119901

12059721198791015840

1205971199102+

1

120588119888119901

16120590lowast11987910158403

infin

3119896lowast

12059721198791015840

120597y2 (7)

Introducing the following dimensionless variables

120578 =

1199101015840

11988001199050

119906 =

1199061015840

1198800

119905 =

1199051015840

1199050

119879 =

1198791015840minus 1198791015840

infin

1198791015840

119908minus 1198791015840

infin

119862 =

1198621015840minus 1198621015840

infin

1198621015840

119908minus 1198621015840

infin

(8)

the governing equations (1) (3) and (7) in dimensionlessform become

120597119906

120597119905

minus 119878

120597119906

120597120578

=

1205972119906

1205971205782+ 119879 + Gm119862 minus119872119906

120597119879

120597119905

minus 119878

120597119879

120597120578

=

(1 + 119873)

Pr1205972119879

1205971205782

120597119862

120597119905

minus 119878

120597119862

120597120578

=

1

119878119888

1205972119862

1205971205782minus 119870119903119862

(9)

where 119878 = V01198800is the suction or blowing parameter Gm =

119892120573119862](1198621015840119908minus 1198621015840

infin)1198803

0is the mass Grashof number 119872 =

1205901198612

0]12058811988020is the magnetic parameter 119873 = 16120590

lowast11987910158403

infin3119896119896lowast

is the thermal radiation parameter Pr = 120588]119888119901119896 is the

Prandtl number Sc = ]119863 is the Schmidt number and119870119903= 119896119903]21198631198802

0is the chemical reaction parameter Here

119878 gt 0 corresponds to suction whereas 119878 lt 0 correspondsto blowing The characteristic time 119905

0and the characteristic

velocity 1198800are defined as

1199050=

]

1198802

0

1198800= [119892120573

119879] (1198791015840119908minus 1198791015840

infin)]

13

(10)

The initial and boundary conditions (4a)ndash(4f) in nondi-mensional form are

119906 = 0 119879 = 0 119862 = 0 for 120578 ge 0 119905 le 0 (11a)

119906 = 1 119862 = 1 at 120578 = 0 for 119905 gt 0 (11b)

119879 = 119905 at 120578 = 0 for 0 lt 119905 le 1 (11c)

119879 = 1 at 120578 = 0 for 119905 gt 1 (11d)

119906 997888rarr 0 119879 997888rarr 0 119862 997888rarr 0 as 120578 997888rarr infin for 119905 gt 0(11e)

The system of partial differential equations (9) subject tothe initial and boundary conditions (11a)ndash(11e) represents themodel for unsteady hydromagnetic free convective flow ofa viscous incompressible electrically conducting and heatradiating fluid past an infinite porous flat plate with rampedwall temperature in the presence of a chemically reactivespecies

4 Mathematical Problems in Engineering

Table 1 A comparison of minus120575119879120575120578|120578=0

when 119878 = 0

119905 119873 Seth et al [37] Present results

02

05 03472 03471791 03007 03006665 01736 017359010 01282 0128204

04

05 04910 04909851 04252 04252065 02455 024549310 01813 0181308

06

05 06013 06013321 05208 05207695 03007 030066610 02221 0222057

08

05 06944 06943581 06013 06013325 03472 034717910 02564 0256409

3 Analytic Solutions

Using the Laplace transform technique on the system ofequations (9) subject to the initial and boundary conditions(11a)ndash(11e) we obtain

(120578 119904) = minus [

Gm1199041198631

1198901205821120578+

(1 minus 119890minus119904)

11990421198632

1198901205822120578]

+

1

119904

+

(1 minus 119890minus119904)

11990421198631

+

Gm1199041198632

1198901205823120578

(12)

119879 (120578 119904) =

(1 minus 119890minus119904)

1199042

1198901205822120578 (13)

119862 (120578 119904) =

1198901205821120578

119904

(14)

where 119862(120578 119904) 119879(120578 119904) (120578 119904) are respectively the Laplacetransforms of 119862(120578 119905) 119879(120578 119905) and 119906(120578 119905) 119904 gt 0 is the Laplacetransform parameter and

1205821= 05 minus119878Sc minus radic1198782Sc2 + 4Sc (119870

119903+ 119904) (15a)

1205822= 05

minus119878(

Pr1 + 119873

) minusradic1198782(

Pr1 + 119873

)

2

+ 4 (

Pr1 + 119873

) 119904

(15b)

1205823= 05 minus119878 minus radic119878

2+ 4 (119872 + 119904) (15c)

1198631= 1205822

1+ 1198781205821minus (119872 + 119904) (15d)

1198632= 1205822

2+ 1198781205822minus (119872 + 119904) (15e)

An exact Laplace transform inversion of (12) can beobtained when there is no suctionblowing that is when

119878 = 0 However for a nonzero 119878 the inversion of (12) isnot possible Thus the inversion of (12)ndash(14) was obtainednumerically using INVLAP routine in Matlab However tocompare the results which are obtained using INVLAP rou-tine the exact inversion of (13) and (14) gave the solutions

119879 (120578 119905) = 119890minus119887120578[119875 (120578 119905) minus 119867 (119905 minus 1) 119875 (120578 119905 minus 1)] (16)

119862 (120578 119905) =

1

2

[119890minus120578(119886minusradic119889) erfc (119905

1) + 119890minus120578(119886+radic119889) erfc (119905

2)] (17)

where erfc is the complementary error function 119867 is theHeaviside unit step function and

119875 (120578 119905) = (

119905

2

+

119898120578

4radic119899

) 119890120578radic119899 erfc (119905

3)

+ (

119905

2

minus

119898120578

4radic119899

) 119890minus120578radic119899 erfc (119905

4)

1199051 1199052= plusmnradic

119889119905

119888

+

120578

2

radic

119888

119905

1199053 1199054= plusmnradic

119899119905

119898

+

120578

2

radic

119898

119905

119886 =

119878Sc2

119887 =

Pr1198782 (1 + 119873)

119888 = Sc

119889 =

1198782Sc2

4

+ Sc119870119903 119898 =

Pr1 + 119873

119899 =

Pr21198782

4(1 + 119873)2

(18)

In the absence of suctionblowing and the thermal radi-ation effect (119878 = 0 and 119873 = 0) the solution 119879(120578 119905) in (16) isin agreement with the solution obtained by Chandran et al[36]

31 Skin Friction Nusselt Number and Sherwood NumberThe nondimensional quantities of engineering interest arethe skin friction 120591 which is a measure of shear stress at theplate theNusselt numberNu whichmeasures the rate of heattransfer and the Sherwood number Sh which measures therate of mass transfer at the plateThe Nusselt number Nu andthe Sherwood number Sh have the exact values

NuRe= minus (1 + 119873)(

120597119879

120597120578

)

120578=0

= minus [119891 (119905) minus 119867 (119905 minus 1) 119891 (119905 minus 1)]

ShRe= minus(

120597119862

120597120578

)

120578=0

= (119886 + radic119889) + radic

119888

120587119905

119890minus119889119905119888

minus radic119889 erfc(radic119889119905119888

)

(19)

Mathematical Problems in Engineering 5

Table 2 Comparison of exact and numerical values of NuRe and ShRe for different values of 119905 and 119878 when119870119903= 05 Sc = 1 Pr = 071 and

119873 = 1

119905 119878

NuRe NuRe ShRe ShReExact Numerical Exact Numerical

02minus1 053388346 0533883 094620854 09462090 060133195 0601332 138566159 13856621 067588346 0675883 194620854 1946209

04minus1 071843936 0718439 064706152 06470620 085041184 0850412 106475703 10647571 100243936 1002439 164706152 1647062

06minus1 084692696 0846927 053359416 05335940 104153758 1041537 093657181 09365721 127292696 1272927 153359416 1533594

08minus1 094692675 0946927 047550301 04755030 120266405 1202664 086753072 08675311 151492675 1514927 147550301 1475503

0 1 2 3 4 50

01

02

03

04

05

06

07

08

Exact solution Numerical solution

S = minus1 0 1

T(120578

120578

t)

(a)

Exact solution Numerical solution

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

S = minus1 0 1

C(120578

120578

t)

(b)

Figure 1 Comparison of numerical and exact solutions of (a) 119879(120578 119905) and (b) 119862(120578 119905) for different values of 119878 when Pr = 071 Sc = 1119873 = 1119870119903= 05 and 119905 = 05

where Re = 1198800119871] is the Reynolds number 119871 is some

characteristic length and

119891 (119905) = (1 + 119873) [(

119898

2radic119899

+ radic119899119905) erfc(radic119899119905119898

)

minus(

119898

2radic119899

+ 119887119905 + radic119899119905) minus radic119898119905

120587

119890minus119899119905119898

]

(20)

The values of skin friction 120591 can be obtained numericallyusing the Matlab INVLAP routine

4 Validation of Numerical Results

In order to validate the numerical results obtained using theMatlab INVLAP routine the values of NuRe are comparedwith the exact values obtained by Seth et al [37] in Table 1Further the numerical values of NuRe and ShRe arecomparedwith the exact analytical values in Table 2Thefluidtemperature and species concentration profiles are plottedusing both the numerical and exact values and are compared

6 Mathematical Problems in Engineering

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

M = 3 5 7

Gm = 1

Suction (S = 1) Blowing (S = minus1)

u(120578t)

120578

(a)

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

M = 3 5 7

Gm = minus1

Suction (S = 1) Blowing (S = minus1)

u(120578t)

120578

(b)

Figure 2 Effect of119872 on 119906(120578 119905) when Sc = 1 119870119903= 05119873 = 1 Pr = 071 and 119905 = 07

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

1

18 2 220

005

01

015

N = 05 10 15

Gm = 1

Suction (S = 1) Blowing (S = minus1)

u(120578t)

u(120578t)

120578

120578

(a)

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

1

N = 05 10 15

Gm = minus1

08 1 12

015

02

025

Suction (S = 1) Blowing (S = minus1)

u(120578t)

u(120578t)

120578

120578

(b)

Figure 3 Effect of119873 on 119906(120578 119905) when Sc = 1 119870119903= 05119872 = 3 Pr = 071 and 119905 = 07

in Figure 1 An excellent agreement between the values isobserved which validates the accuracy of the INVLAP results

5 Results and Discussion

The unsteady hydromagnetic free convective flow of a vis-cous incompressible electrically conducting and heat radi-ating fluid past an infinite vertical flat plate in the presence ofsuctionblowing and chemically reactive species concentra-tion has been studiednumerically using the Matlab INVLAP

routine The fluid velocity 119906(120578 119905) temperature 119879(120578 119905) andspecies concentration 119862(120578 119905) profiles are given in Figures 23 4 5 6 7 8 9 10 and 11 whereas the numerical valuesof the skin friction heat and mass transfer coefficients arepresented in Tables 3 and 4 when 119905 = 07 Gm = 1 (whichcorresponds to assisting buoyancy) and Gm = minus1 (whichcorresponds to opposing buoyancy)

Figures 2 to 7 show the effects of the magnetic fieldthermal radiation suctionblowing chemical reaction massdiffusion and thermal diffusion on the fluid velocity Withassisting mass buoyancy (ie Gm = 1) the fluid velocity

Mathematical Problems in Engineering 7

0 1 2 3 4 5 60

01

02

03

04

05

06

07

08

09

1

S = 1 2 3

S = minus1 minus2 minus3

Gm = 1

u(120578t)

120578

(a)

0 1 2 3 4 5 60

01

02

03

04

05

06

07

08

09

1

S = minus1 minus2 minus3

Gm = minus1

u(120578t)

120578

S = 1 2 3

(b)

Figure 4 Effect of 119878 on 119906(120578 119905) when Sc = 1 119870119903= 05119873 = 1 Pr = 071119872 = 3 and 119905 = 07

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

08 1 12 14 16

015

02

025

03

Gm = 1

u(120578t)

120578

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

Kr = 05 10 15

(a)

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

1

12 14 16

005

01

015

Gm = minus1

u(120578t)

u(120578t)

120578

120578

Suction (S = 1) Blowing (S = minus1)

Kr = 0 10 15

(b)

Figure 5 Effect of119870119903on 119906(120578 119905) when Sc = 1119872 = 3119873 = 1 Pr = 071 and 119905 = 07

decreases with an increase in the magnetic parameter 119872chemical reaction parameter 119870

119903 Schmidt number Sc and

Prandtl number Pr while it increases with the thermalradiation parameter 119873 Since the Schmidt number Sc is theratio of viscous to mass diffusivity an increase in Sc impliesa decrease in the mass diffusion rate Hence the magneticfield and the chemical reaction rate tend to reduce the fluidvelocity whereas the mass diffusivity thermal diffusivity andthermal radiation tend to increase the fluid velocity Withopposing mass buoyancy (ie for Gm = minus1) the behaviourof the fluid velocity with respect to all governing parametersremains the same except with respect to Sc when the fluidvelocity decreases with an increase in mass diffusivity

Figure 4 shows that the fluid velocity decreases with anincrease in suction velocity whereas it increases with anincrease in blowing for both assisting and opposing massbuoyancy This result may be useful in engineering applica-tions where the formation of boundary layer is to be delayedThis can be done by increasing the suction velocity We alsoobserve that the thickness of the momentum boundary layeris less in the presence of opposing mass buoyancy than in thepresence of assisting mass buoyancy

Figures 8 9 and 10 show the effects of thermal radi-ation thermal diffusion and suctionblowing on the fluidtemperature We observe from Figures 8 and 9 that the fluidtemperature increases with119873 while it decreases with Pr The

8 Mathematical Problems in Engineering

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

Sc = 05 10 15

Gm = 1

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

(a)

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

Sc = 05 10 15

Gm = minus1

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

(b)

Figure 6 Effect of Sc on 119906(120578 119905) when119872 = 3 119870119903= 05119873 = 1 Pr = 071 and 119905 = 07

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

1

Pr = 071 1 7

Gm = 1

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

(a)

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

1

Pr = 071 1 7

Gm = minus1

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

(b)

Figure 7 Effect of Pr on 119906(120578 119905) when Sc = 1 119870119903= 05119873 = 1119872 = 3 and 119905 = 07

Prandtl number is the ratio of viscosity to thermal diffusivityand an increase in Pr implies a decrease in thermal diffusivityThermal radiation and diffusivity tend to increase the fluidtemperature Figure 10 shows that an increase in suctionvelocity causes a reduction in the fluid temperature whereasan increase in blowing increases the fluid temperature

Figures 11 12 and 13 show the effects of the chemicalreaction mass diffusion and suction or blowing on thespecies concentration We observe from Figures 11 and 12that the species concentration decreases with an increase inthe chemical reaction parameter or the Schmidt numberThus the chemical reaction rate tends to reduce the species

concentration whereas mass diffusivity increases the speciesconcentration It is clear from Figure 13 that the suctionvelocity reduces the species concentration while blowingcauses an increase in the species concentration We observefrom Figures 11 and 12 that the species concentration is higherin the case of blowing than in the case of suction We alsoobserve from Figures 2ndash11 that the momentum thermal andconcentration boundary layers are thicker in the case ofblowing than in the case of suction

Table 3 shows the effects of the magnetic field thermalradiation suctionblowing chemical reactionmass diffusionand thermal diffusion on the skin friction in the presence

Mathematical Problems in Engineering 9

Table 3 Effect of fluid and physical parameters on the skin friction minus120591 when 119905 = 07

119872 119873 119878 119870119903

Sc Pr minus120591 (119878 gt 0Gm = 1)

minus120591 (119878 lt 0Gm = 1)

minus120591 (119878 gt 0Gm = minus1)

minus120591 (119878 lt 0Gm = minus1)

3 1 mdash 05 1 071 1747428 0788854 2397344 14387705 mdash mdash mdash mdash mdash 2291296 1325411 2865812 18999277 mdash mdash mdash mdash mdash 2737296 1766244 3258082 2287031mdash 05 mdash mdash mdash mdash 1763388 0794076 2413303 1443992mdash 10 mdash mdash mdash mdash 1747428 0788854 2397344 1438770mdash 15 mdash mdash mdash mdash 1735676 0784937 2385592 1434853mdash mdash 1 mdash mdash mdash 1747428 mdash 2397344 mdashmdash mdash 2 mdash mdash mdash 2463830 mdash 3043645 mdashmdash mdash 3 mdash mdash mdash 3291595 mdash 3786658 mdashmdash mdash minus1 mdash mdash mdash mdash 0788854 mdash 1438770mdash mdash minus2 mdash mdash mdash mdash 0540588 mdash 1120403mdash mdash minus3 mdash mdash mdash mdash 0393845 mdash 0888908mdash mdash mdash 05 mdash mdash 1747428 0788854 2397344 1438770mdash mdash mdash 10 mdash mdash 1761672 0803098 2383100 1424526mdash mdash mdash 15 mdash mdash 1774216 0815642 2370555 1411981mdash mdash mdash mdash 05 mdash 1666559 0771682 2478212 1455942mdash mdash mdash mdash 10 mdash 1747428 0788854 2397344 1438770mdash mdash mdash mdash 15 mdash 1796602 0798486 2348170 1429138mdash mdash mdash mdash mdash 071 1747428 0788854 2397344 1438770mdash mdash mdash mdash mdash 10 1766522 0795089 2416438 1445005mdash mdash mdash mdash mdash 70 1884422 0830142 2534338 1480058

0 1 2 3 4 50

01

02

03

04

05

06

07

N = 05 10 15

Suction (S = 1) Blowing (S = minus1)

T(120578

120578

t)

Figure 8 Effect of119873 on 119879(120578 119905) when Pr = 071 and 119905 = 07

of assisting and opposing mass buoyancy In the case ofassisting mass buoyancy an increase in 119872 119870

119903 Sc and Pr

tends to increase the skin friction whereas an increase in119873 reduces the skin friction for both suction and blowingcases Thus the skin friction increases with an increase inthe magnetic field parameter and the chemical reaction rate

0 1 2 3 4 50

01

02

03

04

05

06

07

Pr = 071 1 7

Suction (S = 1) Blowing (S = minus1)

T(120578

120578

t)

Figure 9 Effect of Pr on 119879(120578 119905) when119873 = 1 and 119905 = 07

while it decreaseswith an increase inmass diffusivity thermaldiffusivity and thermal radiation in the case of assisting massbuoyancy However for opposing mass buoyancy the skinfriction increases with mass diffusivity while decreasing withan increase in the rate of the chemical reaction

10 Mathematical Problems in Engineering

Table 4 Values of NuRe and ShRe for different values of governing parameters when 119905 = 07

119873 119878 Pr 119870119903

Sc NuRe NuRe ShRe ShRe(Suction) (Blowing) (Suction) (Blowing)

05 mdash 071 mdash mdash 1249449 0752449 mdash mdash10 mdash 071 mdash mdash 1396642 0899642 mdash mdash15 mdash 071 mdash mdash 1527010 1030010 mdash mdash1 1 071 05 10 1396642 mdash 1500355 mdash1 2 071 05 10 1712939 mdash 2280316 mdash1 3 071 05 10 2069304 mdash 3174187 mdash1 minus1 071 05 10 mdash 0899642 mdash 05003551 minus2 071 05 10 mdash 0718939 mdash 02803161 minus3 071 05 10 mdash 0578304 mdash 01741871 mdash 071 mdash mdash 1396642 0899642 mdash mdash1 mdash 10 mdash mdash 1723721 1023721 mdash mdash1 mdash 70 mdash mdash 6662968 1762968 mdash mdashmdash mdash mdash 05 1 mdash mdash 1500355 0500355mdash mdash mdash 10 1 mdash mdash 1691322 0691322mdash mdash mdash 15 1 mdash mdash 1865496 0865496mdash mdash mdash 05 05 mdash mdash 0921462 0421462mdash mdash mdash 05 10 mdash mdash 1500355 0500355mdash mdash mdash 05 15 mdash mdash 2035801 0535801

0 1 2 3 4 7650

01

02

03

04

05

06

07

S = minus1 minus2 minus3

S = 1 2 3

T(120578

120578

t)

Figure 10 Effect of 119878 on 119879(120578 119905) when Pr = 071119873 = 1 and 119905 = 07

Table 4 shows the effects of thermal radiation suc-tionblowing and thermal diffusion on the heat transfer ratefrom the plate It also shows the effects of suctionblowingchemical reaction rate andmass diffusion on the rate of masstransfer at the plate We observe that the Nusselt numberincreases with 119873 and Pr Thus the thermal radiation tendsto accelerate heat transfer at the plate while thermal diffusionhas reverse effect Table 4 also shows that the rate of heattransfer at the plate increases with an increase in suctionvelocity and decreases with blowing The Sherwood numberincreases with 119870

119903and Sc for both suction and blowing An

increase in the suction velocity causes an increase in therate of mass transfer whereas an increase in blowing has theopposite effect

0 05 1 15 2 25 3 35 4

Suction (S = 1) Blowing (S = minus1)

0

01

02

03

04

05

06

07

08

09

1

C(120578

120578

t)

Kr = 0 05 10

Figure 11 Effect of119872 on 119862(120578 119905) when Sc = 1 and 119905 = 07

6 Conclusions

The unsteady hydromagnetic free convection flow of aviscous incompressible and heat-radiating fluid past animpulsively started vertical porous flat plate with rampedtemperature in the presence of a chemically reactive speciesconcentration was studied For certain parameter valuesexact solutions of the model equations were found usingLaplace transforms The Matlab INVLAP routine was used tofind the numerical results in the general case The numericalresults were found to be in excellent agreement with the

Mathematical Problems in Engineering 11

Suction (S = 1) Blowing (S = minus1)

Sc = 05 10 15

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

C(120578

120578

t)

Figure 12 Effect of Sc on 119862(120578 119905) when 119870119903= 05 and 119905 = 07

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

S = 1 2 3

S = minus1 minus2 minus3

C(120578

120578

t)

Figure 13 Effect of 119878 on 119862(120578 119905) when 119870119903= 05 Sc = 1 and 119905 = 07

results of Seth et al [37] and with the exact results Theimportant findings of the present work may be summarizedas follows

(i) In the case of assisting mass buoyancy force the fluidvelocity is reduced by an increase in themagnetic fieldand the chemical reaction rate

(ii) Blowing causes an acceleration in the fluid velocitywhile suction has reverse effect on it

(iii) An increase in the chemical reaction rate tends toreduce the species concentration whereas the massdiffusivity parameter has reverse effect

(iv) Thermal radiation thermal diffusivity and blowingtend to increase the fluid temperature whereas suc-tion has the opposite effect

(v) In the case of an assisting mass buoyancy force themagnetic field and chemical reaction parameter tendto increase the skin friction while themass diffusivitythermal diffusivity and thermal radiation have thereverse effect

(vi) Thermal diffusivity and blowing tend to reduce therate of heat transfer at the plate whereas thermalradiation and suction have the opposite effect

Acknowledgment

The authors would like to thank the referee for his valuableand constructive suggestions

References

[1] A S Gupta ldquoSteady and transient free convection of anelectrically conducting fluid froma vertical plate in the presenceof a magnetic fieldrdquo Applied Scientific Research vol 9 no 1 pp319ndash333 1960

[2] A S Gupta ldquoLaminar free convection flow of an electricallyconducting fluid from a vertical plate with uniform surface heatflux and variable wall temperature in the presence of amagneticfieldrdquoZeitschrift fur AngewandteMathematik undPhysik vol 13pp 324ndash333 1962

[3] K R Cramer ldquoSeveral magnetohydrodynamic free convectionsolutionsrdquo Journal of Heat Transfer vol 85 pp 35ndash40 1963

[4] I Pop ldquoOn the unsteady hydromagnetic free convection flowpast a vertical infinite flat platerdquo Indian Journal of Physics vol43 pp 196ndash200 1969

[5] H K Kuiken ldquoMagnetohydrodynamic free convection in astrong cross fieldrdquo Journal of Fluid Mechanics vol 40 no 1 pp21ndash38 1970

[6] G Wilks ldquoMagnetohydrodynamic free convection about asemi-infinite vertical plate in a strong cross fieldrdquo Zeitschrift furAngewandte Mathematik und Physik ZAMP vol 27 no 5 pp621ndash631 1976

[7] M A Hossain ldquoEffect of hall current on unsteady hydromag-netic free convection flow near an infinite vertical porous platerdquoJournal of the Physical Society of Japan vol 55 no 7 pp 2183ndash2190 1986

[8] T K Aldoss M A Al-Nimr M A Jarrah and B J Al-SharsquoerldquoMagnetohydrodynamicmixed convection from a vertical plateembedded in a porous mediumrdquo Numerical Heat Transfer PartA vol 28 no 5 pp 635ndash645 1995

[9] K A Helmy ldquoMHD unsteady free convection flow past avertical porous platerdquo Zeitschrift fur Angewandte Mathematikund Mechanik vol 78 no 4 pp 255ndash270 1998

[10] Y J Kim ldquoUnsteadyMHD convective heat transfer past a semi-infinite vertical porous moving plate with variable suctionrdquoInternational Journal of Engineering Science vol 38 no 8 pp833ndash845 2000

[11] H S Takhar S Roy and G Nath ldquoUnsteady free convectionflow over an infinite vertical porous plate due to the combinedeffects of thermal and mass diffusion magnetic field and Hallcurrentsrdquo Heat and Mass Transfer vol 39 no 10 pp 825ndash8342003

[12] N Ahmed H K Sarmah and D Kalita ldquoThermal diffusioneffect on a three-dimensional MHD free convection with masstransfer flow from a porous vertical platerdquo Latin AmericanApplied Research vol 41 pp 165ndash176 2011

12 Mathematical Problems in Engineering

[13] M A Hossain and H S Takhar ldquoRadiation effect on mixedconvection along a vertical plate with uniform surface temper-aturerdquoHeat and Mass Transfer vol 31 no 4 pp 243ndash248 1996

[14] A Y Bakier and R S R Gorla ldquoThermal radiation effect onmixed convection from horizontal surfaces in saturated porousmediardquo Transport in Porous Media vol 23 no 3 pp 357ndash3631996

[15] H S Takhar R S R Gorla and VM Soundalgekar ldquoRadiationeffects onMHD free convection flowof a gas past a semi-infinitevertical platerdquo International Journal of Numerical Methods forHeat and Fluid Flow vol 6 no 2 pp 77ndash83 1996

[16] A J Chamkha ldquoThermal radiation and buoyancy effects onhydromagnetic flow over an accelerating permeable surfacewith heat source or sinkrdquo International Journal of EngineeringScience vol 38 no 15 pp 1699ndash1712 2000

[17] G E-D A Azzam ldquoRadiation effects on the MHDmixed free-forced convective flow past a semi-infinitemoving vertical platefor high temperature differencesrdquo Physica Scripta vol 66 no 1pp 71ndash76 2002

[18] C Israel-Cookey A Ogulu and V B Omubo-Pepple ldquoInflu-ence of viscous dissipation and radiation on unsteady MHDfree-convection flow past an infinite heated vertical plate ina porous medium with time-dependent suctionrdquo InternationalJournal of Heat andMass Transfer vol 46 no 13 pp 2305ndash23112003

[19] M A A Mahmoud ldquoThermal radiation effect on unsteadyMHD free convection flow past a vertical plate with tem-perature-dependent viscosityrdquo Canadian Journal of ChemicalEngineering vol 87 no 1 pp 47ndash52 2009

[20] A R Bestman and S K Adjepong ldquoUnsteady hydromagneticfree-convection flow with radiative heat transfer in a rotatingfluidmdashI Incompressible optically thin fluidrdquo Astrophysics andSpace Science vol 143 no 1 pp 73ndash80 1988

[21] A J Chamkha H S Takhar and V M Soundalgekar ldquoRadia-tion effects on free convection flow past a semi-infinite verticalplate with mass transferrdquo Chemical Engineering Journal vol 84no 3 pp 335ndash342 2001

[22] M A El-Hakiem and M F El-Amin ldquoThermal radiation effecton non-Darcy natural convection with lateral mass transferrdquoHeat and Mass Transfer vol 37 no 2-3 pp 161ndash165 2001

[23] V R Prasad N B Reddy and R Muthucumaraswamy ldquoRadia-tion and mass transfer effects on two-dimensional flow past animpulsively started infinite vertical platerdquo International Journalof Thermal Sciences vol 46 no 12 pp 1251ndash1258 2007

[24] O D Makinde and A Ogulu ldquoThe effect of thermal radiationon the heat and mass transfer flow of a variable viscosity fluidpast a vertical porous plate permeated by a transverse magneticfieldrdquo Chemical Engineering Communications vol 195 no 12pp 1575ndash1584 2008

[25] SMukhopadhyay andG C Layek ldquoEffects of thermal radiationand variable fluid viscosity on free convective flow and heattransfer past a porous stretching surfacerdquo International Journalof Heat andMass Transfer vol 51 no 9-10 pp 2167ndash2178 2008

[26] D Pal ldquoHeat andmass transfer in stagnation-point flow towardsa stretching surface in the presence of buoyancy force andthermal radiationrdquoMeccanica vol 44 no 2 pp 145ndash158 2009

[27] A J Chamkha ldquoMHDflowof a uniformly streched vertical per-meable surface in the presence of heat generationabsorptionand a chemical reactionrdquo International Communications inHeatand Mass Transfer vol 30 no 3 pp 413ndash422 2003

[28] A A Afify ldquoMHD free convective flow and mass transferover a stretching sheet with chemical reactionrdquo Heat and MassTransfer vol 40 no 6-7 pp 495ndash500 2004

[29] A Postelnicu ldquoInfluence of chemical reaction on heat andmasstransfer by natural convection from vertical surfaces in porousmedia considering Soret and Dufour effectsrdquo Heat and MassTransfer vol 43 no 6 pp 595ndash602 2007

[30] R Kandasamy and P G Palanimani ldquoEffects of chemical reac-tions heat and mass transfer on nonlinear magnetohydrody-namic boundary layer flow over a wedge with a porous mediumin the presence of ohmic heating and viscous dissipationrdquoJournal of Porous Media vol 10 no 5 pp 489ndash501 2007

[31] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[32] A A Hayday D A Bowlus and R A McGraw ldquoFree convec-tion from a vertical plate with step discontinuities in surfacetemperaturerdquo ASME Journal of Heat Transfer vol 89 pp 244ndash250 1967

[33] M Kelleher ldquoFree convection from a vertical plate with dis-continuous wall temperaturerdquo Journal of Heat Transfer Trans-actions ASME vol 93 no 4 pp 349ndash356 1971

[34] T-T Kao ldquoLaminar free convective heat transfer response alonga vertical flat plate with step jump in surface temperaturerdquoLetters inHeat andMass Transfer vol 2 no 5 pp 419ndash428 1975

[35] S Lee and M M Yovanovich ldquoLaminar natural convectionfrom a vertical plate with a step change in wall temperaturerdquoJournal of Heat Transfer vol 113 no 2 pp 501ndash504 1991

[36] P Chandran N C Sacheti and A K Singh ldquoNatural convec-tion near a vertical plate with ramped wall temperaturerdquo Heatand Mass Transfer vol 41 no 5 pp 459ndash464 2005

[37] G S Seth M S Ansari and R Nandkeolyar ldquoMHD naturalconvection flow with radiative heat transfer past an impulsivelymoving plate with ramped wall temperaturerdquo Heat and MassTransfer vol 47 no 5 pp 551ndash561 2011

[38] G S Seth R Nandkeolyar and M S Ansari ldquoEffects ofthermal radiation and rotation on unsteady hydromagnetic freeconvection flow past an impulsively moving vertical plate withramped temperature in a porous mediumrdquo Journal of AppliedFluid Mechanics vol 6 no 1 pp 27ndash38 2013

[39] O D Makinde ldquoFree convection flow with thermal radiationand mass transfer past a moving vertical porous platerdquo Interna-tional Communications in Heat and Mass Transfer vol 32 no10 pp 1411ndash1419 2005

[40] I U Mbeledogu and A Ogulu ldquoHeat and mass transfer of anunsteady MHD natural convection flow of a rotating fluid pasta vertical porous flat plate in the presence of radiative heattransferrdquo International Journal of Heat and Mass Transfer vol50 no 9-10 pp 1902ndash1908 2007

[41] A A Afify ldquoSimilarity solution in MHD effects of thermaldiffusion and diffusion thermo on free convective heat andmass transfer over a stretching surface considering suction orinjectionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2202ndash2214 2009

[42] K Cramer and S PaiMagnetofluid Dynamics for Engineers andApplied Physicists McGraw Hill New York NY USA 1973

[43] R C Meyer ldquoOn reducing aerodynamic heat-transfer ratesby magnetohydrodynamic techniquesrdquo Journal of Aero SpaceScience vol 25 pp 561ndash572 1958

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Table 1 A comparison of minus120575119879120575120578|120578=0

when 119878 = 0

119905 119873 Seth et al [37] Present results

02

05 03472 03471791 03007 03006665 01736 017359010 01282 0128204

04

05 04910 04909851 04252 04252065 02455 024549310 01813 0181308

06

05 06013 06013321 05208 05207695 03007 030066610 02221 0222057

08

05 06944 06943581 06013 06013325 03472 034717910 02564 0256409

3 Analytic Solutions

Using the Laplace transform technique on the system ofequations (9) subject to the initial and boundary conditions(11a)ndash(11e) we obtain

(120578 119904) = minus [

Gm1199041198631

1198901205821120578+

(1 minus 119890minus119904)

11990421198632

1198901205822120578]

+

1

119904

+

(1 minus 119890minus119904)

11990421198631

+

Gm1199041198632

1198901205823120578

(12)

119879 (120578 119904) =

(1 minus 119890minus119904)

1199042

1198901205822120578 (13)

119862 (120578 119904) =

1198901205821120578

119904

(14)

where 119862(120578 119904) 119879(120578 119904) (120578 119904) are respectively the Laplacetransforms of 119862(120578 119905) 119879(120578 119905) and 119906(120578 119905) 119904 gt 0 is the Laplacetransform parameter and

1205821= 05 minus119878Sc minus radic1198782Sc2 + 4Sc (119870

119903+ 119904) (15a)

1205822= 05

minus119878(

Pr1 + 119873

) minusradic1198782(

Pr1 + 119873

)

2

+ 4 (

Pr1 + 119873

) 119904

(15b)

1205823= 05 minus119878 minus radic119878

2+ 4 (119872 + 119904) (15c)

1198631= 1205822

1+ 1198781205821minus (119872 + 119904) (15d)

1198632= 1205822

2+ 1198781205822minus (119872 + 119904) (15e)

An exact Laplace transform inversion of (12) can beobtained when there is no suctionblowing that is when

119878 = 0 However for a nonzero 119878 the inversion of (12) isnot possible Thus the inversion of (12)ndash(14) was obtainednumerically using INVLAP routine in Matlab However tocompare the results which are obtained using INVLAP rou-tine the exact inversion of (13) and (14) gave the solutions

119879 (120578 119905) = 119890minus119887120578[119875 (120578 119905) minus 119867 (119905 minus 1) 119875 (120578 119905 minus 1)] (16)

119862 (120578 119905) =

1

2

[119890minus120578(119886minusradic119889) erfc (119905

1) + 119890minus120578(119886+radic119889) erfc (119905

2)] (17)

where erfc is the complementary error function 119867 is theHeaviside unit step function and

119875 (120578 119905) = (

119905

2

+

119898120578

4radic119899

) 119890120578radic119899 erfc (119905

3)

+ (

119905

2

minus

119898120578

4radic119899

) 119890minus120578radic119899 erfc (119905

4)

1199051 1199052= plusmnradic

119889119905

119888

+

120578

2

radic

119888

119905

1199053 1199054= plusmnradic

119899119905

119898

+

120578

2

radic

119898

119905

119886 =

119878Sc2

119887 =

Pr1198782 (1 + 119873)

119888 = Sc

119889 =

1198782Sc2

4

+ Sc119870119903 119898 =

Pr1 + 119873

119899 =

Pr21198782

4(1 + 119873)2

(18)

In the absence of suctionblowing and the thermal radi-ation effect (119878 = 0 and 119873 = 0) the solution 119879(120578 119905) in (16) isin agreement with the solution obtained by Chandran et al[36]

31 Skin Friction Nusselt Number and Sherwood NumberThe nondimensional quantities of engineering interest arethe skin friction 120591 which is a measure of shear stress at theplate theNusselt numberNu whichmeasures the rate of heattransfer and the Sherwood number Sh which measures therate of mass transfer at the plateThe Nusselt number Nu andthe Sherwood number Sh have the exact values

NuRe= minus (1 + 119873)(

120597119879

120597120578

)

120578=0

= minus [119891 (119905) minus 119867 (119905 minus 1) 119891 (119905 minus 1)]

ShRe= minus(

120597119862

120597120578

)

120578=0

= (119886 + radic119889) + radic

119888

120587119905

119890minus119889119905119888

minus radic119889 erfc(radic119889119905119888

)

(19)

Mathematical Problems in Engineering 5

Table 2 Comparison of exact and numerical values of NuRe and ShRe for different values of 119905 and 119878 when119870119903= 05 Sc = 1 Pr = 071 and

119873 = 1

119905 119878

NuRe NuRe ShRe ShReExact Numerical Exact Numerical

02minus1 053388346 0533883 094620854 09462090 060133195 0601332 138566159 13856621 067588346 0675883 194620854 1946209

04minus1 071843936 0718439 064706152 06470620 085041184 0850412 106475703 10647571 100243936 1002439 164706152 1647062

06minus1 084692696 0846927 053359416 05335940 104153758 1041537 093657181 09365721 127292696 1272927 153359416 1533594

08minus1 094692675 0946927 047550301 04755030 120266405 1202664 086753072 08675311 151492675 1514927 147550301 1475503

0 1 2 3 4 50

01

02

03

04

05

06

07

08

Exact solution Numerical solution

S = minus1 0 1

T(120578

120578

t)

(a)

Exact solution Numerical solution

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

S = minus1 0 1

C(120578

120578

t)

(b)

Figure 1 Comparison of numerical and exact solutions of (a) 119879(120578 119905) and (b) 119862(120578 119905) for different values of 119878 when Pr = 071 Sc = 1119873 = 1119870119903= 05 and 119905 = 05

where Re = 1198800119871] is the Reynolds number 119871 is some

characteristic length and

119891 (119905) = (1 + 119873) [(

119898

2radic119899

+ radic119899119905) erfc(radic119899119905119898

)

minus(

119898

2radic119899

+ 119887119905 + radic119899119905) minus radic119898119905

120587

119890minus119899119905119898

]

(20)

The values of skin friction 120591 can be obtained numericallyusing the Matlab INVLAP routine

4 Validation of Numerical Results

In order to validate the numerical results obtained using theMatlab INVLAP routine the values of NuRe are comparedwith the exact values obtained by Seth et al [37] in Table 1Further the numerical values of NuRe and ShRe arecomparedwith the exact analytical values in Table 2Thefluidtemperature and species concentration profiles are plottedusing both the numerical and exact values and are compared

6 Mathematical Problems in Engineering

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

M = 3 5 7

Gm = 1

Suction (S = 1) Blowing (S = minus1)

u(120578t)

120578

(a)

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

M = 3 5 7

Gm = minus1

Suction (S = 1) Blowing (S = minus1)

u(120578t)

120578

(b)

Figure 2 Effect of119872 on 119906(120578 119905) when Sc = 1 119870119903= 05119873 = 1 Pr = 071 and 119905 = 07

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

1

18 2 220

005

01

015

N = 05 10 15

Gm = 1

Suction (S = 1) Blowing (S = minus1)

u(120578t)

u(120578t)

120578

120578

(a)

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

1

N = 05 10 15

Gm = minus1

08 1 12

015

02

025

Suction (S = 1) Blowing (S = minus1)

u(120578t)

u(120578t)

120578

120578

(b)

Figure 3 Effect of119873 on 119906(120578 119905) when Sc = 1 119870119903= 05119872 = 3 Pr = 071 and 119905 = 07

in Figure 1 An excellent agreement between the values isobserved which validates the accuracy of the INVLAP results

5 Results and Discussion

The unsteady hydromagnetic free convective flow of a vis-cous incompressible electrically conducting and heat radi-ating fluid past an infinite vertical flat plate in the presence ofsuctionblowing and chemically reactive species concentra-tion has been studiednumerically using the Matlab INVLAP

routine The fluid velocity 119906(120578 119905) temperature 119879(120578 119905) andspecies concentration 119862(120578 119905) profiles are given in Figures 23 4 5 6 7 8 9 10 and 11 whereas the numerical valuesof the skin friction heat and mass transfer coefficients arepresented in Tables 3 and 4 when 119905 = 07 Gm = 1 (whichcorresponds to assisting buoyancy) and Gm = minus1 (whichcorresponds to opposing buoyancy)

Figures 2 to 7 show the effects of the magnetic fieldthermal radiation suctionblowing chemical reaction massdiffusion and thermal diffusion on the fluid velocity Withassisting mass buoyancy (ie Gm = 1) the fluid velocity

Mathematical Problems in Engineering 7

0 1 2 3 4 5 60

01

02

03

04

05

06

07

08

09

1

S = 1 2 3

S = minus1 minus2 minus3

Gm = 1

u(120578t)

120578

(a)

0 1 2 3 4 5 60

01

02

03

04

05

06

07

08

09

1

S = minus1 minus2 minus3

Gm = minus1

u(120578t)

120578

S = 1 2 3

(b)

Figure 4 Effect of 119878 on 119906(120578 119905) when Sc = 1 119870119903= 05119873 = 1 Pr = 071119872 = 3 and 119905 = 07

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

08 1 12 14 16

015

02

025

03

Gm = 1

u(120578t)

120578

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

Kr = 05 10 15

(a)

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

1

12 14 16

005

01

015

Gm = minus1

u(120578t)

u(120578t)

120578

120578

Suction (S = 1) Blowing (S = minus1)

Kr = 0 10 15

(b)

Figure 5 Effect of119870119903on 119906(120578 119905) when Sc = 1119872 = 3119873 = 1 Pr = 071 and 119905 = 07

decreases with an increase in the magnetic parameter 119872chemical reaction parameter 119870

119903 Schmidt number Sc and

Prandtl number Pr while it increases with the thermalradiation parameter 119873 Since the Schmidt number Sc is theratio of viscous to mass diffusivity an increase in Sc impliesa decrease in the mass diffusion rate Hence the magneticfield and the chemical reaction rate tend to reduce the fluidvelocity whereas the mass diffusivity thermal diffusivity andthermal radiation tend to increase the fluid velocity Withopposing mass buoyancy (ie for Gm = minus1) the behaviourof the fluid velocity with respect to all governing parametersremains the same except with respect to Sc when the fluidvelocity decreases with an increase in mass diffusivity

Figure 4 shows that the fluid velocity decreases with anincrease in suction velocity whereas it increases with anincrease in blowing for both assisting and opposing massbuoyancy This result may be useful in engineering applica-tions where the formation of boundary layer is to be delayedThis can be done by increasing the suction velocity We alsoobserve that the thickness of the momentum boundary layeris less in the presence of opposing mass buoyancy than in thepresence of assisting mass buoyancy

Figures 8 9 and 10 show the effects of thermal radi-ation thermal diffusion and suctionblowing on the fluidtemperature We observe from Figures 8 and 9 that the fluidtemperature increases with119873 while it decreases with Pr The

8 Mathematical Problems in Engineering

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

Sc = 05 10 15

Gm = 1

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

(a)

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

Sc = 05 10 15

Gm = minus1

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

(b)

Figure 6 Effect of Sc on 119906(120578 119905) when119872 = 3 119870119903= 05119873 = 1 Pr = 071 and 119905 = 07

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

1

Pr = 071 1 7

Gm = 1

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

(a)

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

1

Pr = 071 1 7

Gm = minus1

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

(b)

Figure 7 Effect of Pr on 119906(120578 119905) when Sc = 1 119870119903= 05119873 = 1119872 = 3 and 119905 = 07

Prandtl number is the ratio of viscosity to thermal diffusivityand an increase in Pr implies a decrease in thermal diffusivityThermal radiation and diffusivity tend to increase the fluidtemperature Figure 10 shows that an increase in suctionvelocity causes a reduction in the fluid temperature whereasan increase in blowing increases the fluid temperature

Figures 11 12 and 13 show the effects of the chemicalreaction mass diffusion and suction or blowing on thespecies concentration We observe from Figures 11 and 12that the species concentration decreases with an increase inthe chemical reaction parameter or the Schmidt numberThus the chemical reaction rate tends to reduce the species

concentration whereas mass diffusivity increases the speciesconcentration It is clear from Figure 13 that the suctionvelocity reduces the species concentration while blowingcauses an increase in the species concentration We observefrom Figures 11 and 12 that the species concentration is higherin the case of blowing than in the case of suction We alsoobserve from Figures 2ndash11 that the momentum thermal andconcentration boundary layers are thicker in the case ofblowing than in the case of suction

Table 3 shows the effects of the magnetic field thermalradiation suctionblowing chemical reactionmass diffusionand thermal diffusion on the skin friction in the presence

Mathematical Problems in Engineering 9

Table 3 Effect of fluid and physical parameters on the skin friction minus120591 when 119905 = 07

119872 119873 119878 119870119903

Sc Pr minus120591 (119878 gt 0Gm = 1)

minus120591 (119878 lt 0Gm = 1)

minus120591 (119878 gt 0Gm = minus1)

minus120591 (119878 lt 0Gm = minus1)

3 1 mdash 05 1 071 1747428 0788854 2397344 14387705 mdash mdash mdash mdash mdash 2291296 1325411 2865812 18999277 mdash mdash mdash mdash mdash 2737296 1766244 3258082 2287031mdash 05 mdash mdash mdash mdash 1763388 0794076 2413303 1443992mdash 10 mdash mdash mdash mdash 1747428 0788854 2397344 1438770mdash 15 mdash mdash mdash mdash 1735676 0784937 2385592 1434853mdash mdash 1 mdash mdash mdash 1747428 mdash 2397344 mdashmdash mdash 2 mdash mdash mdash 2463830 mdash 3043645 mdashmdash mdash 3 mdash mdash mdash 3291595 mdash 3786658 mdashmdash mdash minus1 mdash mdash mdash mdash 0788854 mdash 1438770mdash mdash minus2 mdash mdash mdash mdash 0540588 mdash 1120403mdash mdash minus3 mdash mdash mdash mdash 0393845 mdash 0888908mdash mdash mdash 05 mdash mdash 1747428 0788854 2397344 1438770mdash mdash mdash 10 mdash mdash 1761672 0803098 2383100 1424526mdash mdash mdash 15 mdash mdash 1774216 0815642 2370555 1411981mdash mdash mdash mdash 05 mdash 1666559 0771682 2478212 1455942mdash mdash mdash mdash 10 mdash 1747428 0788854 2397344 1438770mdash mdash mdash mdash 15 mdash 1796602 0798486 2348170 1429138mdash mdash mdash mdash mdash 071 1747428 0788854 2397344 1438770mdash mdash mdash mdash mdash 10 1766522 0795089 2416438 1445005mdash mdash mdash mdash mdash 70 1884422 0830142 2534338 1480058

0 1 2 3 4 50

01

02

03

04

05

06

07

N = 05 10 15

Suction (S = 1) Blowing (S = minus1)

T(120578

120578

t)

Figure 8 Effect of119873 on 119879(120578 119905) when Pr = 071 and 119905 = 07

of assisting and opposing mass buoyancy In the case ofassisting mass buoyancy an increase in 119872 119870

119903 Sc and Pr

tends to increase the skin friction whereas an increase in119873 reduces the skin friction for both suction and blowingcases Thus the skin friction increases with an increase inthe magnetic field parameter and the chemical reaction rate

0 1 2 3 4 50

01

02

03

04

05

06

07

Pr = 071 1 7

Suction (S = 1) Blowing (S = minus1)

T(120578

120578

t)

Figure 9 Effect of Pr on 119879(120578 119905) when119873 = 1 and 119905 = 07

while it decreaseswith an increase inmass diffusivity thermaldiffusivity and thermal radiation in the case of assisting massbuoyancy However for opposing mass buoyancy the skinfriction increases with mass diffusivity while decreasing withan increase in the rate of the chemical reaction

10 Mathematical Problems in Engineering

Table 4 Values of NuRe and ShRe for different values of governing parameters when 119905 = 07

119873 119878 Pr 119870119903

Sc NuRe NuRe ShRe ShRe(Suction) (Blowing) (Suction) (Blowing)

05 mdash 071 mdash mdash 1249449 0752449 mdash mdash10 mdash 071 mdash mdash 1396642 0899642 mdash mdash15 mdash 071 mdash mdash 1527010 1030010 mdash mdash1 1 071 05 10 1396642 mdash 1500355 mdash1 2 071 05 10 1712939 mdash 2280316 mdash1 3 071 05 10 2069304 mdash 3174187 mdash1 minus1 071 05 10 mdash 0899642 mdash 05003551 minus2 071 05 10 mdash 0718939 mdash 02803161 minus3 071 05 10 mdash 0578304 mdash 01741871 mdash 071 mdash mdash 1396642 0899642 mdash mdash1 mdash 10 mdash mdash 1723721 1023721 mdash mdash1 mdash 70 mdash mdash 6662968 1762968 mdash mdashmdash mdash mdash 05 1 mdash mdash 1500355 0500355mdash mdash mdash 10 1 mdash mdash 1691322 0691322mdash mdash mdash 15 1 mdash mdash 1865496 0865496mdash mdash mdash 05 05 mdash mdash 0921462 0421462mdash mdash mdash 05 10 mdash mdash 1500355 0500355mdash mdash mdash 05 15 mdash mdash 2035801 0535801

0 1 2 3 4 7650

01

02

03

04

05

06

07

S = minus1 minus2 minus3

S = 1 2 3

T(120578

120578

t)

Figure 10 Effect of 119878 on 119879(120578 119905) when Pr = 071119873 = 1 and 119905 = 07

Table 4 shows the effects of thermal radiation suc-tionblowing and thermal diffusion on the heat transfer ratefrom the plate It also shows the effects of suctionblowingchemical reaction rate andmass diffusion on the rate of masstransfer at the plate We observe that the Nusselt numberincreases with 119873 and Pr Thus the thermal radiation tendsto accelerate heat transfer at the plate while thermal diffusionhas reverse effect Table 4 also shows that the rate of heattransfer at the plate increases with an increase in suctionvelocity and decreases with blowing The Sherwood numberincreases with 119870

119903and Sc for both suction and blowing An

increase in the suction velocity causes an increase in therate of mass transfer whereas an increase in blowing has theopposite effect

0 05 1 15 2 25 3 35 4

Suction (S = 1) Blowing (S = minus1)

0

01

02

03

04

05

06

07

08

09

1

C(120578

120578

t)

Kr = 0 05 10

Figure 11 Effect of119872 on 119862(120578 119905) when Sc = 1 and 119905 = 07

6 Conclusions

The unsteady hydromagnetic free convection flow of aviscous incompressible and heat-radiating fluid past animpulsively started vertical porous flat plate with rampedtemperature in the presence of a chemically reactive speciesconcentration was studied For certain parameter valuesexact solutions of the model equations were found usingLaplace transforms The Matlab INVLAP routine was used tofind the numerical results in the general case The numericalresults were found to be in excellent agreement with the

Mathematical Problems in Engineering 11

Suction (S = 1) Blowing (S = minus1)

Sc = 05 10 15

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

C(120578

120578

t)

Figure 12 Effect of Sc on 119862(120578 119905) when 119870119903= 05 and 119905 = 07

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

S = 1 2 3

S = minus1 minus2 minus3

C(120578

120578

t)

Figure 13 Effect of 119878 on 119862(120578 119905) when 119870119903= 05 Sc = 1 and 119905 = 07

results of Seth et al [37] and with the exact results Theimportant findings of the present work may be summarizedas follows

(i) In the case of assisting mass buoyancy force the fluidvelocity is reduced by an increase in themagnetic fieldand the chemical reaction rate

(ii) Blowing causes an acceleration in the fluid velocitywhile suction has reverse effect on it

(iii) An increase in the chemical reaction rate tends toreduce the species concentration whereas the massdiffusivity parameter has reverse effect

(iv) Thermal radiation thermal diffusivity and blowingtend to increase the fluid temperature whereas suc-tion has the opposite effect

(v) In the case of an assisting mass buoyancy force themagnetic field and chemical reaction parameter tendto increase the skin friction while themass diffusivitythermal diffusivity and thermal radiation have thereverse effect

(vi) Thermal diffusivity and blowing tend to reduce therate of heat transfer at the plate whereas thermalradiation and suction have the opposite effect

Acknowledgment

The authors would like to thank the referee for his valuableand constructive suggestions

References

[1] A S Gupta ldquoSteady and transient free convection of anelectrically conducting fluid froma vertical plate in the presenceof a magnetic fieldrdquo Applied Scientific Research vol 9 no 1 pp319ndash333 1960

[2] A S Gupta ldquoLaminar free convection flow of an electricallyconducting fluid from a vertical plate with uniform surface heatflux and variable wall temperature in the presence of amagneticfieldrdquoZeitschrift fur AngewandteMathematik undPhysik vol 13pp 324ndash333 1962

[3] K R Cramer ldquoSeveral magnetohydrodynamic free convectionsolutionsrdquo Journal of Heat Transfer vol 85 pp 35ndash40 1963

[4] I Pop ldquoOn the unsteady hydromagnetic free convection flowpast a vertical infinite flat platerdquo Indian Journal of Physics vol43 pp 196ndash200 1969

[5] H K Kuiken ldquoMagnetohydrodynamic free convection in astrong cross fieldrdquo Journal of Fluid Mechanics vol 40 no 1 pp21ndash38 1970

[6] G Wilks ldquoMagnetohydrodynamic free convection about asemi-infinite vertical plate in a strong cross fieldrdquo Zeitschrift furAngewandte Mathematik und Physik ZAMP vol 27 no 5 pp621ndash631 1976

[7] M A Hossain ldquoEffect of hall current on unsteady hydromag-netic free convection flow near an infinite vertical porous platerdquoJournal of the Physical Society of Japan vol 55 no 7 pp 2183ndash2190 1986

[8] T K Aldoss M A Al-Nimr M A Jarrah and B J Al-SharsquoerldquoMagnetohydrodynamicmixed convection from a vertical plateembedded in a porous mediumrdquo Numerical Heat Transfer PartA vol 28 no 5 pp 635ndash645 1995

[9] K A Helmy ldquoMHD unsteady free convection flow past avertical porous platerdquo Zeitschrift fur Angewandte Mathematikund Mechanik vol 78 no 4 pp 255ndash270 1998

[10] Y J Kim ldquoUnsteadyMHD convective heat transfer past a semi-infinite vertical porous moving plate with variable suctionrdquoInternational Journal of Engineering Science vol 38 no 8 pp833ndash845 2000

[11] H S Takhar S Roy and G Nath ldquoUnsteady free convectionflow over an infinite vertical porous plate due to the combinedeffects of thermal and mass diffusion magnetic field and Hallcurrentsrdquo Heat and Mass Transfer vol 39 no 10 pp 825ndash8342003

[12] N Ahmed H K Sarmah and D Kalita ldquoThermal diffusioneffect on a three-dimensional MHD free convection with masstransfer flow from a porous vertical platerdquo Latin AmericanApplied Research vol 41 pp 165ndash176 2011

12 Mathematical Problems in Engineering

[13] M A Hossain and H S Takhar ldquoRadiation effect on mixedconvection along a vertical plate with uniform surface temper-aturerdquoHeat and Mass Transfer vol 31 no 4 pp 243ndash248 1996

[14] A Y Bakier and R S R Gorla ldquoThermal radiation effect onmixed convection from horizontal surfaces in saturated porousmediardquo Transport in Porous Media vol 23 no 3 pp 357ndash3631996

[15] H S Takhar R S R Gorla and VM Soundalgekar ldquoRadiationeffects onMHD free convection flowof a gas past a semi-infinitevertical platerdquo International Journal of Numerical Methods forHeat and Fluid Flow vol 6 no 2 pp 77ndash83 1996

[16] A J Chamkha ldquoThermal radiation and buoyancy effects onhydromagnetic flow over an accelerating permeable surfacewith heat source or sinkrdquo International Journal of EngineeringScience vol 38 no 15 pp 1699ndash1712 2000

[17] G E-D A Azzam ldquoRadiation effects on the MHDmixed free-forced convective flow past a semi-infinitemoving vertical platefor high temperature differencesrdquo Physica Scripta vol 66 no 1pp 71ndash76 2002

[18] C Israel-Cookey A Ogulu and V B Omubo-Pepple ldquoInflu-ence of viscous dissipation and radiation on unsteady MHDfree-convection flow past an infinite heated vertical plate ina porous medium with time-dependent suctionrdquo InternationalJournal of Heat andMass Transfer vol 46 no 13 pp 2305ndash23112003

[19] M A A Mahmoud ldquoThermal radiation effect on unsteadyMHD free convection flow past a vertical plate with tem-perature-dependent viscosityrdquo Canadian Journal of ChemicalEngineering vol 87 no 1 pp 47ndash52 2009

[20] A R Bestman and S K Adjepong ldquoUnsteady hydromagneticfree-convection flow with radiative heat transfer in a rotatingfluidmdashI Incompressible optically thin fluidrdquo Astrophysics andSpace Science vol 143 no 1 pp 73ndash80 1988

[21] A J Chamkha H S Takhar and V M Soundalgekar ldquoRadia-tion effects on free convection flow past a semi-infinite verticalplate with mass transferrdquo Chemical Engineering Journal vol 84no 3 pp 335ndash342 2001

[22] M A El-Hakiem and M F El-Amin ldquoThermal radiation effecton non-Darcy natural convection with lateral mass transferrdquoHeat and Mass Transfer vol 37 no 2-3 pp 161ndash165 2001

[23] V R Prasad N B Reddy and R Muthucumaraswamy ldquoRadia-tion and mass transfer effects on two-dimensional flow past animpulsively started infinite vertical platerdquo International Journalof Thermal Sciences vol 46 no 12 pp 1251ndash1258 2007

[24] O D Makinde and A Ogulu ldquoThe effect of thermal radiationon the heat and mass transfer flow of a variable viscosity fluidpast a vertical porous plate permeated by a transverse magneticfieldrdquo Chemical Engineering Communications vol 195 no 12pp 1575ndash1584 2008

[25] SMukhopadhyay andG C Layek ldquoEffects of thermal radiationand variable fluid viscosity on free convective flow and heattransfer past a porous stretching surfacerdquo International Journalof Heat andMass Transfer vol 51 no 9-10 pp 2167ndash2178 2008

[26] D Pal ldquoHeat andmass transfer in stagnation-point flow towardsa stretching surface in the presence of buoyancy force andthermal radiationrdquoMeccanica vol 44 no 2 pp 145ndash158 2009

[27] A J Chamkha ldquoMHDflowof a uniformly streched vertical per-meable surface in the presence of heat generationabsorptionand a chemical reactionrdquo International Communications inHeatand Mass Transfer vol 30 no 3 pp 413ndash422 2003

[28] A A Afify ldquoMHD free convective flow and mass transferover a stretching sheet with chemical reactionrdquo Heat and MassTransfer vol 40 no 6-7 pp 495ndash500 2004

[29] A Postelnicu ldquoInfluence of chemical reaction on heat andmasstransfer by natural convection from vertical surfaces in porousmedia considering Soret and Dufour effectsrdquo Heat and MassTransfer vol 43 no 6 pp 595ndash602 2007

[30] R Kandasamy and P G Palanimani ldquoEffects of chemical reac-tions heat and mass transfer on nonlinear magnetohydrody-namic boundary layer flow over a wedge with a porous mediumin the presence of ohmic heating and viscous dissipationrdquoJournal of Porous Media vol 10 no 5 pp 489ndash501 2007

[31] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[32] A A Hayday D A Bowlus and R A McGraw ldquoFree convec-tion from a vertical plate with step discontinuities in surfacetemperaturerdquo ASME Journal of Heat Transfer vol 89 pp 244ndash250 1967

[33] M Kelleher ldquoFree convection from a vertical plate with dis-continuous wall temperaturerdquo Journal of Heat Transfer Trans-actions ASME vol 93 no 4 pp 349ndash356 1971

[34] T-T Kao ldquoLaminar free convective heat transfer response alonga vertical flat plate with step jump in surface temperaturerdquoLetters inHeat andMass Transfer vol 2 no 5 pp 419ndash428 1975

[35] S Lee and M M Yovanovich ldquoLaminar natural convectionfrom a vertical plate with a step change in wall temperaturerdquoJournal of Heat Transfer vol 113 no 2 pp 501ndash504 1991

[36] P Chandran N C Sacheti and A K Singh ldquoNatural convec-tion near a vertical plate with ramped wall temperaturerdquo Heatand Mass Transfer vol 41 no 5 pp 459ndash464 2005

[37] G S Seth M S Ansari and R Nandkeolyar ldquoMHD naturalconvection flow with radiative heat transfer past an impulsivelymoving plate with ramped wall temperaturerdquo Heat and MassTransfer vol 47 no 5 pp 551ndash561 2011

[38] G S Seth R Nandkeolyar and M S Ansari ldquoEffects ofthermal radiation and rotation on unsteady hydromagnetic freeconvection flow past an impulsively moving vertical plate withramped temperature in a porous mediumrdquo Journal of AppliedFluid Mechanics vol 6 no 1 pp 27ndash38 2013

[39] O D Makinde ldquoFree convection flow with thermal radiationand mass transfer past a moving vertical porous platerdquo Interna-tional Communications in Heat and Mass Transfer vol 32 no10 pp 1411ndash1419 2005

[40] I U Mbeledogu and A Ogulu ldquoHeat and mass transfer of anunsteady MHD natural convection flow of a rotating fluid pasta vertical porous flat plate in the presence of radiative heattransferrdquo International Journal of Heat and Mass Transfer vol50 no 9-10 pp 1902ndash1908 2007

[41] A A Afify ldquoSimilarity solution in MHD effects of thermaldiffusion and diffusion thermo on free convective heat andmass transfer over a stretching surface considering suction orinjectionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2202ndash2214 2009

[42] K Cramer and S PaiMagnetofluid Dynamics for Engineers andApplied Physicists McGraw Hill New York NY USA 1973

[43] R C Meyer ldquoOn reducing aerodynamic heat-transfer ratesby magnetohydrodynamic techniquesrdquo Journal of Aero SpaceScience vol 25 pp 561ndash572 1958

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Table 2 Comparison of exact and numerical values of NuRe and ShRe for different values of 119905 and 119878 when119870119903= 05 Sc = 1 Pr = 071 and

119873 = 1

119905 119878

NuRe NuRe ShRe ShReExact Numerical Exact Numerical

02minus1 053388346 0533883 094620854 09462090 060133195 0601332 138566159 13856621 067588346 0675883 194620854 1946209

04minus1 071843936 0718439 064706152 06470620 085041184 0850412 106475703 10647571 100243936 1002439 164706152 1647062

06minus1 084692696 0846927 053359416 05335940 104153758 1041537 093657181 09365721 127292696 1272927 153359416 1533594

08minus1 094692675 0946927 047550301 04755030 120266405 1202664 086753072 08675311 151492675 1514927 147550301 1475503

0 1 2 3 4 50

01

02

03

04

05

06

07

08

Exact solution Numerical solution

S = minus1 0 1

T(120578

120578

t)

(a)

Exact solution Numerical solution

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

S = minus1 0 1

C(120578

120578

t)

(b)

Figure 1 Comparison of numerical and exact solutions of (a) 119879(120578 119905) and (b) 119862(120578 119905) for different values of 119878 when Pr = 071 Sc = 1119873 = 1119870119903= 05 and 119905 = 05

where Re = 1198800119871] is the Reynolds number 119871 is some

characteristic length and

119891 (119905) = (1 + 119873) [(

119898

2radic119899

+ radic119899119905) erfc(radic119899119905119898

)

minus(

119898

2radic119899

+ 119887119905 + radic119899119905) minus radic119898119905

120587

119890minus119899119905119898

]

(20)

The values of skin friction 120591 can be obtained numericallyusing the Matlab INVLAP routine

4 Validation of Numerical Results

In order to validate the numerical results obtained using theMatlab INVLAP routine the values of NuRe are comparedwith the exact values obtained by Seth et al [37] in Table 1Further the numerical values of NuRe and ShRe arecomparedwith the exact analytical values in Table 2Thefluidtemperature and species concentration profiles are plottedusing both the numerical and exact values and are compared

6 Mathematical Problems in Engineering

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

M = 3 5 7

Gm = 1

Suction (S = 1) Blowing (S = minus1)

u(120578t)

120578

(a)

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

M = 3 5 7

Gm = minus1

Suction (S = 1) Blowing (S = minus1)

u(120578t)

120578

(b)

Figure 2 Effect of119872 on 119906(120578 119905) when Sc = 1 119870119903= 05119873 = 1 Pr = 071 and 119905 = 07

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

1

18 2 220

005

01

015

N = 05 10 15

Gm = 1

Suction (S = 1) Blowing (S = minus1)

u(120578t)

u(120578t)

120578

120578

(a)

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

1

N = 05 10 15

Gm = minus1

08 1 12

015

02

025

Suction (S = 1) Blowing (S = minus1)

u(120578t)

u(120578t)

120578

120578

(b)

Figure 3 Effect of119873 on 119906(120578 119905) when Sc = 1 119870119903= 05119872 = 3 Pr = 071 and 119905 = 07

in Figure 1 An excellent agreement between the values isobserved which validates the accuracy of the INVLAP results

5 Results and Discussion

The unsteady hydromagnetic free convective flow of a vis-cous incompressible electrically conducting and heat radi-ating fluid past an infinite vertical flat plate in the presence ofsuctionblowing and chemically reactive species concentra-tion has been studiednumerically using the Matlab INVLAP

routine The fluid velocity 119906(120578 119905) temperature 119879(120578 119905) andspecies concentration 119862(120578 119905) profiles are given in Figures 23 4 5 6 7 8 9 10 and 11 whereas the numerical valuesof the skin friction heat and mass transfer coefficients arepresented in Tables 3 and 4 when 119905 = 07 Gm = 1 (whichcorresponds to assisting buoyancy) and Gm = minus1 (whichcorresponds to opposing buoyancy)

Figures 2 to 7 show the effects of the magnetic fieldthermal radiation suctionblowing chemical reaction massdiffusion and thermal diffusion on the fluid velocity Withassisting mass buoyancy (ie Gm = 1) the fluid velocity

Mathematical Problems in Engineering 7

0 1 2 3 4 5 60

01

02

03

04

05

06

07

08

09

1

S = 1 2 3

S = minus1 minus2 minus3

Gm = 1

u(120578t)

120578

(a)

0 1 2 3 4 5 60

01

02

03

04

05

06

07

08

09

1

S = minus1 minus2 minus3

Gm = minus1

u(120578t)

120578

S = 1 2 3

(b)

Figure 4 Effect of 119878 on 119906(120578 119905) when Sc = 1 119870119903= 05119873 = 1 Pr = 071119872 = 3 and 119905 = 07

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

08 1 12 14 16

015

02

025

03

Gm = 1

u(120578t)

120578

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

Kr = 05 10 15

(a)

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

1

12 14 16

005

01

015

Gm = minus1

u(120578t)

u(120578t)

120578

120578

Suction (S = 1) Blowing (S = minus1)

Kr = 0 10 15

(b)

Figure 5 Effect of119870119903on 119906(120578 119905) when Sc = 1119872 = 3119873 = 1 Pr = 071 and 119905 = 07

decreases with an increase in the magnetic parameter 119872chemical reaction parameter 119870

119903 Schmidt number Sc and

Prandtl number Pr while it increases with the thermalradiation parameter 119873 Since the Schmidt number Sc is theratio of viscous to mass diffusivity an increase in Sc impliesa decrease in the mass diffusion rate Hence the magneticfield and the chemical reaction rate tend to reduce the fluidvelocity whereas the mass diffusivity thermal diffusivity andthermal radiation tend to increase the fluid velocity Withopposing mass buoyancy (ie for Gm = minus1) the behaviourof the fluid velocity with respect to all governing parametersremains the same except with respect to Sc when the fluidvelocity decreases with an increase in mass diffusivity

Figure 4 shows that the fluid velocity decreases with anincrease in suction velocity whereas it increases with anincrease in blowing for both assisting and opposing massbuoyancy This result may be useful in engineering applica-tions where the formation of boundary layer is to be delayedThis can be done by increasing the suction velocity We alsoobserve that the thickness of the momentum boundary layeris less in the presence of opposing mass buoyancy than in thepresence of assisting mass buoyancy

Figures 8 9 and 10 show the effects of thermal radi-ation thermal diffusion and suctionblowing on the fluidtemperature We observe from Figures 8 and 9 that the fluidtemperature increases with119873 while it decreases with Pr The

8 Mathematical Problems in Engineering

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

Sc = 05 10 15

Gm = 1

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

(a)

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

Sc = 05 10 15

Gm = minus1

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

(b)

Figure 6 Effect of Sc on 119906(120578 119905) when119872 = 3 119870119903= 05119873 = 1 Pr = 071 and 119905 = 07

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

1

Pr = 071 1 7

Gm = 1

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

(a)

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

1

Pr = 071 1 7

Gm = minus1

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

(b)

Figure 7 Effect of Pr on 119906(120578 119905) when Sc = 1 119870119903= 05119873 = 1119872 = 3 and 119905 = 07

Prandtl number is the ratio of viscosity to thermal diffusivityand an increase in Pr implies a decrease in thermal diffusivityThermal radiation and diffusivity tend to increase the fluidtemperature Figure 10 shows that an increase in suctionvelocity causes a reduction in the fluid temperature whereasan increase in blowing increases the fluid temperature

Figures 11 12 and 13 show the effects of the chemicalreaction mass diffusion and suction or blowing on thespecies concentration We observe from Figures 11 and 12that the species concentration decreases with an increase inthe chemical reaction parameter or the Schmidt numberThus the chemical reaction rate tends to reduce the species

concentration whereas mass diffusivity increases the speciesconcentration It is clear from Figure 13 that the suctionvelocity reduces the species concentration while blowingcauses an increase in the species concentration We observefrom Figures 11 and 12 that the species concentration is higherin the case of blowing than in the case of suction We alsoobserve from Figures 2ndash11 that the momentum thermal andconcentration boundary layers are thicker in the case ofblowing than in the case of suction

Table 3 shows the effects of the magnetic field thermalradiation suctionblowing chemical reactionmass diffusionand thermal diffusion on the skin friction in the presence

Mathematical Problems in Engineering 9

Table 3 Effect of fluid and physical parameters on the skin friction minus120591 when 119905 = 07

119872 119873 119878 119870119903

Sc Pr minus120591 (119878 gt 0Gm = 1)

minus120591 (119878 lt 0Gm = 1)

minus120591 (119878 gt 0Gm = minus1)

minus120591 (119878 lt 0Gm = minus1)

3 1 mdash 05 1 071 1747428 0788854 2397344 14387705 mdash mdash mdash mdash mdash 2291296 1325411 2865812 18999277 mdash mdash mdash mdash mdash 2737296 1766244 3258082 2287031mdash 05 mdash mdash mdash mdash 1763388 0794076 2413303 1443992mdash 10 mdash mdash mdash mdash 1747428 0788854 2397344 1438770mdash 15 mdash mdash mdash mdash 1735676 0784937 2385592 1434853mdash mdash 1 mdash mdash mdash 1747428 mdash 2397344 mdashmdash mdash 2 mdash mdash mdash 2463830 mdash 3043645 mdashmdash mdash 3 mdash mdash mdash 3291595 mdash 3786658 mdashmdash mdash minus1 mdash mdash mdash mdash 0788854 mdash 1438770mdash mdash minus2 mdash mdash mdash mdash 0540588 mdash 1120403mdash mdash minus3 mdash mdash mdash mdash 0393845 mdash 0888908mdash mdash mdash 05 mdash mdash 1747428 0788854 2397344 1438770mdash mdash mdash 10 mdash mdash 1761672 0803098 2383100 1424526mdash mdash mdash 15 mdash mdash 1774216 0815642 2370555 1411981mdash mdash mdash mdash 05 mdash 1666559 0771682 2478212 1455942mdash mdash mdash mdash 10 mdash 1747428 0788854 2397344 1438770mdash mdash mdash mdash 15 mdash 1796602 0798486 2348170 1429138mdash mdash mdash mdash mdash 071 1747428 0788854 2397344 1438770mdash mdash mdash mdash mdash 10 1766522 0795089 2416438 1445005mdash mdash mdash mdash mdash 70 1884422 0830142 2534338 1480058

0 1 2 3 4 50

01

02

03

04

05

06

07

N = 05 10 15

Suction (S = 1) Blowing (S = minus1)

T(120578

120578

t)

Figure 8 Effect of119873 on 119879(120578 119905) when Pr = 071 and 119905 = 07

of assisting and opposing mass buoyancy In the case ofassisting mass buoyancy an increase in 119872 119870

119903 Sc and Pr

tends to increase the skin friction whereas an increase in119873 reduces the skin friction for both suction and blowingcases Thus the skin friction increases with an increase inthe magnetic field parameter and the chemical reaction rate

0 1 2 3 4 50

01

02

03

04

05

06

07

Pr = 071 1 7

Suction (S = 1) Blowing (S = minus1)

T(120578

120578

t)

Figure 9 Effect of Pr on 119879(120578 119905) when119873 = 1 and 119905 = 07

while it decreaseswith an increase inmass diffusivity thermaldiffusivity and thermal radiation in the case of assisting massbuoyancy However for opposing mass buoyancy the skinfriction increases with mass diffusivity while decreasing withan increase in the rate of the chemical reaction

10 Mathematical Problems in Engineering

Table 4 Values of NuRe and ShRe for different values of governing parameters when 119905 = 07

119873 119878 Pr 119870119903

Sc NuRe NuRe ShRe ShRe(Suction) (Blowing) (Suction) (Blowing)

05 mdash 071 mdash mdash 1249449 0752449 mdash mdash10 mdash 071 mdash mdash 1396642 0899642 mdash mdash15 mdash 071 mdash mdash 1527010 1030010 mdash mdash1 1 071 05 10 1396642 mdash 1500355 mdash1 2 071 05 10 1712939 mdash 2280316 mdash1 3 071 05 10 2069304 mdash 3174187 mdash1 minus1 071 05 10 mdash 0899642 mdash 05003551 minus2 071 05 10 mdash 0718939 mdash 02803161 minus3 071 05 10 mdash 0578304 mdash 01741871 mdash 071 mdash mdash 1396642 0899642 mdash mdash1 mdash 10 mdash mdash 1723721 1023721 mdash mdash1 mdash 70 mdash mdash 6662968 1762968 mdash mdashmdash mdash mdash 05 1 mdash mdash 1500355 0500355mdash mdash mdash 10 1 mdash mdash 1691322 0691322mdash mdash mdash 15 1 mdash mdash 1865496 0865496mdash mdash mdash 05 05 mdash mdash 0921462 0421462mdash mdash mdash 05 10 mdash mdash 1500355 0500355mdash mdash mdash 05 15 mdash mdash 2035801 0535801

0 1 2 3 4 7650

01

02

03

04

05

06

07

S = minus1 minus2 minus3

S = 1 2 3

T(120578

120578

t)

Figure 10 Effect of 119878 on 119879(120578 119905) when Pr = 071119873 = 1 and 119905 = 07

Table 4 shows the effects of thermal radiation suc-tionblowing and thermal diffusion on the heat transfer ratefrom the plate It also shows the effects of suctionblowingchemical reaction rate andmass diffusion on the rate of masstransfer at the plate We observe that the Nusselt numberincreases with 119873 and Pr Thus the thermal radiation tendsto accelerate heat transfer at the plate while thermal diffusionhas reverse effect Table 4 also shows that the rate of heattransfer at the plate increases with an increase in suctionvelocity and decreases with blowing The Sherwood numberincreases with 119870

119903and Sc for both suction and blowing An

increase in the suction velocity causes an increase in therate of mass transfer whereas an increase in blowing has theopposite effect

0 05 1 15 2 25 3 35 4

Suction (S = 1) Blowing (S = minus1)

0

01

02

03

04

05

06

07

08

09

1

C(120578

120578

t)

Kr = 0 05 10

Figure 11 Effect of119872 on 119862(120578 119905) when Sc = 1 and 119905 = 07

6 Conclusions

The unsteady hydromagnetic free convection flow of aviscous incompressible and heat-radiating fluid past animpulsively started vertical porous flat plate with rampedtemperature in the presence of a chemically reactive speciesconcentration was studied For certain parameter valuesexact solutions of the model equations were found usingLaplace transforms The Matlab INVLAP routine was used tofind the numerical results in the general case The numericalresults were found to be in excellent agreement with the

Mathematical Problems in Engineering 11

Suction (S = 1) Blowing (S = minus1)

Sc = 05 10 15

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

C(120578

120578

t)

Figure 12 Effect of Sc on 119862(120578 119905) when 119870119903= 05 and 119905 = 07

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

S = 1 2 3

S = minus1 minus2 minus3

C(120578

120578

t)

Figure 13 Effect of 119878 on 119862(120578 119905) when 119870119903= 05 Sc = 1 and 119905 = 07

results of Seth et al [37] and with the exact results Theimportant findings of the present work may be summarizedas follows

(i) In the case of assisting mass buoyancy force the fluidvelocity is reduced by an increase in themagnetic fieldand the chemical reaction rate

(ii) Blowing causes an acceleration in the fluid velocitywhile suction has reverse effect on it

(iii) An increase in the chemical reaction rate tends toreduce the species concentration whereas the massdiffusivity parameter has reverse effect

(iv) Thermal radiation thermal diffusivity and blowingtend to increase the fluid temperature whereas suc-tion has the opposite effect

(v) In the case of an assisting mass buoyancy force themagnetic field and chemical reaction parameter tendto increase the skin friction while themass diffusivitythermal diffusivity and thermal radiation have thereverse effect

(vi) Thermal diffusivity and blowing tend to reduce therate of heat transfer at the plate whereas thermalradiation and suction have the opposite effect

Acknowledgment

The authors would like to thank the referee for his valuableand constructive suggestions

References

[1] A S Gupta ldquoSteady and transient free convection of anelectrically conducting fluid froma vertical plate in the presenceof a magnetic fieldrdquo Applied Scientific Research vol 9 no 1 pp319ndash333 1960

[2] A S Gupta ldquoLaminar free convection flow of an electricallyconducting fluid from a vertical plate with uniform surface heatflux and variable wall temperature in the presence of amagneticfieldrdquoZeitschrift fur AngewandteMathematik undPhysik vol 13pp 324ndash333 1962

[3] K R Cramer ldquoSeveral magnetohydrodynamic free convectionsolutionsrdquo Journal of Heat Transfer vol 85 pp 35ndash40 1963

[4] I Pop ldquoOn the unsteady hydromagnetic free convection flowpast a vertical infinite flat platerdquo Indian Journal of Physics vol43 pp 196ndash200 1969

[5] H K Kuiken ldquoMagnetohydrodynamic free convection in astrong cross fieldrdquo Journal of Fluid Mechanics vol 40 no 1 pp21ndash38 1970

[6] G Wilks ldquoMagnetohydrodynamic free convection about asemi-infinite vertical plate in a strong cross fieldrdquo Zeitschrift furAngewandte Mathematik und Physik ZAMP vol 27 no 5 pp621ndash631 1976

[7] M A Hossain ldquoEffect of hall current on unsteady hydromag-netic free convection flow near an infinite vertical porous platerdquoJournal of the Physical Society of Japan vol 55 no 7 pp 2183ndash2190 1986

[8] T K Aldoss M A Al-Nimr M A Jarrah and B J Al-SharsquoerldquoMagnetohydrodynamicmixed convection from a vertical plateembedded in a porous mediumrdquo Numerical Heat Transfer PartA vol 28 no 5 pp 635ndash645 1995

[9] K A Helmy ldquoMHD unsteady free convection flow past avertical porous platerdquo Zeitschrift fur Angewandte Mathematikund Mechanik vol 78 no 4 pp 255ndash270 1998

[10] Y J Kim ldquoUnsteadyMHD convective heat transfer past a semi-infinite vertical porous moving plate with variable suctionrdquoInternational Journal of Engineering Science vol 38 no 8 pp833ndash845 2000

[11] H S Takhar S Roy and G Nath ldquoUnsteady free convectionflow over an infinite vertical porous plate due to the combinedeffects of thermal and mass diffusion magnetic field and Hallcurrentsrdquo Heat and Mass Transfer vol 39 no 10 pp 825ndash8342003

[12] N Ahmed H K Sarmah and D Kalita ldquoThermal diffusioneffect on a three-dimensional MHD free convection with masstransfer flow from a porous vertical platerdquo Latin AmericanApplied Research vol 41 pp 165ndash176 2011

12 Mathematical Problems in Engineering

[13] M A Hossain and H S Takhar ldquoRadiation effect on mixedconvection along a vertical plate with uniform surface temper-aturerdquoHeat and Mass Transfer vol 31 no 4 pp 243ndash248 1996

[14] A Y Bakier and R S R Gorla ldquoThermal radiation effect onmixed convection from horizontal surfaces in saturated porousmediardquo Transport in Porous Media vol 23 no 3 pp 357ndash3631996

[15] H S Takhar R S R Gorla and VM Soundalgekar ldquoRadiationeffects onMHD free convection flowof a gas past a semi-infinitevertical platerdquo International Journal of Numerical Methods forHeat and Fluid Flow vol 6 no 2 pp 77ndash83 1996

[16] A J Chamkha ldquoThermal radiation and buoyancy effects onhydromagnetic flow over an accelerating permeable surfacewith heat source or sinkrdquo International Journal of EngineeringScience vol 38 no 15 pp 1699ndash1712 2000

[17] G E-D A Azzam ldquoRadiation effects on the MHDmixed free-forced convective flow past a semi-infinitemoving vertical platefor high temperature differencesrdquo Physica Scripta vol 66 no 1pp 71ndash76 2002

[18] C Israel-Cookey A Ogulu and V B Omubo-Pepple ldquoInflu-ence of viscous dissipation and radiation on unsteady MHDfree-convection flow past an infinite heated vertical plate ina porous medium with time-dependent suctionrdquo InternationalJournal of Heat andMass Transfer vol 46 no 13 pp 2305ndash23112003

[19] M A A Mahmoud ldquoThermal radiation effect on unsteadyMHD free convection flow past a vertical plate with tem-perature-dependent viscosityrdquo Canadian Journal of ChemicalEngineering vol 87 no 1 pp 47ndash52 2009

[20] A R Bestman and S K Adjepong ldquoUnsteady hydromagneticfree-convection flow with radiative heat transfer in a rotatingfluidmdashI Incompressible optically thin fluidrdquo Astrophysics andSpace Science vol 143 no 1 pp 73ndash80 1988

[21] A J Chamkha H S Takhar and V M Soundalgekar ldquoRadia-tion effects on free convection flow past a semi-infinite verticalplate with mass transferrdquo Chemical Engineering Journal vol 84no 3 pp 335ndash342 2001

[22] M A El-Hakiem and M F El-Amin ldquoThermal radiation effecton non-Darcy natural convection with lateral mass transferrdquoHeat and Mass Transfer vol 37 no 2-3 pp 161ndash165 2001

[23] V R Prasad N B Reddy and R Muthucumaraswamy ldquoRadia-tion and mass transfer effects on two-dimensional flow past animpulsively started infinite vertical platerdquo International Journalof Thermal Sciences vol 46 no 12 pp 1251ndash1258 2007

[24] O D Makinde and A Ogulu ldquoThe effect of thermal radiationon the heat and mass transfer flow of a variable viscosity fluidpast a vertical porous plate permeated by a transverse magneticfieldrdquo Chemical Engineering Communications vol 195 no 12pp 1575ndash1584 2008

[25] SMukhopadhyay andG C Layek ldquoEffects of thermal radiationand variable fluid viscosity on free convective flow and heattransfer past a porous stretching surfacerdquo International Journalof Heat andMass Transfer vol 51 no 9-10 pp 2167ndash2178 2008

[26] D Pal ldquoHeat andmass transfer in stagnation-point flow towardsa stretching surface in the presence of buoyancy force andthermal radiationrdquoMeccanica vol 44 no 2 pp 145ndash158 2009

[27] A J Chamkha ldquoMHDflowof a uniformly streched vertical per-meable surface in the presence of heat generationabsorptionand a chemical reactionrdquo International Communications inHeatand Mass Transfer vol 30 no 3 pp 413ndash422 2003

[28] A A Afify ldquoMHD free convective flow and mass transferover a stretching sheet with chemical reactionrdquo Heat and MassTransfer vol 40 no 6-7 pp 495ndash500 2004

[29] A Postelnicu ldquoInfluence of chemical reaction on heat andmasstransfer by natural convection from vertical surfaces in porousmedia considering Soret and Dufour effectsrdquo Heat and MassTransfer vol 43 no 6 pp 595ndash602 2007

[30] R Kandasamy and P G Palanimani ldquoEffects of chemical reac-tions heat and mass transfer on nonlinear magnetohydrody-namic boundary layer flow over a wedge with a porous mediumin the presence of ohmic heating and viscous dissipationrdquoJournal of Porous Media vol 10 no 5 pp 489ndash501 2007

[31] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[32] A A Hayday D A Bowlus and R A McGraw ldquoFree convec-tion from a vertical plate with step discontinuities in surfacetemperaturerdquo ASME Journal of Heat Transfer vol 89 pp 244ndash250 1967

[33] M Kelleher ldquoFree convection from a vertical plate with dis-continuous wall temperaturerdquo Journal of Heat Transfer Trans-actions ASME vol 93 no 4 pp 349ndash356 1971

[34] T-T Kao ldquoLaminar free convective heat transfer response alonga vertical flat plate with step jump in surface temperaturerdquoLetters inHeat andMass Transfer vol 2 no 5 pp 419ndash428 1975

[35] S Lee and M M Yovanovich ldquoLaminar natural convectionfrom a vertical plate with a step change in wall temperaturerdquoJournal of Heat Transfer vol 113 no 2 pp 501ndash504 1991

[36] P Chandran N C Sacheti and A K Singh ldquoNatural convec-tion near a vertical plate with ramped wall temperaturerdquo Heatand Mass Transfer vol 41 no 5 pp 459ndash464 2005

[37] G S Seth M S Ansari and R Nandkeolyar ldquoMHD naturalconvection flow with radiative heat transfer past an impulsivelymoving plate with ramped wall temperaturerdquo Heat and MassTransfer vol 47 no 5 pp 551ndash561 2011

[38] G S Seth R Nandkeolyar and M S Ansari ldquoEffects ofthermal radiation and rotation on unsteady hydromagnetic freeconvection flow past an impulsively moving vertical plate withramped temperature in a porous mediumrdquo Journal of AppliedFluid Mechanics vol 6 no 1 pp 27ndash38 2013

[39] O D Makinde ldquoFree convection flow with thermal radiationand mass transfer past a moving vertical porous platerdquo Interna-tional Communications in Heat and Mass Transfer vol 32 no10 pp 1411ndash1419 2005

[40] I U Mbeledogu and A Ogulu ldquoHeat and mass transfer of anunsteady MHD natural convection flow of a rotating fluid pasta vertical porous flat plate in the presence of radiative heattransferrdquo International Journal of Heat and Mass Transfer vol50 no 9-10 pp 1902ndash1908 2007

[41] A A Afify ldquoSimilarity solution in MHD effects of thermaldiffusion and diffusion thermo on free convective heat andmass transfer over a stretching surface considering suction orinjectionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2202ndash2214 2009

[42] K Cramer and S PaiMagnetofluid Dynamics for Engineers andApplied Physicists McGraw Hill New York NY USA 1973

[43] R C Meyer ldquoOn reducing aerodynamic heat-transfer ratesby magnetohydrodynamic techniquesrdquo Journal of Aero SpaceScience vol 25 pp 561ndash572 1958

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

M = 3 5 7

Gm = 1

Suction (S = 1) Blowing (S = minus1)

u(120578t)

120578

(a)

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

M = 3 5 7

Gm = minus1

Suction (S = 1) Blowing (S = minus1)

u(120578t)

120578

(b)

Figure 2 Effect of119872 on 119906(120578 119905) when Sc = 1 119870119903= 05119873 = 1 Pr = 071 and 119905 = 07

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

1

18 2 220

005

01

015

N = 05 10 15

Gm = 1

Suction (S = 1) Blowing (S = minus1)

u(120578t)

u(120578t)

120578

120578

(a)

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

1

N = 05 10 15

Gm = minus1

08 1 12

015

02

025

Suction (S = 1) Blowing (S = minus1)

u(120578t)

u(120578t)

120578

120578

(b)

Figure 3 Effect of119873 on 119906(120578 119905) when Sc = 1 119870119903= 05119872 = 3 Pr = 071 and 119905 = 07

in Figure 1 An excellent agreement between the values isobserved which validates the accuracy of the INVLAP results

5 Results and Discussion

The unsteady hydromagnetic free convective flow of a vis-cous incompressible electrically conducting and heat radi-ating fluid past an infinite vertical flat plate in the presence ofsuctionblowing and chemically reactive species concentra-tion has been studiednumerically using the Matlab INVLAP

routine The fluid velocity 119906(120578 119905) temperature 119879(120578 119905) andspecies concentration 119862(120578 119905) profiles are given in Figures 23 4 5 6 7 8 9 10 and 11 whereas the numerical valuesof the skin friction heat and mass transfer coefficients arepresented in Tables 3 and 4 when 119905 = 07 Gm = 1 (whichcorresponds to assisting buoyancy) and Gm = minus1 (whichcorresponds to opposing buoyancy)

Figures 2 to 7 show the effects of the magnetic fieldthermal radiation suctionblowing chemical reaction massdiffusion and thermal diffusion on the fluid velocity Withassisting mass buoyancy (ie Gm = 1) the fluid velocity

Mathematical Problems in Engineering 7

0 1 2 3 4 5 60

01

02

03

04

05

06

07

08

09

1

S = 1 2 3

S = minus1 minus2 minus3

Gm = 1

u(120578t)

120578

(a)

0 1 2 3 4 5 60

01

02

03

04

05

06

07

08

09

1

S = minus1 minus2 minus3

Gm = minus1

u(120578t)

120578

S = 1 2 3

(b)

Figure 4 Effect of 119878 on 119906(120578 119905) when Sc = 1 119870119903= 05119873 = 1 Pr = 071119872 = 3 and 119905 = 07

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

08 1 12 14 16

015

02

025

03

Gm = 1

u(120578t)

120578

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

Kr = 05 10 15

(a)

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

1

12 14 16

005

01

015

Gm = minus1

u(120578t)

u(120578t)

120578

120578

Suction (S = 1) Blowing (S = minus1)

Kr = 0 10 15

(b)

Figure 5 Effect of119870119903on 119906(120578 119905) when Sc = 1119872 = 3119873 = 1 Pr = 071 and 119905 = 07

decreases with an increase in the magnetic parameter 119872chemical reaction parameter 119870

119903 Schmidt number Sc and

Prandtl number Pr while it increases with the thermalradiation parameter 119873 Since the Schmidt number Sc is theratio of viscous to mass diffusivity an increase in Sc impliesa decrease in the mass diffusion rate Hence the magneticfield and the chemical reaction rate tend to reduce the fluidvelocity whereas the mass diffusivity thermal diffusivity andthermal radiation tend to increase the fluid velocity Withopposing mass buoyancy (ie for Gm = minus1) the behaviourof the fluid velocity with respect to all governing parametersremains the same except with respect to Sc when the fluidvelocity decreases with an increase in mass diffusivity

Figure 4 shows that the fluid velocity decreases with anincrease in suction velocity whereas it increases with anincrease in blowing for both assisting and opposing massbuoyancy This result may be useful in engineering applica-tions where the formation of boundary layer is to be delayedThis can be done by increasing the suction velocity We alsoobserve that the thickness of the momentum boundary layeris less in the presence of opposing mass buoyancy than in thepresence of assisting mass buoyancy

Figures 8 9 and 10 show the effects of thermal radi-ation thermal diffusion and suctionblowing on the fluidtemperature We observe from Figures 8 and 9 that the fluidtemperature increases with119873 while it decreases with Pr The

8 Mathematical Problems in Engineering

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

Sc = 05 10 15

Gm = 1

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

(a)

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

Sc = 05 10 15

Gm = minus1

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

(b)

Figure 6 Effect of Sc on 119906(120578 119905) when119872 = 3 119870119903= 05119873 = 1 Pr = 071 and 119905 = 07

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

1

Pr = 071 1 7

Gm = 1

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

(a)

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

1

Pr = 071 1 7

Gm = minus1

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

(b)

Figure 7 Effect of Pr on 119906(120578 119905) when Sc = 1 119870119903= 05119873 = 1119872 = 3 and 119905 = 07

Prandtl number is the ratio of viscosity to thermal diffusivityand an increase in Pr implies a decrease in thermal diffusivityThermal radiation and diffusivity tend to increase the fluidtemperature Figure 10 shows that an increase in suctionvelocity causes a reduction in the fluid temperature whereasan increase in blowing increases the fluid temperature

Figures 11 12 and 13 show the effects of the chemicalreaction mass diffusion and suction or blowing on thespecies concentration We observe from Figures 11 and 12that the species concentration decreases with an increase inthe chemical reaction parameter or the Schmidt numberThus the chemical reaction rate tends to reduce the species

concentration whereas mass diffusivity increases the speciesconcentration It is clear from Figure 13 that the suctionvelocity reduces the species concentration while blowingcauses an increase in the species concentration We observefrom Figures 11 and 12 that the species concentration is higherin the case of blowing than in the case of suction We alsoobserve from Figures 2ndash11 that the momentum thermal andconcentration boundary layers are thicker in the case ofblowing than in the case of suction

Table 3 shows the effects of the magnetic field thermalradiation suctionblowing chemical reactionmass diffusionand thermal diffusion on the skin friction in the presence

Mathematical Problems in Engineering 9

Table 3 Effect of fluid and physical parameters on the skin friction minus120591 when 119905 = 07

119872 119873 119878 119870119903

Sc Pr minus120591 (119878 gt 0Gm = 1)

minus120591 (119878 lt 0Gm = 1)

minus120591 (119878 gt 0Gm = minus1)

minus120591 (119878 lt 0Gm = minus1)

3 1 mdash 05 1 071 1747428 0788854 2397344 14387705 mdash mdash mdash mdash mdash 2291296 1325411 2865812 18999277 mdash mdash mdash mdash mdash 2737296 1766244 3258082 2287031mdash 05 mdash mdash mdash mdash 1763388 0794076 2413303 1443992mdash 10 mdash mdash mdash mdash 1747428 0788854 2397344 1438770mdash 15 mdash mdash mdash mdash 1735676 0784937 2385592 1434853mdash mdash 1 mdash mdash mdash 1747428 mdash 2397344 mdashmdash mdash 2 mdash mdash mdash 2463830 mdash 3043645 mdashmdash mdash 3 mdash mdash mdash 3291595 mdash 3786658 mdashmdash mdash minus1 mdash mdash mdash mdash 0788854 mdash 1438770mdash mdash minus2 mdash mdash mdash mdash 0540588 mdash 1120403mdash mdash minus3 mdash mdash mdash mdash 0393845 mdash 0888908mdash mdash mdash 05 mdash mdash 1747428 0788854 2397344 1438770mdash mdash mdash 10 mdash mdash 1761672 0803098 2383100 1424526mdash mdash mdash 15 mdash mdash 1774216 0815642 2370555 1411981mdash mdash mdash mdash 05 mdash 1666559 0771682 2478212 1455942mdash mdash mdash mdash 10 mdash 1747428 0788854 2397344 1438770mdash mdash mdash mdash 15 mdash 1796602 0798486 2348170 1429138mdash mdash mdash mdash mdash 071 1747428 0788854 2397344 1438770mdash mdash mdash mdash mdash 10 1766522 0795089 2416438 1445005mdash mdash mdash mdash mdash 70 1884422 0830142 2534338 1480058

0 1 2 3 4 50

01

02

03

04

05

06

07

N = 05 10 15

Suction (S = 1) Blowing (S = minus1)

T(120578

120578

t)

Figure 8 Effect of119873 on 119879(120578 119905) when Pr = 071 and 119905 = 07

of assisting and opposing mass buoyancy In the case ofassisting mass buoyancy an increase in 119872 119870

119903 Sc and Pr

tends to increase the skin friction whereas an increase in119873 reduces the skin friction for both suction and blowingcases Thus the skin friction increases with an increase inthe magnetic field parameter and the chemical reaction rate

0 1 2 3 4 50

01

02

03

04

05

06

07

Pr = 071 1 7

Suction (S = 1) Blowing (S = minus1)

T(120578

120578

t)

Figure 9 Effect of Pr on 119879(120578 119905) when119873 = 1 and 119905 = 07

while it decreaseswith an increase inmass diffusivity thermaldiffusivity and thermal radiation in the case of assisting massbuoyancy However for opposing mass buoyancy the skinfriction increases with mass diffusivity while decreasing withan increase in the rate of the chemical reaction

10 Mathematical Problems in Engineering

Table 4 Values of NuRe and ShRe for different values of governing parameters when 119905 = 07

119873 119878 Pr 119870119903

Sc NuRe NuRe ShRe ShRe(Suction) (Blowing) (Suction) (Blowing)

05 mdash 071 mdash mdash 1249449 0752449 mdash mdash10 mdash 071 mdash mdash 1396642 0899642 mdash mdash15 mdash 071 mdash mdash 1527010 1030010 mdash mdash1 1 071 05 10 1396642 mdash 1500355 mdash1 2 071 05 10 1712939 mdash 2280316 mdash1 3 071 05 10 2069304 mdash 3174187 mdash1 minus1 071 05 10 mdash 0899642 mdash 05003551 minus2 071 05 10 mdash 0718939 mdash 02803161 minus3 071 05 10 mdash 0578304 mdash 01741871 mdash 071 mdash mdash 1396642 0899642 mdash mdash1 mdash 10 mdash mdash 1723721 1023721 mdash mdash1 mdash 70 mdash mdash 6662968 1762968 mdash mdashmdash mdash mdash 05 1 mdash mdash 1500355 0500355mdash mdash mdash 10 1 mdash mdash 1691322 0691322mdash mdash mdash 15 1 mdash mdash 1865496 0865496mdash mdash mdash 05 05 mdash mdash 0921462 0421462mdash mdash mdash 05 10 mdash mdash 1500355 0500355mdash mdash mdash 05 15 mdash mdash 2035801 0535801

0 1 2 3 4 7650

01

02

03

04

05

06

07

S = minus1 minus2 minus3

S = 1 2 3

T(120578

120578

t)

Figure 10 Effect of 119878 on 119879(120578 119905) when Pr = 071119873 = 1 and 119905 = 07

Table 4 shows the effects of thermal radiation suc-tionblowing and thermal diffusion on the heat transfer ratefrom the plate It also shows the effects of suctionblowingchemical reaction rate andmass diffusion on the rate of masstransfer at the plate We observe that the Nusselt numberincreases with 119873 and Pr Thus the thermal radiation tendsto accelerate heat transfer at the plate while thermal diffusionhas reverse effect Table 4 also shows that the rate of heattransfer at the plate increases with an increase in suctionvelocity and decreases with blowing The Sherwood numberincreases with 119870

119903and Sc for both suction and blowing An

increase in the suction velocity causes an increase in therate of mass transfer whereas an increase in blowing has theopposite effect

0 05 1 15 2 25 3 35 4

Suction (S = 1) Blowing (S = minus1)

0

01

02

03

04

05

06

07

08

09

1

C(120578

120578

t)

Kr = 0 05 10

Figure 11 Effect of119872 on 119862(120578 119905) when Sc = 1 and 119905 = 07

6 Conclusions

The unsteady hydromagnetic free convection flow of aviscous incompressible and heat-radiating fluid past animpulsively started vertical porous flat plate with rampedtemperature in the presence of a chemically reactive speciesconcentration was studied For certain parameter valuesexact solutions of the model equations were found usingLaplace transforms The Matlab INVLAP routine was used tofind the numerical results in the general case The numericalresults were found to be in excellent agreement with the

Mathematical Problems in Engineering 11

Suction (S = 1) Blowing (S = minus1)

Sc = 05 10 15

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

C(120578

120578

t)

Figure 12 Effect of Sc on 119862(120578 119905) when 119870119903= 05 and 119905 = 07

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

S = 1 2 3

S = minus1 minus2 minus3

C(120578

120578

t)

Figure 13 Effect of 119878 on 119862(120578 119905) when 119870119903= 05 Sc = 1 and 119905 = 07

results of Seth et al [37] and with the exact results Theimportant findings of the present work may be summarizedas follows

(i) In the case of assisting mass buoyancy force the fluidvelocity is reduced by an increase in themagnetic fieldand the chemical reaction rate

(ii) Blowing causes an acceleration in the fluid velocitywhile suction has reverse effect on it

(iii) An increase in the chemical reaction rate tends toreduce the species concentration whereas the massdiffusivity parameter has reverse effect

(iv) Thermal radiation thermal diffusivity and blowingtend to increase the fluid temperature whereas suc-tion has the opposite effect

(v) In the case of an assisting mass buoyancy force themagnetic field and chemical reaction parameter tendto increase the skin friction while themass diffusivitythermal diffusivity and thermal radiation have thereverse effect

(vi) Thermal diffusivity and blowing tend to reduce therate of heat transfer at the plate whereas thermalradiation and suction have the opposite effect

Acknowledgment

The authors would like to thank the referee for his valuableand constructive suggestions

References

[1] A S Gupta ldquoSteady and transient free convection of anelectrically conducting fluid froma vertical plate in the presenceof a magnetic fieldrdquo Applied Scientific Research vol 9 no 1 pp319ndash333 1960

[2] A S Gupta ldquoLaminar free convection flow of an electricallyconducting fluid from a vertical plate with uniform surface heatflux and variable wall temperature in the presence of amagneticfieldrdquoZeitschrift fur AngewandteMathematik undPhysik vol 13pp 324ndash333 1962

[3] K R Cramer ldquoSeveral magnetohydrodynamic free convectionsolutionsrdquo Journal of Heat Transfer vol 85 pp 35ndash40 1963

[4] I Pop ldquoOn the unsteady hydromagnetic free convection flowpast a vertical infinite flat platerdquo Indian Journal of Physics vol43 pp 196ndash200 1969

[5] H K Kuiken ldquoMagnetohydrodynamic free convection in astrong cross fieldrdquo Journal of Fluid Mechanics vol 40 no 1 pp21ndash38 1970

[6] G Wilks ldquoMagnetohydrodynamic free convection about asemi-infinite vertical plate in a strong cross fieldrdquo Zeitschrift furAngewandte Mathematik und Physik ZAMP vol 27 no 5 pp621ndash631 1976

[7] M A Hossain ldquoEffect of hall current on unsteady hydromag-netic free convection flow near an infinite vertical porous platerdquoJournal of the Physical Society of Japan vol 55 no 7 pp 2183ndash2190 1986

[8] T K Aldoss M A Al-Nimr M A Jarrah and B J Al-SharsquoerldquoMagnetohydrodynamicmixed convection from a vertical plateembedded in a porous mediumrdquo Numerical Heat Transfer PartA vol 28 no 5 pp 635ndash645 1995

[9] K A Helmy ldquoMHD unsteady free convection flow past avertical porous platerdquo Zeitschrift fur Angewandte Mathematikund Mechanik vol 78 no 4 pp 255ndash270 1998

[10] Y J Kim ldquoUnsteadyMHD convective heat transfer past a semi-infinite vertical porous moving plate with variable suctionrdquoInternational Journal of Engineering Science vol 38 no 8 pp833ndash845 2000

[11] H S Takhar S Roy and G Nath ldquoUnsteady free convectionflow over an infinite vertical porous plate due to the combinedeffects of thermal and mass diffusion magnetic field and Hallcurrentsrdquo Heat and Mass Transfer vol 39 no 10 pp 825ndash8342003

[12] N Ahmed H K Sarmah and D Kalita ldquoThermal diffusioneffect on a three-dimensional MHD free convection with masstransfer flow from a porous vertical platerdquo Latin AmericanApplied Research vol 41 pp 165ndash176 2011

12 Mathematical Problems in Engineering

[13] M A Hossain and H S Takhar ldquoRadiation effect on mixedconvection along a vertical plate with uniform surface temper-aturerdquoHeat and Mass Transfer vol 31 no 4 pp 243ndash248 1996

[14] A Y Bakier and R S R Gorla ldquoThermal radiation effect onmixed convection from horizontal surfaces in saturated porousmediardquo Transport in Porous Media vol 23 no 3 pp 357ndash3631996

[15] H S Takhar R S R Gorla and VM Soundalgekar ldquoRadiationeffects onMHD free convection flowof a gas past a semi-infinitevertical platerdquo International Journal of Numerical Methods forHeat and Fluid Flow vol 6 no 2 pp 77ndash83 1996

[16] A J Chamkha ldquoThermal radiation and buoyancy effects onhydromagnetic flow over an accelerating permeable surfacewith heat source or sinkrdquo International Journal of EngineeringScience vol 38 no 15 pp 1699ndash1712 2000

[17] G E-D A Azzam ldquoRadiation effects on the MHDmixed free-forced convective flow past a semi-infinitemoving vertical platefor high temperature differencesrdquo Physica Scripta vol 66 no 1pp 71ndash76 2002

[18] C Israel-Cookey A Ogulu and V B Omubo-Pepple ldquoInflu-ence of viscous dissipation and radiation on unsteady MHDfree-convection flow past an infinite heated vertical plate ina porous medium with time-dependent suctionrdquo InternationalJournal of Heat andMass Transfer vol 46 no 13 pp 2305ndash23112003

[19] M A A Mahmoud ldquoThermal radiation effect on unsteadyMHD free convection flow past a vertical plate with tem-perature-dependent viscosityrdquo Canadian Journal of ChemicalEngineering vol 87 no 1 pp 47ndash52 2009

[20] A R Bestman and S K Adjepong ldquoUnsteady hydromagneticfree-convection flow with radiative heat transfer in a rotatingfluidmdashI Incompressible optically thin fluidrdquo Astrophysics andSpace Science vol 143 no 1 pp 73ndash80 1988

[21] A J Chamkha H S Takhar and V M Soundalgekar ldquoRadia-tion effects on free convection flow past a semi-infinite verticalplate with mass transferrdquo Chemical Engineering Journal vol 84no 3 pp 335ndash342 2001

[22] M A El-Hakiem and M F El-Amin ldquoThermal radiation effecton non-Darcy natural convection with lateral mass transferrdquoHeat and Mass Transfer vol 37 no 2-3 pp 161ndash165 2001

[23] V R Prasad N B Reddy and R Muthucumaraswamy ldquoRadia-tion and mass transfer effects on two-dimensional flow past animpulsively started infinite vertical platerdquo International Journalof Thermal Sciences vol 46 no 12 pp 1251ndash1258 2007

[24] O D Makinde and A Ogulu ldquoThe effect of thermal radiationon the heat and mass transfer flow of a variable viscosity fluidpast a vertical porous plate permeated by a transverse magneticfieldrdquo Chemical Engineering Communications vol 195 no 12pp 1575ndash1584 2008

[25] SMukhopadhyay andG C Layek ldquoEffects of thermal radiationand variable fluid viscosity on free convective flow and heattransfer past a porous stretching surfacerdquo International Journalof Heat andMass Transfer vol 51 no 9-10 pp 2167ndash2178 2008

[26] D Pal ldquoHeat andmass transfer in stagnation-point flow towardsa stretching surface in the presence of buoyancy force andthermal radiationrdquoMeccanica vol 44 no 2 pp 145ndash158 2009

[27] A J Chamkha ldquoMHDflowof a uniformly streched vertical per-meable surface in the presence of heat generationabsorptionand a chemical reactionrdquo International Communications inHeatand Mass Transfer vol 30 no 3 pp 413ndash422 2003

[28] A A Afify ldquoMHD free convective flow and mass transferover a stretching sheet with chemical reactionrdquo Heat and MassTransfer vol 40 no 6-7 pp 495ndash500 2004

[29] A Postelnicu ldquoInfluence of chemical reaction on heat andmasstransfer by natural convection from vertical surfaces in porousmedia considering Soret and Dufour effectsrdquo Heat and MassTransfer vol 43 no 6 pp 595ndash602 2007

[30] R Kandasamy and P G Palanimani ldquoEffects of chemical reac-tions heat and mass transfer on nonlinear magnetohydrody-namic boundary layer flow over a wedge with a porous mediumin the presence of ohmic heating and viscous dissipationrdquoJournal of Porous Media vol 10 no 5 pp 489ndash501 2007

[31] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[32] A A Hayday D A Bowlus and R A McGraw ldquoFree convec-tion from a vertical plate with step discontinuities in surfacetemperaturerdquo ASME Journal of Heat Transfer vol 89 pp 244ndash250 1967

[33] M Kelleher ldquoFree convection from a vertical plate with dis-continuous wall temperaturerdquo Journal of Heat Transfer Trans-actions ASME vol 93 no 4 pp 349ndash356 1971

[34] T-T Kao ldquoLaminar free convective heat transfer response alonga vertical flat plate with step jump in surface temperaturerdquoLetters inHeat andMass Transfer vol 2 no 5 pp 419ndash428 1975

[35] S Lee and M M Yovanovich ldquoLaminar natural convectionfrom a vertical plate with a step change in wall temperaturerdquoJournal of Heat Transfer vol 113 no 2 pp 501ndash504 1991

[36] P Chandran N C Sacheti and A K Singh ldquoNatural convec-tion near a vertical plate with ramped wall temperaturerdquo Heatand Mass Transfer vol 41 no 5 pp 459ndash464 2005

[37] G S Seth M S Ansari and R Nandkeolyar ldquoMHD naturalconvection flow with radiative heat transfer past an impulsivelymoving plate with ramped wall temperaturerdquo Heat and MassTransfer vol 47 no 5 pp 551ndash561 2011

[38] G S Seth R Nandkeolyar and M S Ansari ldquoEffects ofthermal radiation and rotation on unsteady hydromagnetic freeconvection flow past an impulsively moving vertical plate withramped temperature in a porous mediumrdquo Journal of AppliedFluid Mechanics vol 6 no 1 pp 27ndash38 2013

[39] O D Makinde ldquoFree convection flow with thermal radiationand mass transfer past a moving vertical porous platerdquo Interna-tional Communications in Heat and Mass Transfer vol 32 no10 pp 1411ndash1419 2005

[40] I U Mbeledogu and A Ogulu ldquoHeat and mass transfer of anunsteady MHD natural convection flow of a rotating fluid pasta vertical porous flat plate in the presence of radiative heattransferrdquo International Journal of Heat and Mass Transfer vol50 no 9-10 pp 1902ndash1908 2007

[41] A A Afify ldquoSimilarity solution in MHD effects of thermaldiffusion and diffusion thermo on free convective heat andmass transfer over a stretching surface considering suction orinjectionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2202ndash2214 2009

[42] K Cramer and S PaiMagnetofluid Dynamics for Engineers andApplied Physicists McGraw Hill New York NY USA 1973

[43] R C Meyer ldquoOn reducing aerodynamic heat-transfer ratesby magnetohydrodynamic techniquesrdquo Journal of Aero SpaceScience vol 25 pp 561ndash572 1958

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

0 1 2 3 4 5 60

01

02

03

04

05

06

07

08

09

1

S = 1 2 3

S = minus1 minus2 minus3

Gm = 1

u(120578t)

120578

(a)

0 1 2 3 4 5 60

01

02

03

04

05

06

07

08

09

1

S = minus1 minus2 minus3

Gm = minus1

u(120578t)

120578

S = 1 2 3

(b)

Figure 4 Effect of 119878 on 119906(120578 119905) when Sc = 1 119870119903= 05119873 = 1 Pr = 071119872 = 3 and 119905 = 07

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

08 1 12 14 16

015

02

025

03

Gm = 1

u(120578t)

120578

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

Kr = 05 10 15

(a)

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

1

12 14 16

005

01

015

Gm = minus1

u(120578t)

u(120578t)

120578

120578

Suction (S = 1) Blowing (S = minus1)

Kr = 0 10 15

(b)

Figure 5 Effect of119870119903on 119906(120578 119905) when Sc = 1119872 = 3119873 = 1 Pr = 071 and 119905 = 07

decreases with an increase in the magnetic parameter 119872chemical reaction parameter 119870

119903 Schmidt number Sc and

Prandtl number Pr while it increases with the thermalradiation parameter 119873 Since the Schmidt number Sc is theratio of viscous to mass diffusivity an increase in Sc impliesa decrease in the mass diffusion rate Hence the magneticfield and the chemical reaction rate tend to reduce the fluidvelocity whereas the mass diffusivity thermal diffusivity andthermal radiation tend to increase the fluid velocity Withopposing mass buoyancy (ie for Gm = minus1) the behaviourof the fluid velocity with respect to all governing parametersremains the same except with respect to Sc when the fluidvelocity decreases with an increase in mass diffusivity

Figure 4 shows that the fluid velocity decreases with anincrease in suction velocity whereas it increases with anincrease in blowing for both assisting and opposing massbuoyancy This result may be useful in engineering applica-tions where the formation of boundary layer is to be delayedThis can be done by increasing the suction velocity We alsoobserve that the thickness of the momentum boundary layeris less in the presence of opposing mass buoyancy than in thepresence of assisting mass buoyancy

Figures 8 9 and 10 show the effects of thermal radi-ation thermal diffusion and suctionblowing on the fluidtemperature We observe from Figures 8 and 9 that the fluidtemperature increases with119873 while it decreases with Pr The

8 Mathematical Problems in Engineering

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

Sc = 05 10 15

Gm = 1

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

(a)

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

Sc = 05 10 15

Gm = minus1

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

(b)

Figure 6 Effect of Sc on 119906(120578 119905) when119872 = 3 119870119903= 05119873 = 1 Pr = 071 and 119905 = 07

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

1

Pr = 071 1 7

Gm = 1

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

(a)

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

1

Pr = 071 1 7

Gm = minus1

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

(b)

Figure 7 Effect of Pr on 119906(120578 119905) when Sc = 1 119870119903= 05119873 = 1119872 = 3 and 119905 = 07

Prandtl number is the ratio of viscosity to thermal diffusivityand an increase in Pr implies a decrease in thermal diffusivityThermal radiation and diffusivity tend to increase the fluidtemperature Figure 10 shows that an increase in suctionvelocity causes a reduction in the fluid temperature whereasan increase in blowing increases the fluid temperature

Figures 11 12 and 13 show the effects of the chemicalreaction mass diffusion and suction or blowing on thespecies concentration We observe from Figures 11 and 12that the species concentration decreases with an increase inthe chemical reaction parameter or the Schmidt numberThus the chemical reaction rate tends to reduce the species

concentration whereas mass diffusivity increases the speciesconcentration It is clear from Figure 13 that the suctionvelocity reduces the species concentration while blowingcauses an increase in the species concentration We observefrom Figures 11 and 12 that the species concentration is higherin the case of blowing than in the case of suction We alsoobserve from Figures 2ndash11 that the momentum thermal andconcentration boundary layers are thicker in the case ofblowing than in the case of suction

Table 3 shows the effects of the magnetic field thermalradiation suctionblowing chemical reactionmass diffusionand thermal diffusion on the skin friction in the presence

Mathematical Problems in Engineering 9

Table 3 Effect of fluid and physical parameters on the skin friction minus120591 when 119905 = 07

119872 119873 119878 119870119903

Sc Pr minus120591 (119878 gt 0Gm = 1)

minus120591 (119878 lt 0Gm = 1)

minus120591 (119878 gt 0Gm = minus1)

minus120591 (119878 lt 0Gm = minus1)

3 1 mdash 05 1 071 1747428 0788854 2397344 14387705 mdash mdash mdash mdash mdash 2291296 1325411 2865812 18999277 mdash mdash mdash mdash mdash 2737296 1766244 3258082 2287031mdash 05 mdash mdash mdash mdash 1763388 0794076 2413303 1443992mdash 10 mdash mdash mdash mdash 1747428 0788854 2397344 1438770mdash 15 mdash mdash mdash mdash 1735676 0784937 2385592 1434853mdash mdash 1 mdash mdash mdash 1747428 mdash 2397344 mdashmdash mdash 2 mdash mdash mdash 2463830 mdash 3043645 mdashmdash mdash 3 mdash mdash mdash 3291595 mdash 3786658 mdashmdash mdash minus1 mdash mdash mdash mdash 0788854 mdash 1438770mdash mdash minus2 mdash mdash mdash mdash 0540588 mdash 1120403mdash mdash minus3 mdash mdash mdash mdash 0393845 mdash 0888908mdash mdash mdash 05 mdash mdash 1747428 0788854 2397344 1438770mdash mdash mdash 10 mdash mdash 1761672 0803098 2383100 1424526mdash mdash mdash 15 mdash mdash 1774216 0815642 2370555 1411981mdash mdash mdash mdash 05 mdash 1666559 0771682 2478212 1455942mdash mdash mdash mdash 10 mdash 1747428 0788854 2397344 1438770mdash mdash mdash mdash 15 mdash 1796602 0798486 2348170 1429138mdash mdash mdash mdash mdash 071 1747428 0788854 2397344 1438770mdash mdash mdash mdash mdash 10 1766522 0795089 2416438 1445005mdash mdash mdash mdash mdash 70 1884422 0830142 2534338 1480058

0 1 2 3 4 50

01

02

03

04

05

06

07

N = 05 10 15

Suction (S = 1) Blowing (S = minus1)

T(120578

120578

t)

Figure 8 Effect of119873 on 119879(120578 119905) when Pr = 071 and 119905 = 07

of assisting and opposing mass buoyancy In the case ofassisting mass buoyancy an increase in 119872 119870

119903 Sc and Pr

tends to increase the skin friction whereas an increase in119873 reduces the skin friction for both suction and blowingcases Thus the skin friction increases with an increase inthe magnetic field parameter and the chemical reaction rate

0 1 2 3 4 50

01

02

03

04

05

06

07

Pr = 071 1 7

Suction (S = 1) Blowing (S = minus1)

T(120578

120578

t)

Figure 9 Effect of Pr on 119879(120578 119905) when119873 = 1 and 119905 = 07

while it decreaseswith an increase inmass diffusivity thermaldiffusivity and thermal radiation in the case of assisting massbuoyancy However for opposing mass buoyancy the skinfriction increases with mass diffusivity while decreasing withan increase in the rate of the chemical reaction

10 Mathematical Problems in Engineering

Table 4 Values of NuRe and ShRe for different values of governing parameters when 119905 = 07

119873 119878 Pr 119870119903

Sc NuRe NuRe ShRe ShRe(Suction) (Blowing) (Suction) (Blowing)

05 mdash 071 mdash mdash 1249449 0752449 mdash mdash10 mdash 071 mdash mdash 1396642 0899642 mdash mdash15 mdash 071 mdash mdash 1527010 1030010 mdash mdash1 1 071 05 10 1396642 mdash 1500355 mdash1 2 071 05 10 1712939 mdash 2280316 mdash1 3 071 05 10 2069304 mdash 3174187 mdash1 minus1 071 05 10 mdash 0899642 mdash 05003551 minus2 071 05 10 mdash 0718939 mdash 02803161 minus3 071 05 10 mdash 0578304 mdash 01741871 mdash 071 mdash mdash 1396642 0899642 mdash mdash1 mdash 10 mdash mdash 1723721 1023721 mdash mdash1 mdash 70 mdash mdash 6662968 1762968 mdash mdashmdash mdash mdash 05 1 mdash mdash 1500355 0500355mdash mdash mdash 10 1 mdash mdash 1691322 0691322mdash mdash mdash 15 1 mdash mdash 1865496 0865496mdash mdash mdash 05 05 mdash mdash 0921462 0421462mdash mdash mdash 05 10 mdash mdash 1500355 0500355mdash mdash mdash 05 15 mdash mdash 2035801 0535801

0 1 2 3 4 7650

01

02

03

04

05

06

07

S = minus1 minus2 minus3

S = 1 2 3

T(120578

120578

t)

Figure 10 Effect of 119878 on 119879(120578 119905) when Pr = 071119873 = 1 and 119905 = 07

Table 4 shows the effects of thermal radiation suc-tionblowing and thermal diffusion on the heat transfer ratefrom the plate It also shows the effects of suctionblowingchemical reaction rate andmass diffusion on the rate of masstransfer at the plate We observe that the Nusselt numberincreases with 119873 and Pr Thus the thermal radiation tendsto accelerate heat transfer at the plate while thermal diffusionhas reverse effect Table 4 also shows that the rate of heattransfer at the plate increases with an increase in suctionvelocity and decreases with blowing The Sherwood numberincreases with 119870

119903and Sc for both suction and blowing An

increase in the suction velocity causes an increase in therate of mass transfer whereas an increase in blowing has theopposite effect

0 05 1 15 2 25 3 35 4

Suction (S = 1) Blowing (S = minus1)

0

01

02

03

04

05

06

07

08

09

1

C(120578

120578

t)

Kr = 0 05 10

Figure 11 Effect of119872 on 119862(120578 119905) when Sc = 1 and 119905 = 07

6 Conclusions

The unsteady hydromagnetic free convection flow of aviscous incompressible and heat-radiating fluid past animpulsively started vertical porous flat plate with rampedtemperature in the presence of a chemically reactive speciesconcentration was studied For certain parameter valuesexact solutions of the model equations were found usingLaplace transforms The Matlab INVLAP routine was used tofind the numerical results in the general case The numericalresults were found to be in excellent agreement with the

Mathematical Problems in Engineering 11

Suction (S = 1) Blowing (S = minus1)

Sc = 05 10 15

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

C(120578

120578

t)

Figure 12 Effect of Sc on 119862(120578 119905) when 119870119903= 05 and 119905 = 07

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

S = 1 2 3

S = minus1 minus2 minus3

C(120578

120578

t)

Figure 13 Effect of 119878 on 119862(120578 119905) when 119870119903= 05 Sc = 1 and 119905 = 07

results of Seth et al [37] and with the exact results Theimportant findings of the present work may be summarizedas follows

(i) In the case of assisting mass buoyancy force the fluidvelocity is reduced by an increase in themagnetic fieldand the chemical reaction rate

(ii) Blowing causes an acceleration in the fluid velocitywhile suction has reverse effect on it

(iii) An increase in the chemical reaction rate tends toreduce the species concentration whereas the massdiffusivity parameter has reverse effect

(iv) Thermal radiation thermal diffusivity and blowingtend to increase the fluid temperature whereas suc-tion has the opposite effect

(v) In the case of an assisting mass buoyancy force themagnetic field and chemical reaction parameter tendto increase the skin friction while themass diffusivitythermal diffusivity and thermal radiation have thereverse effect

(vi) Thermal diffusivity and blowing tend to reduce therate of heat transfer at the plate whereas thermalradiation and suction have the opposite effect

Acknowledgment

The authors would like to thank the referee for his valuableand constructive suggestions

References

[1] A S Gupta ldquoSteady and transient free convection of anelectrically conducting fluid froma vertical plate in the presenceof a magnetic fieldrdquo Applied Scientific Research vol 9 no 1 pp319ndash333 1960

[2] A S Gupta ldquoLaminar free convection flow of an electricallyconducting fluid from a vertical plate with uniform surface heatflux and variable wall temperature in the presence of amagneticfieldrdquoZeitschrift fur AngewandteMathematik undPhysik vol 13pp 324ndash333 1962

[3] K R Cramer ldquoSeveral magnetohydrodynamic free convectionsolutionsrdquo Journal of Heat Transfer vol 85 pp 35ndash40 1963

[4] I Pop ldquoOn the unsteady hydromagnetic free convection flowpast a vertical infinite flat platerdquo Indian Journal of Physics vol43 pp 196ndash200 1969

[5] H K Kuiken ldquoMagnetohydrodynamic free convection in astrong cross fieldrdquo Journal of Fluid Mechanics vol 40 no 1 pp21ndash38 1970

[6] G Wilks ldquoMagnetohydrodynamic free convection about asemi-infinite vertical plate in a strong cross fieldrdquo Zeitschrift furAngewandte Mathematik und Physik ZAMP vol 27 no 5 pp621ndash631 1976

[7] M A Hossain ldquoEffect of hall current on unsteady hydromag-netic free convection flow near an infinite vertical porous platerdquoJournal of the Physical Society of Japan vol 55 no 7 pp 2183ndash2190 1986

[8] T K Aldoss M A Al-Nimr M A Jarrah and B J Al-SharsquoerldquoMagnetohydrodynamicmixed convection from a vertical plateembedded in a porous mediumrdquo Numerical Heat Transfer PartA vol 28 no 5 pp 635ndash645 1995

[9] K A Helmy ldquoMHD unsteady free convection flow past avertical porous platerdquo Zeitschrift fur Angewandte Mathematikund Mechanik vol 78 no 4 pp 255ndash270 1998

[10] Y J Kim ldquoUnsteadyMHD convective heat transfer past a semi-infinite vertical porous moving plate with variable suctionrdquoInternational Journal of Engineering Science vol 38 no 8 pp833ndash845 2000

[11] H S Takhar S Roy and G Nath ldquoUnsteady free convectionflow over an infinite vertical porous plate due to the combinedeffects of thermal and mass diffusion magnetic field and Hallcurrentsrdquo Heat and Mass Transfer vol 39 no 10 pp 825ndash8342003

[12] N Ahmed H K Sarmah and D Kalita ldquoThermal diffusioneffect on a three-dimensional MHD free convection with masstransfer flow from a porous vertical platerdquo Latin AmericanApplied Research vol 41 pp 165ndash176 2011

12 Mathematical Problems in Engineering

[13] M A Hossain and H S Takhar ldquoRadiation effect on mixedconvection along a vertical plate with uniform surface temper-aturerdquoHeat and Mass Transfer vol 31 no 4 pp 243ndash248 1996

[14] A Y Bakier and R S R Gorla ldquoThermal radiation effect onmixed convection from horizontal surfaces in saturated porousmediardquo Transport in Porous Media vol 23 no 3 pp 357ndash3631996

[15] H S Takhar R S R Gorla and VM Soundalgekar ldquoRadiationeffects onMHD free convection flowof a gas past a semi-infinitevertical platerdquo International Journal of Numerical Methods forHeat and Fluid Flow vol 6 no 2 pp 77ndash83 1996

[16] A J Chamkha ldquoThermal radiation and buoyancy effects onhydromagnetic flow over an accelerating permeable surfacewith heat source or sinkrdquo International Journal of EngineeringScience vol 38 no 15 pp 1699ndash1712 2000

[17] G E-D A Azzam ldquoRadiation effects on the MHDmixed free-forced convective flow past a semi-infinitemoving vertical platefor high temperature differencesrdquo Physica Scripta vol 66 no 1pp 71ndash76 2002

[18] C Israel-Cookey A Ogulu and V B Omubo-Pepple ldquoInflu-ence of viscous dissipation and radiation on unsteady MHDfree-convection flow past an infinite heated vertical plate ina porous medium with time-dependent suctionrdquo InternationalJournal of Heat andMass Transfer vol 46 no 13 pp 2305ndash23112003

[19] M A A Mahmoud ldquoThermal radiation effect on unsteadyMHD free convection flow past a vertical plate with tem-perature-dependent viscosityrdquo Canadian Journal of ChemicalEngineering vol 87 no 1 pp 47ndash52 2009

[20] A R Bestman and S K Adjepong ldquoUnsteady hydromagneticfree-convection flow with radiative heat transfer in a rotatingfluidmdashI Incompressible optically thin fluidrdquo Astrophysics andSpace Science vol 143 no 1 pp 73ndash80 1988

[21] A J Chamkha H S Takhar and V M Soundalgekar ldquoRadia-tion effects on free convection flow past a semi-infinite verticalplate with mass transferrdquo Chemical Engineering Journal vol 84no 3 pp 335ndash342 2001

[22] M A El-Hakiem and M F El-Amin ldquoThermal radiation effecton non-Darcy natural convection with lateral mass transferrdquoHeat and Mass Transfer vol 37 no 2-3 pp 161ndash165 2001

[23] V R Prasad N B Reddy and R Muthucumaraswamy ldquoRadia-tion and mass transfer effects on two-dimensional flow past animpulsively started infinite vertical platerdquo International Journalof Thermal Sciences vol 46 no 12 pp 1251ndash1258 2007

[24] O D Makinde and A Ogulu ldquoThe effect of thermal radiationon the heat and mass transfer flow of a variable viscosity fluidpast a vertical porous plate permeated by a transverse magneticfieldrdquo Chemical Engineering Communications vol 195 no 12pp 1575ndash1584 2008

[25] SMukhopadhyay andG C Layek ldquoEffects of thermal radiationand variable fluid viscosity on free convective flow and heattransfer past a porous stretching surfacerdquo International Journalof Heat andMass Transfer vol 51 no 9-10 pp 2167ndash2178 2008

[26] D Pal ldquoHeat andmass transfer in stagnation-point flow towardsa stretching surface in the presence of buoyancy force andthermal radiationrdquoMeccanica vol 44 no 2 pp 145ndash158 2009

[27] A J Chamkha ldquoMHDflowof a uniformly streched vertical per-meable surface in the presence of heat generationabsorptionand a chemical reactionrdquo International Communications inHeatand Mass Transfer vol 30 no 3 pp 413ndash422 2003

[28] A A Afify ldquoMHD free convective flow and mass transferover a stretching sheet with chemical reactionrdquo Heat and MassTransfer vol 40 no 6-7 pp 495ndash500 2004

[29] A Postelnicu ldquoInfluence of chemical reaction on heat andmasstransfer by natural convection from vertical surfaces in porousmedia considering Soret and Dufour effectsrdquo Heat and MassTransfer vol 43 no 6 pp 595ndash602 2007

[30] R Kandasamy and P G Palanimani ldquoEffects of chemical reac-tions heat and mass transfer on nonlinear magnetohydrody-namic boundary layer flow over a wedge with a porous mediumin the presence of ohmic heating and viscous dissipationrdquoJournal of Porous Media vol 10 no 5 pp 489ndash501 2007

[31] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[32] A A Hayday D A Bowlus and R A McGraw ldquoFree convec-tion from a vertical plate with step discontinuities in surfacetemperaturerdquo ASME Journal of Heat Transfer vol 89 pp 244ndash250 1967

[33] M Kelleher ldquoFree convection from a vertical plate with dis-continuous wall temperaturerdquo Journal of Heat Transfer Trans-actions ASME vol 93 no 4 pp 349ndash356 1971

[34] T-T Kao ldquoLaminar free convective heat transfer response alonga vertical flat plate with step jump in surface temperaturerdquoLetters inHeat andMass Transfer vol 2 no 5 pp 419ndash428 1975

[35] S Lee and M M Yovanovich ldquoLaminar natural convectionfrom a vertical plate with a step change in wall temperaturerdquoJournal of Heat Transfer vol 113 no 2 pp 501ndash504 1991

[36] P Chandran N C Sacheti and A K Singh ldquoNatural convec-tion near a vertical plate with ramped wall temperaturerdquo Heatand Mass Transfer vol 41 no 5 pp 459ndash464 2005

[37] G S Seth M S Ansari and R Nandkeolyar ldquoMHD naturalconvection flow with radiative heat transfer past an impulsivelymoving plate with ramped wall temperaturerdquo Heat and MassTransfer vol 47 no 5 pp 551ndash561 2011

[38] G S Seth R Nandkeolyar and M S Ansari ldquoEffects ofthermal radiation and rotation on unsteady hydromagnetic freeconvection flow past an impulsively moving vertical plate withramped temperature in a porous mediumrdquo Journal of AppliedFluid Mechanics vol 6 no 1 pp 27ndash38 2013

[39] O D Makinde ldquoFree convection flow with thermal radiationand mass transfer past a moving vertical porous platerdquo Interna-tional Communications in Heat and Mass Transfer vol 32 no10 pp 1411ndash1419 2005

[40] I U Mbeledogu and A Ogulu ldquoHeat and mass transfer of anunsteady MHD natural convection flow of a rotating fluid pasta vertical porous flat plate in the presence of radiative heattransferrdquo International Journal of Heat and Mass Transfer vol50 no 9-10 pp 1902ndash1908 2007

[41] A A Afify ldquoSimilarity solution in MHD effects of thermaldiffusion and diffusion thermo on free convective heat andmass transfer over a stretching surface considering suction orinjectionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2202ndash2214 2009

[42] K Cramer and S PaiMagnetofluid Dynamics for Engineers andApplied Physicists McGraw Hill New York NY USA 1973

[43] R C Meyer ldquoOn reducing aerodynamic heat-transfer ratesby magnetohydrodynamic techniquesrdquo Journal of Aero SpaceScience vol 25 pp 561ndash572 1958

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

Sc = 05 10 15

Gm = 1

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

(a)

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

Sc = 05 10 15

Gm = minus1

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

(b)

Figure 6 Effect of Sc on 119906(120578 119905) when119872 = 3 119870119903= 05119873 = 1 Pr = 071 and 119905 = 07

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

1

Pr = 071 1 7

Gm = 1

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

(a)

0 05 1 15 2 25 3 35 40

01

02

03

04

05

06

07

08

09

1

Pr = 071 1 7

Gm = minus1

u(120578t)

120578

Suction (S = 1) Blowing (S = minus1)

(b)

Figure 7 Effect of Pr on 119906(120578 119905) when Sc = 1 119870119903= 05119873 = 1119872 = 3 and 119905 = 07

Prandtl number is the ratio of viscosity to thermal diffusivityand an increase in Pr implies a decrease in thermal diffusivityThermal radiation and diffusivity tend to increase the fluidtemperature Figure 10 shows that an increase in suctionvelocity causes a reduction in the fluid temperature whereasan increase in blowing increases the fluid temperature

Figures 11 12 and 13 show the effects of the chemicalreaction mass diffusion and suction or blowing on thespecies concentration We observe from Figures 11 and 12that the species concentration decreases with an increase inthe chemical reaction parameter or the Schmidt numberThus the chemical reaction rate tends to reduce the species

concentration whereas mass diffusivity increases the speciesconcentration It is clear from Figure 13 that the suctionvelocity reduces the species concentration while blowingcauses an increase in the species concentration We observefrom Figures 11 and 12 that the species concentration is higherin the case of blowing than in the case of suction We alsoobserve from Figures 2ndash11 that the momentum thermal andconcentration boundary layers are thicker in the case ofblowing than in the case of suction

Table 3 shows the effects of the magnetic field thermalradiation suctionblowing chemical reactionmass diffusionand thermal diffusion on the skin friction in the presence

Mathematical Problems in Engineering 9

Table 3 Effect of fluid and physical parameters on the skin friction minus120591 when 119905 = 07

119872 119873 119878 119870119903

Sc Pr minus120591 (119878 gt 0Gm = 1)

minus120591 (119878 lt 0Gm = 1)

minus120591 (119878 gt 0Gm = minus1)

minus120591 (119878 lt 0Gm = minus1)

3 1 mdash 05 1 071 1747428 0788854 2397344 14387705 mdash mdash mdash mdash mdash 2291296 1325411 2865812 18999277 mdash mdash mdash mdash mdash 2737296 1766244 3258082 2287031mdash 05 mdash mdash mdash mdash 1763388 0794076 2413303 1443992mdash 10 mdash mdash mdash mdash 1747428 0788854 2397344 1438770mdash 15 mdash mdash mdash mdash 1735676 0784937 2385592 1434853mdash mdash 1 mdash mdash mdash 1747428 mdash 2397344 mdashmdash mdash 2 mdash mdash mdash 2463830 mdash 3043645 mdashmdash mdash 3 mdash mdash mdash 3291595 mdash 3786658 mdashmdash mdash minus1 mdash mdash mdash mdash 0788854 mdash 1438770mdash mdash minus2 mdash mdash mdash mdash 0540588 mdash 1120403mdash mdash minus3 mdash mdash mdash mdash 0393845 mdash 0888908mdash mdash mdash 05 mdash mdash 1747428 0788854 2397344 1438770mdash mdash mdash 10 mdash mdash 1761672 0803098 2383100 1424526mdash mdash mdash 15 mdash mdash 1774216 0815642 2370555 1411981mdash mdash mdash mdash 05 mdash 1666559 0771682 2478212 1455942mdash mdash mdash mdash 10 mdash 1747428 0788854 2397344 1438770mdash mdash mdash mdash 15 mdash 1796602 0798486 2348170 1429138mdash mdash mdash mdash mdash 071 1747428 0788854 2397344 1438770mdash mdash mdash mdash mdash 10 1766522 0795089 2416438 1445005mdash mdash mdash mdash mdash 70 1884422 0830142 2534338 1480058

0 1 2 3 4 50

01

02

03

04

05

06

07

N = 05 10 15

Suction (S = 1) Blowing (S = minus1)

T(120578

120578

t)

Figure 8 Effect of119873 on 119879(120578 119905) when Pr = 071 and 119905 = 07

of assisting and opposing mass buoyancy In the case ofassisting mass buoyancy an increase in 119872 119870

119903 Sc and Pr

tends to increase the skin friction whereas an increase in119873 reduces the skin friction for both suction and blowingcases Thus the skin friction increases with an increase inthe magnetic field parameter and the chemical reaction rate

0 1 2 3 4 50

01

02

03

04

05

06

07

Pr = 071 1 7

Suction (S = 1) Blowing (S = minus1)

T(120578

120578

t)

Figure 9 Effect of Pr on 119879(120578 119905) when119873 = 1 and 119905 = 07

while it decreaseswith an increase inmass diffusivity thermaldiffusivity and thermal radiation in the case of assisting massbuoyancy However for opposing mass buoyancy the skinfriction increases with mass diffusivity while decreasing withan increase in the rate of the chemical reaction

10 Mathematical Problems in Engineering

Table 4 Values of NuRe and ShRe for different values of governing parameters when 119905 = 07

119873 119878 Pr 119870119903

Sc NuRe NuRe ShRe ShRe(Suction) (Blowing) (Suction) (Blowing)

05 mdash 071 mdash mdash 1249449 0752449 mdash mdash10 mdash 071 mdash mdash 1396642 0899642 mdash mdash15 mdash 071 mdash mdash 1527010 1030010 mdash mdash1 1 071 05 10 1396642 mdash 1500355 mdash1 2 071 05 10 1712939 mdash 2280316 mdash1 3 071 05 10 2069304 mdash 3174187 mdash1 minus1 071 05 10 mdash 0899642 mdash 05003551 minus2 071 05 10 mdash 0718939 mdash 02803161 minus3 071 05 10 mdash 0578304 mdash 01741871 mdash 071 mdash mdash 1396642 0899642 mdash mdash1 mdash 10 mdash mdash 1723721 1023721 mdash mdash1 mdash 70 mdash mdash 6662968 1762968 mdash mdashmdash mdash mdash 05 1 mdash mdash 1500355 0500355mdash mdash mdash 10 1 mdash mdash 1691322 0691322mdash mdash mdash 15 1 mdash mdash 1865496 0865496mdash mdash mdash 05 05 mdash mdash 0921462 0421462mdash mdash mdash 05 10 mdash mdash 1500355 0500355mdash mdash mdash 05 15 mdash mdash 2035801 0535801

0 1 2 3 4 7650

01

02

03

04

05

06

07

S = minus1 minus2 minus3

S = 1 2 3

T(120578

120578

t)

Figure 10 Effect of 119878 on 119879(120578 119905) when Pr = 071119873 = 1 and 119905 = 07

Table 4 shows the effects of thermal radiation suc-tionblowing and thermal diffusion on the heat transfer ratefrom the plate It also shows the effects of suctionblowingchemical reaction rate andmass diffusion on the rate of masstransfer at the plate We observe that the Nusselt numberincreases with 119873 and Pr Thus the thermal radiation tendsto accelerate heat transfer at the plate while thermal diffusionhas reverse effect Table 4 also shows that the rate of heattransfer at the plate increases with an increase in suctionvelocity and decreases with blowing The Sherwood numberincreases with 119870

119903and Sc for both suction and blowing An

increase in the suction velocity causes an increase in therate of mass transfer whereas an increase in blowing has theopposite effect

0 05 1 15 2 25 3 35 4

Suction (S = 1) Blowing (S = minus1)

0

01

02

03

04

05

06

07

08

09

1

C(120578

120578

t)

Kr = 0 05 10

Figure 11 Effect of119872 on 119862(120578 119905) when Sc = 1 and 119905 = 07

6 Conclusions

The unsteady hydromagnetic free convection flow of aviscous incompressible and heat-radiating fluid past animpulsively started vertical porous flat plate with rampedtemperature in the presence of a chemically reactive speciesconcentration was studied For certain parameter valuesexact solutions of the model equations were found usingLaplace transforms The Matlab INVLAP routine was used tofind the numerical results in the general case The numericalresults were found to be in excellent agreement with the

Mathematical Problems in Engineering 11

Suction (S = 1) Blowing (S = minus1)

Sc = 05 10 15

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

C(120578

120578

t)

Figure 12 Effect of Sc on 119862(120578 119905) when 119870119903= 05 and 119905 = 07

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

S = 1 2 3

S = minus1 minus2 minus3

C(120578

120578

t)

Figure 13 Effect of 119878 on 119862(120578 119905) when 119870119903= 05 Sc = 1 and 119905 = 07

results of Seth et al [37] and with the exact results Theimportant findings of the present work may be summarizedas follows

(i) In the case of assisting mass buoyancy force the fluidvelocity is reduced by an increase in themagnetic fieldand the chemical reaction rate

(ii) Blowing causes an acceleration in the fluid velocitywhile suction has reverse effect on it

(iii) An increase in the chemical reaction rate tends toreduce the species concentration whereas the massdiffusivity parameter has reverse effect

(iv) Thermal radiation thermal diffusivity and blowingtend to increase the fluid temperature whereas suc-tion has the opposite effect

(v) In the case of an assisting mass buoyancy force themagnetic field and chemical reaction parameter tendto increase the skin friction while themass diffusivitythermal diffusivity and thermal radiation have thereverse effect

(vi) Thermal diffusivity and blowing tend to reduce therate of heat transfer at the plate whereas thermalradiation and suction have the opposite effect

Acknowledgment

The authors would like to thank the referee for his valuableand constructive suggestions

References

[1] A S Gupta ldquoSteady and transient free convection of anelectrically conducting fluid froma vertical plate in the presenceof a magnetic fieldrdquo Applied Scientific Research vol 9 no 1 pp319ndash333 1960

[2] A S Gupta ldquoLaminar free convection flow of an electricallyconducting fluid from a vertical plate with uniform surface heatflux and variable wall temperature in the presence of amagneticfieldrdquoZeitschrift fur AngewandteMathematik undPhysik vol 13pp 324ndash333 1962

[3] K R Cramer ldquoSeveral magnetohydrodynamic free convectionsolutionsrdquo Journal of Heat Transfer vol 85 pp 35ndash40 1963

[4] I Pop ldquoOn the unsteady hydromagnetic free convection flowpast a vertical infinite flat platerdquo Indian Journal of Physics vol43 pp 196ndash200 1969

[5] H K Kuiken ldquoMagnetohydrodynamic free convection in astrong cross fieldrdquo Journal of Fluid Mechanics vol 40 no 1 pp21ndash38 1970

[6] G Wilks ldquoMagnetohydrodynamic free convection about asemi-infinite vertical plate in a strong cross fieldrdquo Zeitschrift furAngewandte Mathematik und Physik ZAMP vol 27 no 5 pp621ndash631 1976

[7] M A Hossain ldquoEffect of hall current on unsteady hydromag-netic free convection flow near an infinite vertical porous platerdquoJournal of the Physical Society of Japan vol 55 no 7 pp 2183ndash2190 1986

[8] T K Aldoss M A Al-Nimr M A Jarrah and B J Al-SharsquoerldquoMagnetohydrodynamicmixed convection from a vertical plateembedded in a porous mediumrdquo Numerical Heat Transfer PartA vol 28 no 5 pp 635ndash645 1995

[9] K A Helmy ldquoMHD unsteady free convection flow past avertical porous platerdquo Zeitschrift fur Angewandte Mathematikund Mechanik vol 78 no 4 pp 255ndash270 1998

[10] Y J Kim ldquoUnsteadyMHD convective heat transfer past a semi-infinite vertical porous moving plate with variable suctionrdquoInternational Journal of Engineering Science vol 38 no 8 pp833ndash845 2000

[11] H S Takhar S Roy and G Nath ldquoUnsteady free convectionflow over an infinite vertical porous plate due to the combinedeffects of thermal and mass diffusion magnetic field and Hallcurrentsrdquo Heat and Mass Transfer vol 39 no 10 pp 825ndash8342003

[12] N Ahmed H K Sarmah and D Kalita ldquoThermal diffusioneffect on a three-dimensional MHD free convection with masstransfer flow from a porous vertical platerdquo Latin AmericanApplied Research vol 41 pp 165ndash176 2011

12 Mathematical Problems in Engineering

[13] M A Hossain and H S Takhar ldquoRadiation effect on mixedconvection along a vertical plate with uniform surface temper-aturerdquoHeat and Mass Transfer vol 31 no 4 pp 243ndash248 1996

[14] A Y Bakier and R S R Gorla ldquoThermal radiation effect onmixed convection from horizontal surfaces in saturated porousmediardquo Transport in Porous Media vol 23 no 3 pp 357ndash3631996

[15] H S Takhar R S R Gorla and VM Soundalgekar ldquoRadiationeffects onMHD free convection flowof a gas past a semi-infinitevertical platerdquo International Journal of Numerical Methods forHeat and Fluid Flow vol 6 no 2 pp 77ndash83 1996

[16] A J Chamkha ldquoThermal radiation and buoyancy effects onhydromagnetic flow over an accelerating permeable surfacewith heat source or sinkrdquo International Journal of EngineeringScience vol 38 no 15 pp 1699ndash1712 2000

[17] G E-D A Azzam ldquoRadiation effects on the MHDmixed free-forced convective flow past a semi-infinitemoving vertical platefor high temperature differencesrdquo Physica Scripta vol 66 no 1pp 71ndash76 2002

[18] C Israel-Cookey A Ogulu and V B Omubo-Pepple ldquoInflu-ence of viscous dissipation and radiation on unsteady MHDfree-convection flow past an infinite heated vertical plate ina porous medium with time-dependent suctionrdquo InternationalJournal of Heat andMass Transfer vol 46 no 13 pp 2305ndash23112003

[19] M A A Mahmoud ldquoThermal radiation effect on unsteadyMHD free convection flow past a vertical plate with tem-perature-dependent viscosityrdquo Canadian Journal of ChemicalEngineering vol 87 no 1 pp 47ndash52 2009

[20] A R Bestman and S K Adjepong ldquoUnsteady hydromagneticfree-convection flow with radiative heat transfer in a rotatingfluidmdashI Incompressible optically thin fluidrdquo Astrophysics andSpace Science vol 143 no 1 pp 73ndash80 1988

[21] A J Chamkha H S Takhar and V M Soundalgekar ldquoRadia-tion effects on free convection flow past a semi-infinite verticalplate with mass transferrdquo Chemical Engineering Journal vol 84no 3 pp 335ndash342 2001

[22] M A El-Hakiem and M F El-Amin ldquoThermal radiation effecton non-Darcy natural convection with lateral mass transferrdquoHeat and Mass Transfer vol 37 no 2-3 pp 161ndash165 2001

[23] V R Prasad N B Reddy and R Muthucumaraswamy ldquoRadia-tion and mass transfer effects on two-dimensional flow past animpulsively started infinite vertical platerdquo International Journalof Thermal Sciences vol 46 no 12 pp 1251ndash1258 2007

[24] O D Makinde and A Ogulu ldquoThe effect of thermal radiationon the heat and mass transfer flow of a variable viscosity fluidpast a vertical porous plate permeated by a transverse magneticfieldrdquo Chemical Engineering Communications vol 195 no 12pp 1575ndash1584 2008

[25] SMukhopadhyay andG C Layek ldquoEffects of thermal radiationand variable fluid viscosity on free convective flow and heattransfer past a porous stretching surfacerdquo International Journalof Heat andMass Transfer vol 51 no 9-10 pp 2167ndash2178 2008

[26] D Pal ldquoHeat andmass transfer in stagnation-point flow towardsa stretching surface in the presence of buoyancy force andthermal radiationrdquoMeccanica vol 44 no 2 pp 145ndash158 2009

[27] A J Chamkha ldquoMHDflowof a uniformly streched vertical per-meable surface in the presence of heat generationabsorptionand a chemical reactionrdquo International Communications inHeatand Mass Transfer vol 30 no 3 pp 413ndash422 2003

[28] A A Afify ldquoMHD free convective flow and mass transferover a stretching sheet with chemical reactionrdquo Heat and MassTransfer vol 40 no 6-7 pp 495ndash500 2004

[29] A Postelnicu ldquoInfluence of chemical reaction on heat andmasstransfer by natural convection from vertical surfaces in porousmedia considering Soret and Dufour effectsrdquo Heat and MassTransfer vol 43 no 6 pp 595ndash602 2007

[30] R Kandasamy and P G Palanimani ldquoEffects of chemical reac-tions heat and mass transfer on nonlinear magnetohydrody-namic boundary layer flow over a wedge with a porous mediumin the presence of ohmic heating and viscous dissipationrdquoJournal of Porous Media vol 10 no 5 pp 489ndash501 2007

[31] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[32] A A Hayday D A Bowlus and R A McGraw ldquoFree convec-tion from a vertical plate with step discontinuities in surfacetemperaturerdquo ASME Journal of Heat Transfer vol 89 pp 244ndash250 1967

[33] M Kelleher ldquoFree convection from a vertical plate with dis-continuous wall temperaturerdquo Journal of Heat Transfer Trans-actions ASME vol 93 no 4 pp 349ndash356 1971

[34] T-T Kao ldquoLaminar free convective heat transfer response alonga vertical flat plate with step jump in surface temperaturerdquoLetters inHeat andMass Transfer vol 2 no 5 pp 419ndash428 1975

[35] S Lee and M M Yovanovich ldquoLaminar natural convectionfrom a vertical plate with a step change in wall temperaturerdquoJournal of Heat Transfer vol 113 no 2 pp 501ndash504 1991

[36] P Chandran N C Sacheti and A K Singh ldquoNatural convec-tion near a vertical plate with ramped wall temperaturerdquo Heatand Mass Transfer vol 41 no 5 pp 459ndash464 2005

[37] G S Seth M S Ansari and R Nandkeolyar ldquoMHD naturalconvection flow with radiative heat transfer past an impulsivelymoving plate with ramped wall temperaturerdquo Heat and MassTransfer vol 47 no 5 pp 551ndash561 2011

[38] G S Seth R Nandkeolyar and M S Ansari ldquoEffects ofthermal radiation and rotation on unsteady hydromagnetic freeconvection flow past an impulsively moving vertical plate withramped temperature in a porous mediumrdquo Journal of AppliedFluid Mechanics vol 6 no 1 pp 27ndash38 2013

[39] O D Makinde ldquoFree convection flow with thermal radiationand mass transfer past a moving vertical porous platerdquo Interna-tional Communications in Heat and Mass Transfer vol 32 no10 pp 1411ndash1419 2005

[40] I U Mbeledogu and A Ogulu ldquoHeat and mass transfer of anunsteady MHD natural convection flow of a rotating fluid pasta vertical porous flat plate in the presence of radiative heattransferrdquo International Journal of Heat and Mass Transfer vol50 no 9-10 pp 1902ndash1908 2007

[41] A A Afify ldquoSimilarity solution in MHD effects of thermaldiffusion and diffusion thermo on free convective heat andmass transfer over a stretching surface considering suction orinjectionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2202ndash2214 2009

[42] K Cramer and S PaiMagnetofluid Dynamics for Engineers andApplied Physicists McGraw Hill New York NY USA 1973

[43] R C Meyer ldquoOn reducing aerodynamic heat-transfer ratesby magnetohydrodynamic techniquesrdquo Journal of Aero SpaceScience vol 25 pp 561ndash572 1958

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Table 3 Effect of fluid and physical parameters on the skin friction minus120591 when 119905 = 07

119872 119873 119878 119870119903

Sc Pr minus120591 (119878 gt 0Gm = 1)

minus120591 (119878 lt 0Gm = 1)

minus120591 (119878 gt 0Gm = minus1)

minus120591 (119878 lt 0Gm = minus1)

3 1 mdash 05 1 071 1747428 0788854 2397344 14387705 mdash mdash mdash mdash mdash 2291296 1325411 2865812 18999277 mdash mdash mdash mdash mdash 2737296 1766244 3258082 2287031mdash 05 mdash mdash mdash mdash 1763388 0794076 2413303 1443992mdash 10 mdash mdash mdash mdash 1747428 0788854 2397344 1438770mdash 15 mdash mdash mdash mdash 1735676 0784937 2385592 1434853mdash mdash 1 mdash mdash mdash 1747428 mdash 2397344 mdashmdash mdash 2 mdash mdash mdash 2463830 mdash 3043645 mdashmdash mdash 3 mdash mdash mdash 3291595 mdash 3786658 mdashmdash mdash minus1 mdash mdash mdash mdash 0788854 mdash 1438770mdash mdash minus2 mdash mdash mdash mdash 0540588 mdash 1120403mdash mdash minus3 mdash mdash mdash mdash 0393845 mdash 0888908mdash mdash mdash 05 mdash mdash 1747428 0788854 2397344 1438770mdash mdash mdash 10 mdash mdash 1761672 0803098 2383100 1424526mdash mdash mdash 15 mdash mdash 1774216 0815642 2370555 1411981mdash mdash mdash mdash 05 mdash 1666559 0771682 2478212 1455942mdash mdash mdash mdash 10 mdash 1747428 0788854 2397344 1438770mdash mdash mdash mdash 15 mdash 1796602 0798486 2348170 1429138mdash mdash mdash mdash mdash 071 1747428 0788854 2397344 1438770mdash mdash mdash mdash mdash 10 1766522 0795089 2416438 1445005mdash mdash mdash mdash mdash 70 1884422 0830142 2534338 1480058

0 1 2 3 4 50

01

02

03

04

05

06

07

N = 05 10 15

Suction (S = 1) Blowing (S = minus1)

T(120578

120578

t)

Figure 8 Effect of119873 on 119879(120578 119905) when Pr = 071 and 119905 = 07

of assisting and opposing mass buoyancy In the case ofassisting mass buoyancy an increase in 119872 119870

119903 Sc and Pr

tends to increase the skin friction whereas an increase in119873 reduces the skin friction for both suction and blowingcases Thus the skin friction increases with an increase inthe magnetic field parameter and the chemical reaction rate

0 1 2 3 4 50

01

02

03

04

05

06

07

Pr = 071 1 7

Suction (S = 1) Blowing (S = minus1)

T(120578

120578

t)

Figure 9 Effect of Pr on 119879(120578 119905) when119873 = 1 and 119905 = 07

while it decreaseswith an increase inmass diffusivity thermaldiffusivity and thermal radiation in the case of assisting massbuoyancy However for opposing mass buoyancy the skinfriction increases with mass diffusivity while decreasing withan increase in the rate of the chemical reaction

10 Mathematical Problems in Engineering

Table 4 Values of NuRe and ShRe for different values of governing parameters when 119905 = 07

119873 119878 Pr 119870119903

Sc NuRe NuRe ShRe ShRe(Suction) (Blowing) (Suction) (Blowing)

05 mdash 071 mdash mdash 1249449 0752449 mdash mdash10 mdash 071 mdash mdash 1396642 0899642 mdash mdash15 mdash 071 mdash mdash 1527010 1030010 mdash mdash1 1 071 05 10 1396642 mdash 1500355 mdash1 2 071 05 10 1712939 mdash 2280316 mdash1 3 071 05 10 2069304 mdash 3174187 mdash1 minus1 071 05 10 mdash 0899642 mdash 05003551 minus2 071 05 10 mdash 0718939 mdash 02803161 minus3 071 05 10 mdash 0578304 mdash 01741871 mdash 071 mdash mdash 1396642 0899642 mdash mdash1 mdash 10 mdash mdash 1723721 1023721 mdash mdash1 mdash 70 mdash mdash 6662968 1762968 mdash mdashmdash mdash mdash 05 1 mdash mdash 1500355 0500355mdash mdash mdash 10 1 mdash mdash 1691322 0691322mdash mdash mdash 15 1 mdash mdash 1865496 0865496mdash mdash mdash 05 05 mdash mdash 0921462 0421462mdash mdash mdash 05 10 mdash mdash 1500355 0500355mdash mdash mdash 05 15 mdash mdash 2035801 0535801

0 1 2 3 4 7650

01

02

03

04

05

06

07

S = minus1 minus2 minus3

S = 1 2 3

T(120578

120578

t)

Figure 10 Effect of 119878 on 119879(120578 119905) when Pr = 071119873 = 1 and 119905 = 07

Table 4 shows the effects of thermal radiation suc-tionblowing and thermal diffusion on the heat transfer ratefrom the plate It also shows the effects of suctionblowingchemical reaction rate andmass diffusion on the rate of masstransfer at the plate We observe that the Nusselt numberincreases with 119873 and Pr Thus the thermal radiation tendsto accelerate heat transfer at the plate while thermal diffusionhas reverse effect Table 4 also shows that the rate of heattransfer at the plate increases with an increase in suctionvelocity and decreases with blowing The Sherwood numberincreases with 119870

119903and Sc for both suction and blowing An

increase in the suction velocity causes an increase in therate of mass transfer whereas an increase in blowing has theopposite effect

0 05 1 15 2 25 3 35 4

Suction (S = 1) Blowing (S = minus1)

0

01

02

03

04

05

06

07

08

09

1

C(120578

120578

t)

Kr = 0 05 10

Figure 11 Effect of119872 on 119862(120578 119905) when Sc = 1 and 119905 = 07

6 Conclusions

The unsteady hydromagnetic free convection flow of aviscous incompressible and heat-radiating fluid past animpulsively started vertical porous flat plate with rampedtemperature in the presence of a chemically reactive speciesconcentration was studied For certain parameter valuesexact solutions of the model equations were found usingLaplace transforms The Matlab INVLAP routine was used tofind the numerical results in the general case The numericalresults were found to be in excellent agreement with the

Mathematical Problems in Engineering 11

Suction (S = 1) Blowing (S = minus1)

Sc = 05 10 15

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

C(120578

120578

t)

Figure 12 Effect of Sc on 119862(120578 119905) when 119870119903= 05 and 119905 = 07

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

S = 1 2 3

S = minus1 minus2 minus3

C(120578

120578

t)

Figure 13 Effect of 119878 on 119862(120578 119905) when 119870119903= 05 Sc = 1 and 119905 = 07

results of Seth et al [37] and with the exact results Theimportant findings of the present work may be summarizedas follows

(i) In the case of assisting mass buoyancy force the fluidvelocity is reduced by an increase in themagnetic fieldand the chemical reaction rate

(ii) Blowing causes an acceleration in the fluid velocitywhile suction has reverse effect on it

(iii) An increase in the chemical reaction rate tends toreduce the species concentration whereas the massdiffusivity parameter has reverse effect

(iv) Thermal radiation thermal diffusivity and blowingtend to increase the fluid temperature whereas suc-tion has the opposite effect

(v) In the case of an assisting mass buoyancy force themagnetic field and chemical reaction parameter tendto increase the skin friction while themass diffusivitythermal diffusivity and thermal radiation have thereverse effect

(vi) Thermal diffusivity and blowing tend to reduce therate of heat transfer at the plate whereas thermalradiation and suction have the opposite effect

Acknowledgment

The authors would like to thank the referee for his valuableand constructive suggestions

References

[1] A S Gupta ldquoSteady and transient free convection of anelectrically conducting fluid froma vertical plate in the presenceof a magnetic fieldrdquo Applied Scientific Research vol 9 no 1 pp319ndash333 1960

[2] A S Gupta ldquoLaminar free convection flow of an electricallyconducting fluid from a vertical plate with uniform surface heatflux and variable wall temperature in the presence of amagneticfieldrdquoZeitschrift fur AngewandteMathematik undPhysik vol 13pp 324ndash333 1962

[3] K R Cramer ldquoSeveral magnetohydrodynamic free convectionsolutionsrdquo Journal of Heat Transfer vol 85 pp 35ndash40 1963

[4] I Pop ldquoOn the unsteady hydromagnetic free convection flowpast a vertical infinite flat platerdquo Indian Journal of Physics vol43 pp 196ndash200 1969

[5] H K Kuiken ldquoMagnetohydrodynamic free convection in astrong cross fieldrdquo Journal of Fluid Mechanics vol 40 no 1 pp21ndash38 1970

[6] G Wilks ldquoMagnetohydrodynamic free convection about asemi-infinite vertical plate in a strong cross fieldrdquo Zeitschrift furAngewandte Mathematik und Physik ZAMP vol 27 no 5 pp621ndash631 1976

[7] M A Hossain ldquoEffect of hall current on unsteady hydromag-netic free convection flow near an infinite vertical porous platerdquoJournal of the Physical Society of Japan vol 55 no 7 pp 2183ndash2190 1986

[8] T K Aldoss M A Al-Nimr M A Jarrah and B J Al-SharsquoerldquoMagnetohydrodynamicmixed convection from a vertical plateembedded in a porous mediumrdquo Numerical Heat Transfer PartA vol 28 no 5 pp 635ndash645 1995

[9] K A Helmy ldquoMHD unsteady free convection flow past avertical porous platerdquo Zeitschrift fur Angewandte Mathematikund Mechanik vol 78 no 4 pp 255ndash270 1998

[10] Y J Kim ldquoUnsteadyMHD convective heat transfer past a semi-infinite vertical porous moving plate with variable suctionrdquoInternational Journal of Engineering Science vol 38 no 8 pp833ndash845 2000

[11] H S Takhar S Roy and G Nath ldquoUnsteady free convectionflow over an infinite vertical porous plate due to the combinedeffects of thermal and mass diffusion magnetic field and Hallcurrentsrdquo Heat and Mass Transfer vol 39 no 10 pp 825ndash8342003

[12] N Ahmed H K Sarmah and D Kalita ldquoThermal diffusioneffect on a three-dimensional MHD free convection with masstransfer flow from a porous vertical platerdquo Latin AmericanApplied Research vol 41 pp 165ndash176 2011

12 Mathematical Problems in Engineering

[13] M A Hossain and H S Takhar ldquoRadiation effect on mixedconvection along a vertical plate with uniform surface temper-aturerdquoHeat and Mass Transfer vol 31 no 4 pp 243ndash248 1996

[14] A Y Bakier and R S R Gorla ldquoThermal radiation effect onmixed convection from horizontal surfaces in saturated porousmediardquo Transport in Porous Media vol 23 no 3 pp 357ndash3631996

[15] H S Takhar R S R Gorla and VM Soundalgekar ldquoRadiationeffects onMHD free convection flowof a gas past a semi-infinitevertical platerdquo International Journal of Numerical Methods forHeat and Fluid Flow vol 6 no 2 pp 77ndash83 1996

[16] A J Chamkha ldquoThermal radiation and buoyancy effects onhydromagnetic flow over an accelerating permeable surfacewith heat source or sinkrdquo International Journal of EngineeringScience vol 38 no 15 pp 1699ndash1712 2000

[17] G E-D A Azzam ldquoRadiation effects on the MHDmixed free-forced convective flow past a semi-infinitemoving vertical platefor high temperature differencesrdquo Physica Scripta vol 66 no 1pp 71ndash76 2002

[18] C Israel-Cookey A Ogulu and V B Omubo-Pepple ldquoInflu-ence of viscous dissipation and radiation on unsteady MHDfree-convection flow past an infinite heated vertical plate ina porous medium with time-dependent suctionrdquo InternationalJournal of Heat andMass Transfer vol 46 no 13 pp 2305ndash23112003

[19] M A A Mahmoud ldquoThermal radiation effect on unsteadyMHD free convection flow past a vertical plate with tem-perature-dependent viscosityrdquo Canadian Journal of ChemicalEngineering vol 87 no 1 pp 47ndash52 2009

[20] A R Bestman and S K Adjepong ldquoUnsteady hydromagneticfree-convection flow with radiative heat transfer in a rotatingfluidmdashI Incompressible optically thin fluidrdquo Astrophysics andSpace Science vol 143 no 1 pp 73ndash80 1988

[21] A J Chamkha H S Takhar and V M Soundalgekar ldquoRadia-tion effects on free convection flow past a semi-infinite verticalplate with mass transferrdquo Chemical Engineering Journal vol 84no 3 pp 335ndash342 2001

[22] M A El-Hakiem and M F El-Amin ldquoThermal radiation effecton non-Darcy natural convection with lateral mass transferrdquoHeat and Mass Transfer vol 37 no 2-3 pp 161ndash165 2001

[23] V R Prasad N B Reddy and R Muthucumaraswamy ldquoRadia-tion and mass transfer effects on two-dimensional flow past animpulsively started infinite vertical platerdquo International Journalof Thermal Sciences vol 46 no 12 pp 1251ndash1258 2007

[24] O D Makinde and A Ogulu ldquoThe effect of thermal radiationon the heat and mass transfer flow of a variable viscosity fluidpast a vertical porous plate permeated by a transverse magneticfieldrdquo Chemical Engineering Communications vol 195 no 12pp 1575ndash1584 2008

[25] SMukhopadhyay andG C Layek ldquoEffects of thermal radiationand variable fluid viscosity on free convective flow and heattransfer past a porous stretching surfacerdquo International Journalof Heat andMass Transfer vol 51 no 9-10 pp 2167ndash2178 2008

[26] D Pal ldquoHeat andmass transfer in stagnation-point flow towardsa stretching surface in the presence of buoyancy force andthermal radiationrdquoMeccanica vol 44 no 2 pp 145ndash158 2009

[27] A J Chamkha ldquoMHDflowof a uniformly streched vertical per-meable surface in the presence of heat generationabsorptionand a chemical reactionrdquo International Communications inHeatand Mass Transfer vol 30 no 3 pp 413ndash422 2003

[28] A A Afify ldquoMHD free convective flow and mass transferover a stretching sheet with chemical reactionrdquo Heat and MassTransfer vol 40 no 6-7 pp 495ndash500 2004

[29] A Postelnicu ldquoInfluence of chemical reaction on heat andmasstransfer by natural convection from vertical surfaces in porousmedia considering Soret and Dufour effectsrdquo Heat and MassTransfer vol 43 no 6 pp 595ndash602 2007

[30] R Kandasamy and P G Palanimani ldquoEffects of chemical reac-tions heat and mass transfer on nonlinear magnetohydrody-namic boundary layer flow over a wedge with a porous mediumin the presence of ohmic heating and viscous dissipationrdquoJournal of Porous Media vol 10 no 5 pp 489ndash501 2007

[31] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[32] A A Hayday D A Bowlus and R A McGraw ldquoFree convec-tion from a vertical plate with step discontinuities in surfacetemperaturerdquo ASME Journal of Heat Transfer vol 89 pp 244ndash250 1967

[33] M Kelleher ldquoFree convection from a vertical plate with dis-continuous wall temperaturerdquo Journal of Heat Transfer Trans-actions ASME vol 93 no 4 pp 349ndash356 1971

[34] T-T Kao ldquoLaminar free convective heat transfer response alonga vertical flat plate with step jump in surface temperaturerdquoLetters inHeat andMass Transfer vol 2 no 5 pp 419ndash428 1975

[35] S Lee and M M Yovanovich ldquoLaminar natural convectionfrom a vertical plate with a step change in wall temperaturerdquoJournal of Heat Transfer vol 113 no 2 pp 501ndash504 1991

[36] P Chandran N C Sacheti and A K Singh ldquoNatural convec-tion near a vertical plate with ramped wall temperaturerdquo Heatand Mass Transfer vol 41 no 5 pp 459ndash464 2005

[37] G S Seth M S Ansari and R Nandkeolyar ldquoMHD naturalconvection flow with radiative heat transfer past an impulsivelymoving plate with ramped wall temperaturerdquo Heat and MassTransfer vol 47 no 5 pp 551ndash561 2011

[38] G S Seth R Nandkeolyar and M S Ansari ldquoEffects ofthermal radiation and rotation on unsteady hydromagnetic freeconvection flow past an impulsively moving vertical plate withramped temperature in a porous mediumrdquo Journal of AppliedFluid Mechanics vol 6 no 1 pp 27ndash38 2013

[39] O D Makinde ldquoFree convection flow with thermal radiationand mass transfer past a moving vertical porous platerdquo Interna-tional Communications in Heat and Mass Transfer vol 32 no10 pp 1411ndash1419 2005

[40] I U Mbeledogu and A Ogulu ldquoHeat and mass transfer of anunsteady MHD natural convection flow of a rotating fluid pasta vertical porous flat plate in the presence of radiative heattransferrdquo International Journal of Heat and Mass Transfer vol50 no 9-10 pp 1902ndash1908 2007

[41] A A Afify ldquoSimilarity solution in MHD effects of thermaldiffusion and diffusion thermo on free convective heat andmass transfer over a stretching surface considering suction orinjectionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2202ndash2214 2009

[42] K Cramer and S PaiMagnetofluid Dynamics for Engineers andApplied Physicists McGraw Hill New York NY USA 1973

[43] R C Meyer ldquoOn reducing aerodynamic heat-transfer ratesby magnetohydrodynamic techniquesrdquo Journal of Aero SpaceScience vol 25 pp 561ndash572 1958

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

Table 4 Values of NuRe and ShRe for different values of governing parameters when 119905 = 07

119873 119878 Pr 119870119903

Sc NuRe NuRe ShRe ShRe(Suction) (Blowing) (Suction) (Blowing)

05 mdash 071 mdash mdash 1249449 0752449 mdash mdash10 mdash 071 mdash mdash 1396642 0899642 mdash mdash15 mdash 071 mdash mdash 1527010 1030010 mdash mdash1 1 071 05 10 1396642 mdash 1500355 mdash1 2 071 05 10 1712939 mdash 2280316 mdash1 3 071 05 10 2069304 mdash 3174187 mdash1 minus1 071 05 10 mdash 0899642 mdash 05003551 minus2 071 05 10 mdash 0718939 mdash 02803161 minus3 071 05 10 mdash 0578304 mdash 01741871 mdash 071 mdash mdash 1396642 0899642 mdash mdash1 mdash 10 mdash mdash 1723721 1023721 mdash mdash1 mdash 70 mdash mdash 6662968 1762968 mdash mdashmdash mdash mdash 05 1 mdash mdash 1500355 0500355mdash mdash mdash 10 1 mdash mdash 1691322 0691322mdash mdash mdash 15 1 mdash mdash 1865496 0865496mdash mdash mdash 05 05 mdash mdash 0921462 0421462mdash mdash mdash 05 10 mdash mdash 1500355 0500355mdash mdash mdash 05 15 mdash mdash 2035801 0535801

0 1 2 3 4 7650

01

02

03

04

05

06

07

S = minus1 minus2 minus3

S = 1 2 3

T(120578

120578

t)

Figure 10 Effect of 119878 on 119879(120578 119905) when Pr = 071119873 = 1 and 119905 = 07

Table 4 shows the effects of thermal radiation suc-tionblowing and thermal diffusion on the heat transfer ratefrom the plate It also shows the effects of suctionblowingchemical reaction rate andmass diffusion on the rate of masstransfer at the plate We observe that the Nusselt numberincreases with 119873 and Pr Thus the thermal radiation tendsto accelerate heat transfer at the plate while thermal diffusionhas reverse effect Table 4 also shows that the rate of heattransfer at the plate increases with an increase in suctionvelocity and decreases with blowing The Sherwood numberincreases with 119870

119903and Sc for both suction and blowing An

increase in the suction velocity causes an increase in therate of mass transfer whereas an increase in blowing has theopposite effect

0 05 1 15 2 25 3 35 4

Suction (S = 1) Blowing (S = minus1)

0

01

02

03

04

05

06

07

08

09

1

C(120578

120578

t)

Kr = 0 05 10

Figure 11 Effect of119872 on 119862(120578 119905) when Sc = 1 and 119905 = 07

6 Conclusions

The unsteady hydromagnetic free convection flow of aviscous incompressible and heat-radiating fluid past animpulsively started vertical porous flat plate with rampedtemperature in the presence of a chemically reactive speciesconcentration was studied For certain parameter valuesexact solutions of the model equations were found usingLaplace transforms The Matlab INVLAP routine was used tofind the numerical results in the general case The numericalresults were found to be in excellent agreement with the

Mathematical Problems in Engineering 11

Suction (S = 1) Blowing (S = minus1)

Sc = 05 10 15

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

C(120578

120578

t)

Figure 12 Effect of Sc on 119862(120578 119905) when 119870119903= 05 and 119905 = 07

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

S = 1 2 3

S = minus1 minus2 minus3

C(120578

120578

t)

Figure 13 Effect of 119878 on 119862(120578 119905) when 119870119903= 05 Sc = 1 and 119905 = 07

results of Seth et al [37] and with the exact results Theimportant findings of the present work may be summarizedas follows

(i) In the case of assisting mass buoyancy force the fluidvelocity is reduced by an increase in themagnetic fieldand the chemical reaction rate

(ii) Blowing causes an acceleration in the fluid velocitywhile suction has reverse effect on it

(iii) An increase in the chemical reaction rate tends toreduce the species concentration whereas the massdiffusivity parameter has reverse effect

(iv) Thermal radiation thermal diffusivity and blowingtend to increase the fluid temperature whereas suc-tion has the opposite effect

(v) In the case of an assisting mass buoyancy force themagnetic field and chemical reaction parameter tendto increase the skin friction while themass diffusivitythermal diffusivity and thermal radiation have thereverse effect

(vi) Thermal diffusivity and blowing tend to reduce therate of heat transfer at the plate whereas thermalradiation and suction have the opposite effect

Acknowledgment

The authors would like to thank the referee for his valuableand constructive suggestions

References

[1] A S Gupta ldquoSteady and transient free convection of anelectrically conducting fluid froma vertical plate in the presenceof a magnetic fieldrdquo Applied Scientific Research vol 9 no 1 pp319ndash333 1960

[2] A S Gupta ldquoLaminar free convection flow of an electricallyconducting fluid from a vertical plate with uniform surface heatflux and variable wall temperature in the presence of amagneticfieldrdquoZeitschrift fur AngewandteMathematik undPhysik vol 13pp 324ndash333 1962

[3] K R Cramer ldquoSeveral magnetohydrodynamic free convectionsolutionsrdquo Journal of Heat Transfer vol 85 pp 35ndash40 1963

[4] I Pop ldquoOn the unsteady hydromagnetic free convection flowpast a vertical infinite flat platerdquo Indian Journal of Physics vol43 pp 196ndash200 1969

[5] H K Kuiken ldquoMagnetohydrodynamic free convection in astrong cross fieldrdquo Journal of Fluid Mechanics vol 40 no 1 pp21ndash38 1970

[6] G Wilks ldquoMagnetohydrodynamic free convection about asemi-infinite vertical plate in a strong cross fieldrdquo Zeitschrift furAngewandte Mathematik und Physik ZAMP vol 27 no 5 pp621ndash631 1976

[7] M A Hossain ldquoEffect of hall current on unsteady hydromag-netic free convection flow near an infinite vertical porous platerdquoJournal of the Physical Society of Japan vol 55 no 7 pp 2183ndash2190 1986

[8] T K Aldoss M A Al-Nimr M A Jarrah and B J Al-SharsquoerldquoMagnetohydrodynamicmixed convection from a vertical plateembedded in a porous mediumrdquo Numerical Heat Transfer PartA vol 28 no 5 pp 635ndash645 1995

[9] K A Helmy ldquoMHD unsteady free convection flow past avertical porous platerdquo Zeitschrift fur Angewandte Mathematikund Mechanik vol 78 no 4 pp 255ndash270 1998

[10] Y J Kim ldquoUnsteadyMHD convective heat transfer past a semi-infinite vertical porous moving plate with variable suctionrdquoInternational Journal of Engineering Science vol 38 no 8 pp833ndash845 2000

[11] H S Takhar S Roy and G Nath ldquoUnsteady free convectionflow over an infinite vertical porous plate due to the combinedeffects of thermal and mass diffusion magnetic field and Hallcurrentsrdquo Heat and Mass Transfer vol 39 no 10 pp 825ndash8342003

[12] N Ahmed H K Sarmah and D Kalita ldquoThermal diffusioneffect on a three-dimensional MHD free convection with masstransfer flow from a porous vertical platerdquo Latin AmericanApplied Research vol 41 pp 165ndash176 2011

12 Mathematical Problems in Engineering

[13] M A Hossain and H S Takhar ldquoRadiation effect on mixedconvection along a vertical plate with uniform surface temper-aturerdquoHeat and Mass Transfer vol 31 no 4 pp 243ndash248 1996

[14] A Y Bakier and R S R Gorla ldquoThermal radiation effect onmixed convection from horizontal surfaces in saturated porousmediardquo Transport in Porous Media vol 23 no 3 pp 357ndash3631996

[15] H S Takhar R S R Gorla and VM Soundalgekar ldquoRadiationeffects onMHD free convection flowof a gas past a semi-infinitevertical platerdquo International Journal of Numerical Methods forHeat and Fluid Flow vol 6 no 2 pp 77ndash83 1996

[16] A J Chamkha ldquoThermal radiation and buoyancy effects onhydromagnetic flow over an accelerating permeable surfacewith heat source or sinkrdquo International Journal of EngineeringScience vol 38 no 15 pp 1699ndash1712 2000

[17] G E-D A Azzam ldquoRadiation effects on the MHDmixed free-forced convective flow past a semi-infinitemoving vertical platefor high temperature differencesrdquo Physica Scripta vol 66 no 1pp 71ndash76 2002

[18] C Israel-Cookey A Ogulu and V B Omubo-Pepple ldquoInflu-ence of viscous dissipation and radiation on unsteady MHDfree-convection flow past an infinite heated vertical plate ina porous medium with time-dependent suctionrdquo InternationalJournal of Heat andMass Transfer vol 46 no 13 pp 2305ndash23112003

[19] M A A Mahmoud ldquoThermal radiation effect on unsteadyMHD free convection flow past a vertical plate with tem-perature-dependent viscosityrdquo Canadian Journal of ChemicalEngineering vol 87 no 1 pp 47ndash52 2009

[20] A R Bestman and S K Adjepong ldquoUnsteady hydromagneticfree-convection flow with radiative heat transfer in a rotatingfluidmdashI Incompressible optically thin fluidrdquo Astrophysics andSpace Science vol 143 no 1 pp 73ndash80 1988

[21] A J Chamkha H S Takhar and V M Soundalgekar ldquoRadia-tion effects on free convection flow past a semi-infinite verticalplate with mass transferrdquo Chemical Engineering Journal vol 84no 3 pp 335ndash342 2001

[22] M A El-Hakiem and M F El-Amin ldquoThermal radiation effecton non-Darcy natural convection with lateral mass transferrdquoHeat and Mass Transfer vol 37 no 2-3 pp 161ndash165 2001

[23] V R Prasad N B Reddy and R Muthucumaraswamy ldquoRadia-tion and mass transfer effects on two-dimensional flow past animpulsively started infinite vertical platerdquo International Journalof Thermal Sciences vol 46 no 12 pp 1251ndash1258 2007

[24] O D Makinde and A Ogulu ldquoThe effect of thermal radiationon the heat and mass transfer flow of a variable viscosity fluidpast a vertical porous plate permeated by a transverse magneticfieldrdquo Chemical Engineering Communications vol 195 no 12pp 1575ndash1584 2008

[25] SMukhopadhyay andG C Layek ldquoEffects of thermal radiationand variable fluid viscosity on free convective flow and heattransfer past a porous stretching surfacerdquo International Journalof Heat andMass Transfer vol 51 no 9-10 pp 2167ndash2178 2008

[26] D Pal ldquoHeat andmass transfer in stagnation-point flow towardsa stretching surface in the presence of buoyancy force andthermal radiationrdquoMeccanica vol 44 no 2 pp 145ndash158 2009

[27] A J Chamkha ldquoMHDflowof a uniformly streched vertical per-meable surface in the presence of heat generationabsorptionand a chemical reactionrdquo International Communications inHeatand Mass Transfer vol 30 no 3 pp 413ndash422 2003

[28] A A Afify ldquoMHD free convective flow and mass transferover a stretching sheet with chemical reactionrdquo Heat and MassTransfer vol 40 no 6-7 pp 495ndash500 2004

[29] A Postelnicu ldquoInfluence of chemical reaction on heat andmasstransfer by natural convection from vertical surfaces in porousmedia considering Soret and Dufour effectsrdquo Heat and MassTransfer vol 43 no 6 pp 595ndash602 2007

[30] R Kandasamy and P G Palanimani ldquoEffects of chemical reac-tions heat and mass transfer on nonlinear magnetohydrody-namic boundary layer flow over a wedge with a porous mediumin the presence of ohmic heating and viscous dissipationrdquoJournal of Porous Media vol 10 no 5 pp 489ndash501 2007

[31] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[32] A A Hayday D A Bowlus and R A McGraw ldquoFree convec-tion from a vertical plate with step discontinuities in surfacetemperaturerdquo ASME Journal of Heat Transfer vol 89 pp 244ndash250 1967

[33] M Kelleher ldquoFree convection from a vertical plate with dis-continuous wall temperaturerdquo Journal of Heat Transfer Trans-actions ASME vol 93 no 4 pp 349ndash356 1971

[34] T-T Kao ldquoLaminar free convective heat transfer response alonga vertical flat plate with step jump in surface temperaturerdquoLetters inHeat andMass Transfer vol 2 no 5 pp 419ndash428 1975

[35] S Lee and M M Yovanovich ldquoLaminar natural convectionfrom a vertical plate with a step change in wall temperaturerdquoJournal of Heat Transfer vol 113 no 2 pp 501ndash504 1991

[36] P Chandran N C Sacheti and A K Singh ldquoNatural convec-tion near a vertical plate with ramped wall temperaturerdquo Heatand Mass Transfer vol 41 no 5 pp 459ndash464 2005

[37] G S Seth M S Ansari and R Nandkeolyar ldquoMHD naturalconvection flow with radiative heat transfer past an impulsivelymoving plate with ramped wall temperaturerdquo Heat and MassTransfer vol 47 no 5 pp 551ndash561 2011

[38] G S Seth R Nandkeolyar and M S Ansari ldquoEffects ofthermal radiation and rotation on unsteady hydromagnetic freeconvection flow past an impulsively moving vertical plate withramped temperature in a porous mediumrdquo Journal of AppliedFluid Mechanics vol 6 no 1 pp 27ndash38 2013

[39] O D Makinde ldquoFree convection flow with thermal radiationand mass transfer past a moving vertical porous platerdquo Interna-tional Communications in Heat and Mass Transfer vol 32 no10 pp 1411ndash1419 2005

[40] I U Mbeledogu and A Ogulu ldquoHeat and mass transfer of anunsteady MHD natural convection flow of a rotating fluid pasta vertical porous flat plate in the presence of radiative heattransferrdquo International Journal of Heat and Mass Transfer vol50 no 9-10 pp 1902ndash1908 2007

[41] A A Afify ldquoSimilarity solution in MHD effects of thermaldiffusion and diffusion thermo on free convective heat andmass transfer over a stretching surface considering suction orinjectionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2202ndash2214 2009

[42] K Cramer and S PaiMagnetofluid Dynamics for Engineers andApplied Physicists McGraw Hill New York NY USA 1973

[43] R C Meyer ldquoOn reducing aerodynamic heat-transfer ratesby magnetohydrodynamic techniquesrdquo Journal of Aero SpaceScience vol 25 pp 561ndash572 1958

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

Suction (S = 1) Blowing (S = minus1)

Sc = 05 10 15

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

C(120578

120578

t)

Figure 12 Effect of Sc on 119862(120578 119905) when 119870119903= 05 and 119905 = 07

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

S = 1 2 3

S = minus1 minus2 minus3

C(120578

120578

t)

Figure 13 Effect of 119878 on 119862(120578 119905) when 119870119903= 05 Sc = 1 and 119905 = 07

results of Seth et al [37] and with the exact results Theimportant findings of the present work may be summarizedas follows

(i) In the case of assisting mass buoyancy force the fluidvelocity is reduced by an increase in themagnetic fieldand the chemical reaction rate

(ii) Blowing causes an acceleration in the fluid velocitywhile suction has reverse effect on it

(iii) An increase in the chemical reaction rate tends toreduce the species concentration whereas the massdiffusivity parameter has reverse effect

(iv) Thermal radiation thermal diffusivity and blowingtend to increase the fluid temperature whereas suc-tion has the opposite effect

(v) In the case of an assisting mass buoyancy force themagnetic field and chemical reaction parameter tendto increase the skin friction while themass diffusivitythermal diffusivity and thermal radiation have thereverse effect

(vi) Thermal diffusivity and blowing tend to reduce therate of heat transfer at the plate whereas thermalradiation and suction have the opposite effect

Acknowledgment

The authors would like to thank the referee for his valuableand constructive suggestions

References

[1] A S Gupta ldquoSteady and transient free convection of anelectrically conducting fluid froma vertical plate in the presenceof a magnetic fieldrdquo Applied Scientific Research vol 9 no 1 pp319ndash333 1960

[2] A S Gupta ldquoLaminar free convection flow of an electricallyconducting fluid from a vertical plate with uniform surface heatflux and variable wall temperature in the presence of amagneticfieldrdquoZeitschrift fur AngewandteMathematik undPhysik vol 13pp 324ndash333 1962

[3] K R Cramer ldquoSeveral magnetohydrodynamic free convectionsolutionsrdquo Journal of Heat Transfer vol 85 pp 35ndash40 1963

[4] I Pop ldquoOn the unsteady hydromagnetic free convection flowpast a vertical infinite flat platerdquo Indian Journal of Physics vol43 pp 196ndash200 1969

[5] H K Kuiken ldquoMagnetohydrodynamic free convection in astrong cross fieldrdquo Journal of Fluid Mechanics vol 40 no 1 pp21ndash38 1970

[6] G Wilks ldquoMagnetohydrodynamic free convection about asemi-infinite vertical plate in a strong cross fieldrdquo Zeitschrift furAngewandte Mathematik und Physik ZAMP vol 27 no 5 pp621ndash631 1976

[7] M A Hossain ldquoEffect of hall current on unsteady hydromag-netic free convection flow near an infinite vertical porous platerdquoJournal of the Physical Society of Japan vol 55 no 7 pp 2183ndash2190 1986

[8] T K Aldoss M A Al-Nimr M A Jarrah and B J Al-SharsquoerldquoMagnetohydrodynamicmixed convection from a vertical plateembedded in a porous mediumrdquo Numerical Heat Transfer PartA vol 28 no 5 pp 635ndash645 1995

[9] K A Helmy ldquoMHD unsteady free convection flow past avertical porous platerdquo Zeitschrift fur Angewandte Mathematikund Mechanik vol 78 no 4 pp 255ndash270 1998

[10] Y J Kim ldquoUnsteadyMHD convective heat transfer past a semi-infinite vertical porous moving plate with variable suctionrdquoInternational Journal of Engineering Science vol 38 no 8 pp833ndash845 2000

[11] H S Takhar S Roy and G Nath ldquoUnsteady free convectionflow over an infinite vertical porous plate due to the combinedeffects of thermal and mass diffusion magnetic field and Hallcurrentsrdquo Heat and Mass Transfer vol 39 no 10 pp 825ndash8342003

[12] N Ahmed H K Sarmah and D Kalita ldquoThermal diffusioneffect on a three-dimensional MHD free convection with masstransfer flow from a porous vertical platerdquo Latin AmericanApplied Research vol 41 pp 165ndash176 2011

12 Mathematical Problems in Engineering

[13] M A Hossain and H S Takhar ldquoRadiation effect on mixedconvection along a vertical plate with uniform surface temper-aturerdquoHeat and Mass Transfer vol 31 no 4 pp 243ndash248 1996

[14] A Y Bakier and R S R Gorla ldquoThermal radiation effect onmixed convection from horizontal surfaces in saturated porousmediardquo Transport in Porous Media vol 23 no 3 pp 357ndash3631996

[15] H S Takhar R S R Gorla and VM Soundalgekar ldquoRadiationeffects onMHD free convection flowof a gas past a semi-infinitevertical platerdquo International Journal of Numerical Methods forHeat and Fluid Flow vol 6 no 2 pp 77ndash83 1996

[16] A J Chamkha ldquoThermal radiation and buoyancy effects onhydromagnetic flow over an accelerating permeable surfacewith heat source or sinkrdquo International Journal of EngineeringScience vol 38 no 15 pp 1699ndash1712 2000

[17] G E-D A Azzam ldquoRadiation effects on the MHDmixed free-forced convective flow past a semi-infinitemoving vertical platefor high temperature differencesrdquo Physica Scripta vol 66 no 1pp 71ndash76 2002

[18] C Israel-Cookey A Ogulu and V B Omubo-Pepple ldquoInflu-ence of viscous dissipation and radiation on unsteady MHDfree-convection flow past an infinite heated vertical plate ina porous medium with time-dependent suctionrdquo InternationalJournal of Heat andMass Transfer vol 46 no 13 pp 2305ndash23112003

[19] M A A Mahmoud ldquoThermal radiation effect on unsteadyMHD free convection flow past a vertical plate with tem-perature-dependent viscosityrdquo Canadian Journal of ChemicalEngineering vol 87 no 1 pp 47ndash52 2009

[20] A R Bestman and S K Adjepong ldquoUnsteady hydromagneticfree-convection flow with radiative heat transfer in a rotatingfluidmdashI Incompressible optically thin fluidrdquo Astrophysics andSpace Science vol 143 no 1 pp 73ndash80 1988

[21] A J Chamkha H S Takhar and V M Soundalgekar ldquoRadia-tion effects on free convection flow past a semi-infinite verticalplate with mass transferrdquo Chemical Engineering Journal vol 84no 3 pp 335ndash342 2001

[22] M A El-Hakiem and M F El-Amin ldquoThermal radiation effecton non-Darcy natural convection with lateral mass transferrdquoHeat and Mass Transfer vol 37 no 2-3 pp 161ndash165 2001

[23] V R Prasad N B Reddy and R Muthucumaraswamy ldquoRadia-tion and mass transfer effects on two-dimensional flow past animpulsively started infinite vertical platerdquo International Journalof Thermal Sciences vol 46 no 12 pp 1251ndash1258 2007

[24] O D Makinde and A Ogulu ldquoThe effect of thermal radiationon the heat and mass transfer flow of a variable viscosity fluidpast a vertical porous plate permeated by a transverse magneticfieldrdquo Chemical Engineering Communications vol 195 no 12pp 1575ndash1584 2008

[25] SMukhopadhyay andG C Layek ldquoEffects of thermal radiationand variable fluid viscosity on free convective flow and heattransfer past a porous stretching surfacerdquo International Journalof Heat andMass Transfer vol 51 no 9-10 pp 2167ndash2178 2008

[26] D Pal ldquoHeat andmass transfer in stagnation-point flow towardsa stretching surface in the presence of buoyancy force andthermal radiationrdquoMeccanica vol 44 no 2 pp 145ndash158 2009

[27] A J Chamkha ldquoMHDflowof a uniformly streched vertical per-meable surface in the presence of heat generationabsorptionand a chemical reactionrdquo International Communications inHeatand Mass Transfer vol 30 no 3 pp 413ndash422 2003

[28] A A Afify ldquoMHD free convective flow and mass transferover a stretching sheet with chemical reactionrdquo Heat and MassTransfer vol 40 no 6-7 pp 495ndash500 2004

[29] A Postelnicu ldquoInfluence of chemical reaction on heat andmasstransfer by natural convection from vertical surfaces in porousmedia considering Soret and Dufour effectsrdquo Heat and MassTransfer vol 43 no 6 pp 595ndash602 2007

[30] R Kandasamy and P G Palanimani ldquoEffects of chemical reac-tions heat and mass transfer on nonlinear magnetohydrody-namic boundary layer flow over a wedge with a porous mediumin the presence of ohmic heating and viscous dissipationrdquoJournal of Porous Media vol 10 no 5 pp 489ndash501 2007

[31] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[32] A A Hayday D A Bowlus and R A McGraw ldquoFree convec-tion from a vertical plate with step discontinuities in surfacetemperaturerdquo ASME Journal of Heat Transfer vol 89 pp 244ndash250 1967

[33] M Kelleher ldquoFree convection from a vertical plate with dis-continuous wall temperaturerdquo Journal of Heat Transfer Trans-actions ASME vol 93 no 4 pp 349ndash356 1971

[34] T-T Kao ldquoLaminar free convective heat transfer response alonga vertical flat plate with step jump in surface temperaturerdquoLetters inHeat andMass Transfer vol 2 no 5 pp 419ndash428 1975

[35] S Lee and M M Yovanovich ldquoLaminar natural convectionfrom a vertical plate with a step change in wall temperaturerdquoJournal of Heat Transfer vol 113 no 2 pp 501ndash504 1991

[36] P Chandran N C Sacheti and A K Singh ldquoNatural convec-tion near a vertical plate with ramped wall temperaturerdquo Heatand Mass Transfer vol 41 no 5 pp 459ndash464 2005

[37] G S Seth M S Ansari and R Nandkeolyar ldquoMHD naturalconvection flow with radiative heat transfer past an impulsivelymoving plate with ramped wall temperaturerdquo Heat and MassTransfer vol 47 no 5 pp 551ndash561 2011

[38] G S Seth R Nandkeolyar and M S Ansari ldquoEffects ofthermal radiation and rotation on unsteady hydromagnetic freeconvection flow past an impulsively moving vertical plate withramped temperature in a porous mediumrdquo Journal of AppliedFluid Mechanics vol 6 no 1 pp 27ndash38 2013

[39] O D Makinde ldquoFree convection flow with thermal radiationand mass transfer past a moving vertical porous platerdquo Interna-tional Communications in Heat and Mass Transfer vol 32 no10 pp 1411ndash1419 2005

[40] I U Mbeledogu and A Ogulu ldquoHeat and mass transfer of anunsteady MHD natural convection flow of a rotating fluid pasta vertical porous flat plate in the presence of radiative heattransferrdquo International Journal of Heat and Mass Transfer vol50 no 9-10 pp 1902ndash1908 2007

[41] A A Afify ldquoSimilarity solution in MHD effects of thermaldiffusion and diffusion thermo on free convective heat andmass transfer over a stretching surface considering suction orinjectionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2202ndash2214 2009

[42] K Cramer and S PaiMagnetofluid Dynamics for Engineers andApplied Physicists McGraw Hill New York NY USA 1973

[43] R C Meyer ldquoOn reducing aerodynamic heat-transfer ratesby magnetohydrodynamic techniquesrdquo Journal of Aero SpaceScience vol 25 pp 561ndash572 1958

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

[13] M A Hossain and H S Takhar ldquoRadiation effect on mixedconvection along a vertical plate with uniform surface temper-aturerdquoHeat and Mass Transfer vol 31 no 4 pp 243ndash248 1996

[14] A Y Bakier and R S R Gorla ldquoThermal radiation effect onmixed convection from horizontal surfaces in saturated porousmediardquo Transport in Porous Media vol 23 no 3 pp 357ndash3631996

[15] H S Takhar R S R Gorla and VM Soundalgekar ldquoRadiationeffects onMHD free convection flowof a gas past a semi-infinitevertical platerdquo International Journal of Numerical Methods forHeat and Fluid Flow vol 6 no 2 pp 77ndash83 1996

[16] A J Chamkha ldquoThermal radiation and buoyancy effects onhydromagnetic flow over an accelerating permeable surfacewith heat source or sinkrdquo International Journal of EngineeringScience vol 38 no 15 pp 1699ndash1712 2000

[17] G E-D A Azzam ldquoRadiation effects on the MHDmixed free-forced convective flow past a semi-infinitemoving vertical platefor high temperature differencesrdquo Physica Scripta vol 66 no 1pp 71ndash76 2002

[18] C Israel-Cookey A Ogulu and V B Omubo-Pepple ldquoInflu-ence of viscous dissipation and radiation on unsteady MHDfree-convection flow past an infinite heated vertical plate ina porous medium with time-dependent suctionrdquo InternationalJournal of Heat andMass Transfer vol 46 no 13 pp 2305ndash23112003

[19] M A A Mahmoud ldquoThermal radiation effect on unsteadyMHD free convection flow past a vertical plate with tem-perature-dependent viscosityrdquo Canadian Journal of ChemicalEngineering vol 87 no 1 pp 47ndash52 2009

[20] A R Bestman and S K Adjepong ldquoUnsteady hydromagneticfree-convection flow with radiative heat transfer in a rotatingfluidmdashI Incompressible optically thin fluidrdquo Astrophysics andSpace Science vol 143 no 1 pp 73ndash80 1988

[21] A J Chamkha H S Takhar and V M Soundalgekar ldquoRadia-tion effects on free convection flow past a semi-infinite verticalplate with mass transferrdquo Chemical Engineering Journal vol 84no 3 pp 335ndash342 2001

[22] M A El-Hakiem and M F El-Amin ldquoThermal radiation effecton non-Darcy natural convection with lateral mass transferrdquoHeat and Mass Transfer vol 37 no 2-3 pp 161ndash165 2001

[23] V R Prasad N B Reddy and R Muthucumaraswamy ldquoRadia-tion and mass transfer effects on two-dimensional flow past animpulsively started infinite vertical platerdquo International Journalof Thermal Sciences vol 46 no 12 pp 1251ndash1258 2007

[24] O D Makinde and A Ogulu ldquoThe effect of thermal radiationon the heat and mass transfer flow of a variable viscosity fluidpast a vertical porous plate permeated by a transverse magneticfieldrdquo Chemical Engineering Communications vol 195 no 12pp 1575ndash1584 2008

[25] SMukhopadhyay andG C Layek ldquoEffects of thermal radiationand variable fluid viscosity on free convective flow and heattransfer past a porous stretching surfacerdquo International Journalof Heat andMass Transfer vol 51 no 9-10 pp 2167ndash2178 2008

[26] D Pal ldquoHeat andmass transfer in stagnation-point flow towardsa stretching surface in the presence of buoyancy force andthermal radiationrdquoMeccanica vol 44 no 2 pp 145ndash158 2009

[27] A J Chamkha ldquoMHDflowof a uniformly streched vertical per-meable surface in the presence of heat generationabsorptionand a chemical reactionrdquo International Communications inHeatand Mass Transfer vol 30 no 3 pp 413ndash422 2003

[28] A A Afify ldquoMHD free convective flow and mass transferover a stretching sheet with chemical reactionrdquo Heat and MassTransfer vol 40 no 6-7 pp 495ndash500 2004

[29] A Postelnicu ldquoInfluence of chemical reaction on heat andmasstransfer by natural convection from vertical surfaces in porousmedia considering Soret and Dufour effectsrdquo Heat and MassTransfer vol 43 no 6 pp 595ndash602 2007

[30] R Kandasamy and P G Palanimani ldquoEffects of chemical reac-tions heat and mass transfer on nonlinear magnetohydrody-namic boundary layer flow over a wedge with a porous mediumin the presence of ohmic heating and viscous dissipationrdquoJournal of Porous Media vol 10 no 5 pp 489ndash501 2007

[31] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[32] A A Hayday D A Bowlus and R A McGraw ldquoFree convec-tion from a vertical plate with step discontinuities in surfacetemperaturerdquo ASME Journal of Heat Transfer vol 89 pp 244ndash250 1967

[33] M Kelleher ldquoFree convection from a vertical plate with dis-continuous wall temperaturerdquo Journal of Heat Transfer Trans-actions ASME vol 93 no 4 pp 349ndash356 1971

[34] T-T Kao ldquoLaminar free convective heat transfer response alonga vertical flat plate with step jump in surface temperaturerdquoLetters inHeat andMass Transfer vol 2 no 5 pp 419ndash428 1975

[35] S Lee and M M Yovanovich ldquoLaminar natural convectionfrom a vertical plate with a step change in wall temperaturerdquoJournal of Heat Transfer vol 113 no 2 pp 501ndash504 1991

[36] P Chandran N C Sacheti and A K Singh ldquoNatural convec-tion near a vertical plate with ramped wall temperaturerdquo Heatand Mass Transfer vol 41 no 5 pp 459ndash464 2005

[37] G S Seth M S Ansari and R Nandkeolyar ldquoMHD naturalconvection flow with radiative heat transfer past an impulsivelymoving plate with ramped wall temperaturerdquo Heat and MassTransfer vol 47 no 5 pp 551ndash561 2011

[38] G S Seth R Nandkeolyar and M S Ansari ldquoEffects ofthermal radiation and rotation on unsteady hydromagnetic freeconvection flow past an impulsively moving vertical plate withramped temperature in a porous mediumrdquo Journal of AppliedFluid Mechanics vol 6 no 1 pp 27ndash38 2013

[39] O D Makinde ldquoFree convection flow with thermal radiationand mass transfer past a moving vertical porous platerdquo Interna-tional Communications in Heat and Mass Transfer vol 32 no10 pp 1411ndash1419 2005

[40] I U Mbeledogu and A Ogulu ldquoHeat and mass transfer of anunsteady MHD natural convection flow of a rotating fluid pasta vertical porous flat plate in the presence of radiative heattransferrdquo International Journal of Heat and Mass Transfer vol50 no 9-10 pp 1902ndash1908 2007

[41] A A Afify ldquoSimilarity solution in MHD effects of thermaldiffusion and diffusion thermo on free convective heat andmass transfer over a stretching surface considering suction orinjectionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2202ndash2214 2009

[42] K Cramer and S PaiMagnetofluid Dynamics for Engineers andApplied Physicists McGraw Hill New York NY USA 1973

[43] R C Meyer ldquoOn reducing aerodynamic heat-transfer ratesby magnetohydrodynamic techniquesrdquo Journal of Aero SpaceScience vol 25 pp 561ndash572 1958

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of