research article similarity solution of heat and mass...

Hindawi Publishing CorporationISRNThermodynamicsVolume 2013 Article ID 790604 10 pageshttpdxdoiorg1011552013790604

Research ArticleSimilarity Solution of Heat and Mass Transfer forNatural Convection over a Moving Vertical Plate with InternalHeat Generation and a Convective Boundary Condition in thePresence of Thermal Radiation Viscous Dissipation andChemical Reaction

S Mohammed Ibrahim1 and N Bhashar Reddy2

1 Department of Mathematics Priyadarshini College of Engineering and Technology Nellore 524004 India2Department of Mathematics Sri Venkateswara University Tirupat 517501 India

Correspondence should be addressed to S Mohammed Ibrahim ibrahimsvugmailcom

Received 14 April 2013 Accepted 7 July 2013

Academic Editors R R Burnette T M Inerbaev and P Trens

Copyright copy 2013 S M Ibrahim and N Bhashar Reddy This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Steady laminar natural convection flow over a semi-infinite moving vertical plate with internal heat generation and convectivesurface boundary condition in the presence of thermal radiation viscous dissipation and chemical reaction is examined in thispaper In the analysis we assumed that the left surface of the plate is in contact with a hot fluidwhile the cold fluid on the right surfaceof the plate contains a heat source that decays exponentially with the classical similarity variable We utilized similarity variableto transform the governing nonlinear partial differential equations into a system of ordinary differential equations which aresolved numerically by applying shooting iteration technique along fourth-order Runge-Kutta method The effects of the local Biotnumber Prandtl number buoyancy forces the internal heat generation the thermal radiation Eckert number viscous dissipationand chemical reaction on the velocity temperature and concentration profiles are illustrated and interpreted in physical terms Acomparison with previously published results on the similar special cases showed an excellent agreement Finally numerical valuesof physical quantities such as the local skin-friction coefficient the local Nusselt number and the local Sherwood number arepresented in tabular form

1 Introduction

Convective flows with simultaneous heat and mass transferunder the influence of the chemical reaction arise in manytransport processes both naturally and artificially in manybranches of science and engineering applications This phe-nomenon plays an important role in the chemical industrypower and cooling industry for drying chemical vapourdeposition on surfaces cooling of nuclear reactors andpetroleum industries

Natural convection flow occurs frequently in nature Itoccurs due to temperature differences as well as due toconcentration differences or the combination of these twofor example in atmospheric flows there exist differences in

water concentration and hence the flow is influenced by suchconcentration difference

Changes in fluid density gradients may be caused bynonreversible chemical reaction in the system as well as bythe differences in the molecular weight between values ofthe reactants and the products Chemical reactions can bemodeled as either homogenous or heterogeneous processesThis depends on whether they occur at an interface or asa single phase value reaction A homogeneous reaction isone that occurs uniformly throughout a given phase Onthe other hand a heterogeneous reaction takes place in arestricted area or within the boundary of a phase In mostcases of chemical reaction the reaction rate depends on theconcentration of the species itself A reaction is said to be of

2 ISRNThermodynamics

first order if the rate of reaction is directly proportional to theconcentration itself (Cussler [1]) For example the formationof smog is a first-order homogeneous reaction Consider theemission of nitrogen dioxide from automobiles and othersmog stacks This nitrogen dioxide reacts chemically in theatmosphere with unburned hydrocarbons (aided by sunlight)and produces peroxyacetylnitrate which forms an envelopewhich is turned photochemical smog The study of heat andmass transfer with chemical reaction is of great practicalimportance in many branches of science and engineeringDas et al [2] studied the effects of mass transfer flow past animpulsively started infinite vertical plate with constant heatflux and chemical reaction Anjalidevi and Kandasamy [3]found the effects of chemical reaction heat andmass transferon laminar flow along a semi-infinite horizontal plate Morerecently intensive studies have been carried out to investigatethe effects of chemical reaction and different flow types (seeSeddeek et al [4] Salem and Abd El-Aziz [5] Mohamed [6]and Ibrahim et al [7])

The study of heat generation or absorption inmoving flu-ids is important in problems dealing with chemical reactionsand those concernedwith dissociating fluidsHeat generationeffects may alter the temperature distribution and these inturn can affect the particle deposition rate in nuclear reactorselectronic chips and semiconductorrsquos wafers Although exactmodeling of internal heat generation or absorption is quitedifficult some simple mathematical models can be used toexpress its general behavior formost physical situations Heatgeneration or absorption can be assumed to be constantspace dependent or temperature dependent Crepeau andClarksean [8] have used a space-dependent exponentiallydecaying heat generation or absorption in their study onflow and heat transfer from vertical plate Several interestingcomputational studies of reactiveMHD boundary layer flowswith heat andmass transfer in the presence of heat generationor absorption have appeared in recent years (see Patil andKulkarni [9] Salem and Abd El-Aziz [5] Mohamed [6] andMahdy [10])

Convective heat transfer studies are very important inprocesses involving high temperatures such as gas turbinesnuclear plants and thermal energy storage Ishak [11] exam-ined the similarity solutions for flow and heat transfer overa permeable surface with convective boundary conditionMoreover Aziz [12] studied a similarity solution for laminarthermal boundary layer over a flat plate with a convectivesurface boundary condition and also studied hydrodynamicand thermal slip flow boundary layers over a flat plate with aconstant heat flux boundary condition Makinde and Olan-rewaju [13] investigated the buoyancy effects on a thermalboundary layer over a vertical plate with a convective surfaceboundary condition More recently Makinde [14] studiedsimilarity solution for natural convection from a movingvertical plate with internal heat generation and a convectiveboundary condition Olanrewaju et al [15] examined theeffects of internal heat generation thermal radiation andbuoyancy force on a boundary layer over a vertical platewith a convective surface boundary condition Makinde andOlanrewaju [16] investigated the combined effects of internalheat generation and buoyancy force on boundary layer flow

over a vertical plate with a convective surface boundarycondition

Viscous dissipation changes the temperature distribu-tions by playing a role like an energy source which leadsto the affected heat transfer rates The merit of the effect ofviscous dissipation depends on whether the plate is beingcooled or heated Heat transfer analysis over porous surfaceis of much practical interest due to its abundant applicationsTo be more specific heat-treated materials traveling betweena feed roll and wind-up roll or materials manufacturedby extrusion glass-fiber and paper production cooling ofmetallic sheets or electronic chips and crystal growingjust to name a few In these cases the final product ofdesired characteristics depends on the rate of cooling in theprocess and the process of stretching The work of Sonthet al [17] deals with the effect of the viscous dissipationterm along with temperature-dependent heat sourcesink onmomentum and heat and mass transfer in a viscoelastic fluidflow over an accelerating surface Chen [18] examined theeffect of combined heat and mass transfer on MHD-freeconvection from a vertical surface with ohmic heating andviscous dissipationThe effect of viscous dissipation and Jouleheating on MHD-free convection flow past a semi-infinitevertical flat plate in the presence of the combined effect ofHall and nonslip currents for the case of power-law variationof the wall temperature is analyzed by Abo-Eldahab and ElAziz [19]

In many new engineering areas processes such as fossilfuel combustion energy processes solar power technologyastrophysical flows gas turbines and the various propulsiondevices for aircraft missiles satellites and space vehiclereentry occur at high temperatures so knowledge of radiationheat transfer beside the convective heat transfer plays a veryimportant role and hence its effect cannot be neglected Alsothermal radiation is of major importance in many processesin engineering areas which occur at a very high temperaturefor the design of many advanced energy conversion systemsand pertinent equipment The Rosseland approximation isused to describe the radiative heat flux in the energy equa-tion Pal and Mondal [20] investigate the unsteady two-dimensional MHD non-Darcian mixed convection heat andmass transfer past a semi-infinite vertical permeable plateembedded in a porous medium by taking into account Soretand Dufour effects in the presence of suction or injectionthermal radiation and first-order chemical reaction Uwanta[21] studied the effects of chemical reaction and radiation onheat and mass transfer past a semi-infinite vertical porousplate with constant mass flux and dissipation Olanrewaju etal [22] found the effects of internal heat generation thermalradiation and buoyancy force on a boundary layer over avertical plate with a convective surface boundary condition

The objective of this paper was to explore the effectsof thermal radiation heat generation viscous dissipationand chemical reaction on the similarity solution for naturalconvection from a moving vertical plate under a convectiveboundary condition which is an extension of Makinde[14] with the addition of thermal radiation viscous dissi-pation and chemical reaction parameter for more physicalimplications Using a similarity approach the governing

ISRNThermodynamics 3

equations are transformed into nonlinear ordinary equationsand solved numerically using shooting iteration techniquetogether with fourth-order Runge-Kutta integration schemeThe pertinent results are displayed graphically and discussedquantitatively

2 Mathematical Formulation

We consider the steady laminar incompressible naturalconvection boundary layer flows over the right surface ofa vertical flat plate moving with uniform velocity 119880

0in

contact with a quiescence cold fluid at temperature 119879infin

andconcentration 119862

infin The cold fluid on the right surface of the

plate generates heat internally at the volumetric rate 119902The leftsurface of the plate is heated by convection from a hot fluid attemperature 119879

119891which provides a heat transfer coefficient ℎ

119891

as shown in Figure 1 Under the Boussinesq for fluid densityvariation the continuity momentum energy equation andmass diffusion equations describing the flow can be writtenas

120597119906

120597119909+120597V

120597119910= 0 (1)

119906120597119906

120597119909+ V

120597119906

120597119910= ]

1205972119906

1205971199102+ 119892120573 (119879 minus 119879

infin) + 119892120573

lowast(119862 minus 119862

infin) (2)

120588119862119901(119906

120597119906

120597119909+ V

120597119906

120597119910) = 119896

1205972119879

1205971199102+ 119902 minus

120597119902119903

120597119910+ 120583(

120597119906

120597119910)

2

(3)

119906120597119862

120597119909+ V

120597119862

120597119910= 119863

1205972119862

1205971199102minus 1198701015840

119903(119862 minus 119862

infin) (4)

where 119906 and V are the 119909 (along the plate) and 119910 (normalto the plate) components of the velocities respectively 119879is the temperature 119862 is the concentration 120583 is the fluidviscosity V is the kinematics viscosity of the fluid 119896 is thethermal conductivity of the fluid 120573 is the thermal expansioncoefficient 120573lowast is concentration expansion coefficient 119902 isthe internally generated heat at volumetric rate 119892 is thegravitational acceleration 119902

119903is the radiative heat flux 119863 is

the diffusion coefficient and 1198701015840

119903is the chemical reaction

parameterThe boundary conditions at the plate surface and for the

cold fluid may be written as

119906 (119909 0) = 1198800 V (119909 0) = 0

minus119870120597119879

120597119910(119909 0) = ℎ

119891[119879119891minus 119879 (119909 0)]

119862119891(119909 0) = 119860119909

120582+ 119862infin

119906 (119909infin) = 0 119879 (119909infin) = 119879infin 119862 (119909infin) = 119862

infin

(5)

g

y

x

Tinfin

Cinfin

u T C

Cf = Ax120582 + Cinfin

minusk120597T

120597y= hf(Tf minus T)

u = U0 = 0

q

Figure 1 Flow configuration and coordinate system

By using the Rosseland diffusion approximation Hossain etal [23] and following Raptis [24] among other researchers theradiative heat flux 119902

119903is given by

119902119903= minus

4120590lowast

3119870119904

1205971198794

120597119910 (6)

where 120590lowast and 119870

119904are the Stefan-Boltzmann constant and

the mean absorption coefficient respectively FollowingChamkha [25] we assume that the temperature differenceswithin the flow are sufficiently small so that 1198794 can beexpressed as a linear function after using the Taylor seriesto expand 119879

4 about the free stream temperature 119879infin

andneglecting higher-order terms This result is the followingapproximation

1198794asymp 41198793

infin119879 minus 3119879

4

infin (7)

Using (6) and (7) in (3) we obtain

120597119902119903

120597119910= minus

16120590lowast

3119870

12059721198794

1205971199102 (8)

Introducing a similarity variable 120578 and a dimensionlessstream function 119891(120578) temperature 120579(120578) and concentration120601(120578) as

120578 = 119910radic1198800

V119909=119910

119909radicRe119909

119906

1198800

= 1198911015840 V =

1

2119909radicRe119909(1205781198911015840minus 119891)

120579 (120578) =119879 minus 119879infin

119879119891minus 119879infin

120601 (120578) =119862 minus 119862

infin

119862119891minus 119862infin

(9)

where prime symbol denotes differentiation with respect to120578 and Re

119909= 1198800119909] is the local Reynolds number These

nonlinear partial differential equations are then transformed


by similarity transformation into a system of ordinary differ-ential equations given as

119891101584010158401015840+1

21198911198911015840+ Gr 120579 + Gc120601 = 0

12057910158401015840[1 +

4

3119877] +

1

2Pr1198911205791015840 + Pr119876119890minus120578 + Ec Pr(11989110158401015840)

2

= 0

12060110158401015840+1

2Sc1198911206011015840 minus Kr Sc120601 = 0

119891 (0) = 0 1198911015840(0) = 1

1205791015840(0) = minusBi [1 minus 120579 (0)] 120601 (0) = 1

1198911015840(infin) = 1 120579 (infin) = 0 120601 (infin) = 0

(10)

where

Bi =ℎ119891

119896radic]119909

1198800

Pr = ]

120572

Gr =119909119892120573 (119879

119891minus 119879infin)

1198802

0

Gc =119909119892120573lowast(119862119891minus 119862infin)

1198802

0

Ra =4120572

lowast

1205901198793

infin

119896119870 119876 =

1199092119902119890120578

119896Re119909(119879119891minus 119879infin)

Ec =1198802

0

119896 (119879119891minus 119879infin)

Sc = ]

119863 Kr = Kr1015840119909

1198800

(11)

Bi is the local Biot number Pr is the Prandtl numberGr is local Grashof number Gc is modified local Grashofnumber Ra is the radiation parameter Q is the internal heatgeneration parameter Ec is the Eckert number Sc is theSchmidt number and Kr is the chemical reaction parameter

For the momentum and energy equations to have asimilarity solution the parameters Gr Gc 119876 and Bi

119909must

be constants and not functions of 119909 as in (11) This conditioncan be met if the heat transfer coefficient ℎ

119891is proportional

to 119909minus12 the thermal expansion coefficient 120573 is proportionalto 119909minus1 and the internal generation 119902 is proportional to 119909minus1We therefore assume

ℎ119891= 119888119909minus12

120573 = 119898119909minus1119902 = 119897119909

minus1119890minus120578 (12)

where 119888119898 and 119897 are constants Substituting (12) into (13) wehave

Bi = 119888

119896radic

]

1198800

Gr =

119898119892 (119879119891minus 119879infin)

1198802

0

Gc =

119898119892 (119862119891minus 119862infin)

1198802

0

119876 =119897]

1198961198800(119879119891minus 119879infin)

(13)

With Bi 119876 and Gr Gc is defined by (13) The solutionsof (10) yield the similarity solutions However the solutions

generated are the local similarity solutions whenever 119894 119876Grand Gc are defined as in (13)

The coupled nonlinear boundary value problems repre-sented by (10) have been solved numerically using the shoot-ing techniques with the fourth-order Runge-Kutta methodFrom the numerical computations the plate surface temper-ature local skin-friction coefficient the localNusselt numberand the local Sherwood number which are respectivelyproportional to 120579(0) 119891

10158401015840(0) minus120579

1015840(0) and minus120601

1015840(0) are worked

out and their numerical values are presented in a tabularform

3 Results and Discussion

To analyze the results numerical computation has beencarried out using the method described in the previousparagraph for various governing parameters namely thermalGrashof number Gr modified Grashof number Gc Prandtlnumber Pr thermal radiation parameter 119877 heat generationparameter 119876 Eckert number Ec Schmidt number Sc chem-ical reaction parameter Kr and convective parameter Bi Inthe present study the following default parameter values areadopted for computations Gr = 10 Gc = 10 Pr = 072119877 = 05 119876 = 05 Ec = 05 Sc = 06 Kr = 0 and Bi =01 All graphs therefore correspond to these values unlessspecifically indicated on the appropriate graph

Table 1 shows the comparison of Makinde [14] work withthe present work for Ec = 119877 = Sc = Kr = Gc = 0 andit is noteworthy that there is a perfect agreement Table 2shows the values of the skin-friction coefficient the Nusseltnumber the surface temperature and the Sherwood numberin terms of 11989110158401015840(0) 1205791015840(0) 120579(0) and 120601

1015840(0) respectively for

various values embedded flow parameters From Table 2 itis understood that the skin friction the rate of heat transferwall surface temperature at the plate surface and the rate ofmass transfer increase with an increase in local Biot numberAn increase in buoyancy forces thermal radiation internalheat generation Eckert number there is an increase in skin-friction surface temperature and the Sherwood number butdecrease in the Nusselt number An increase in the Prandtlnumber there is decrease in skin friction surface temperatureand the Sherwood number but increases the Nusselt numberHowever an increase in the Schmidt number and chemicalreaction parameter causes a decrease in the skin friction theNusselt number and surface temperature and increase in thesurface mass transfer rate that is the Sherwood number

31 Velocity Profiles Figures 2ndash10 depict the effects of variousthermophysical parameters on the fluid velocity profile Itwas observed that generally the fluid velocity increasesgradually away from the plate attain its peak value withinthe boundary layer and the decreases to the free stream zerovalue satisfying the boundary conditions From Figures 2 and3 we observed that the velocity boundary layer thicknessincreases with an increase in the values of local Grashofnumber (Gr) andmodified local Grashof number (Gc) due tobuoyancy effect In Figure 4 the influence of Prandtl numberon the fluid velocity was displayed and it is interesting to


Table 1 Computations showing comparison with Makinde [14] results for Gc = 0 Ec = 0 119877 = 0 Sc = 0 and 119870119903= 0

Bi Gr Pr Q 11989110158401015840(0)Makinde [14] 120579

1015840(0)Makinde [14] 120579(0)Makinde [14] 119891

10158401015840(0) present 120579

1015840(0) present 120579(0) present

01 01 072 10 minus02000518 0076578477 176578477 minus0253226 00353022 13530210 01 072 10 minus02459676 0281651449 128165144 minus0279242 0128217 11282210 01 072 10 minus02695171 0382952717 103829527 minus0280211 0173623 10173601 05 072 10 04221216 0048257030 148257030 0250851 0016076 11607601 10 072 10 09895493 0034011263 134011263 0717892 000569792 10569801 01 30 10 minus03748695 minus0023814576 076185423 minus026024 0100968 20096901 01 710 10 minus04138825 minus0057164001 042835998 minus0258586 0153863 25386301 01 072 5 03741286 0576670381 676670381 0198617 041344 51344401 01 072 10 09010790 1106605802 120660580 0620946 0818126 918126

Table 2 Computation showing 11989110158401015840(0) 1205791015840(0) 120579(0) and 1206011015840(0) for different embedded flow parameter values

Bi Gr Gc Pr R Q Ec Sc 119870119903

11989110158401015840(0) 120579

1015840(0) 120579(0) 120601

1015840(0)

01 10 10 10 05 01 01 06 05 0617935 00690068 0309932 069090410 10 10 10 05 01 01 06 05 0995463 024145 075855 0705335100 10 10 10 05 01 01 06 05 116294 0325667 0967433 07113701 20 10 10 05 01 01 06 05 0914867 00688259 0311741 070301701 30 10 10 05 01 01 06 05 121966 00680544 0319456 071474701 10 20 10 05 01 01 06 05 127296 0067557 032443 071175501 10 30 10 05 01 01 06 05 18969 0064555 035445 072982301 10 10 10 05 01 01 06 05 0598557 00696409 0303591 068967901 10 10 30 05 01 01 06 05 0537444 0070888 029112 068573601 10 10 10 10 01 01 06 05 0635229 00684205 0315795 069199501 10 10 10 15 01 01 06 05 0645997 00680532 0319468 069267601 10 10 10 05 02 01 06 05 0694807 00621171 0378829 069427101 10 10 10 05 03 01 06 05 0769222 00553443 0446557 069747101 10 10 10 05 01 02 06 05 0654059 00662619 0337381 069265301 10 10 10 05 01 03 06 05 0693151 00632311 0367689 069452101 10 10 10 05 01 01 078 05 0562746 00687271 0312729 078661601 10 10 10 05 01 01 10 05 0511741 0068638 0311362 088992801 10 10 10 05 01 01 06 10 0514718 00688634 0311366 087701101 10 10 10 05 01 01 06 15 0488687 00687467 0312533 103195

note that velocity boundary layer thickness decreases withan increase in the Prandtl number Figure 5 depicts theinfluence of thermal radiation on the fluid velocity and itis interesting to note that increases the radiation parameterthickness the velocity boundary layer thickness away fromthe plate surface Figure 6 depicts the effects of local internalheat generation parameter on the fluid velocity An increasein the exponentially decaying internal heat generation causesa further increase in the velocity boundary layer thicknessFigure 7 represents the curve of fluid velocity against span-wise co-ordinate 120578 for various values of Eckert number whichshows that the increase in Eckert number leads to a suddenincrease in the fluid velocity immediately away from thewall plate before satisfying the boundary conditions It isinteresting to note that it thickens the velocity boundarylayer thickness close to the wall plate when the velocityprofile attains itsmaximumvalue point Figure 8 and Figure 9show the variation of the boundary layer velocity with theSchmidt number and chemical reaction parameter From this

Figures we observed a slight decrease in the fluid velocitywith an increase in Schmidt number and chemical reactionparameter Figure 10 depicts the effects of the variation ofthe boundary layer velocity with the intensity of local Biotnumber (Bi) From this it was observed that the velocityboundary layer thicknesses slightly increase with an increasein the local Biot number due to convective heat transfer theplate surface

32 Temperature Profiles Figures 11 12 13 14 and 15 illustratethe fluid temperature profiles within the boundary layerGenerally the fluid temperature is maximum at the platesurface and decreases exponentially to zero value far awayfrom the plate satisfying the boundary conditions From thesefigures it is noteworthy that the thermal boundary layerthickness increases with an increase in the exponentiallydecaying internal heat generation thermal radiation Eckertnumber and local Biot number and decreases with anincrease in the values of Prandtl number At high Prandtl


0 1 2 3 4 5 6 70

05

1

15

2

25

3

Gr = 05 1 15 2

120578

f998400

Figure 2 Effects of local Grashof number on velocity profile

number has low velocity which in turn also implies thatat lower fluid velocity the species diffusion is comparativelylower and hence higher species concentration is observed athigh Prandtl number

33 Concentration Profiles Figures 16 and 17 depict chemicalspecies concentration profiles against spanwise coordinate 120578for varying values of physical parameters in the boundarylayerThe species concentration is highest at the plate surfaceand decreases to zero far away from the plate satisfying theboundary condition From these figures it is noteworthy thatthe concentration boundary layer thickness decreases withan increase in the Schmidt number and chemical reactionparameter

Table 2 shows the values of the skin-friction coefficientthe Nusselt number surface temperature and the Sherwoodnumber in terms of 11989110158401015840(0) 1205791015840(0) 120579(0) and 120601

1015840(0) respec-

tively for various values of embedded flow parameters It isunderstood that the skin friction the rate of heat transferwall surface temperature at the plate surface and the rateof mass transfer increase with an increase in the local Biotnumber An increase in buoyancy forces (Gr Gc) thermalradiation internal heat generation Eckert number there is anincrease in skin-friction surface temperature and Sherwoodnumber and decrease in the Nusselt number An increase inthe Prandtl number there is decrease in skin-friction surfacetemperature and Sherwood number but increases theNusseltnumber However an increase in the Schmidt number andchemical reaction parameter causes a decrease in the skinfriction the Nusselt number and surface temperature andincreases the surface mass transfer rate

4 Conclusions

The similarity solution for natural convection from a movingvertical plate with internal heat generation and a convec-tive boundary condition in the presence of thermal radia-tion viscous dissipation and chemical reaction is studied

0

05

1

15

2

0 2 4 6 8

Gc = 05 1 15 2

120578

f998400

Figure 3 Effects of local modified Grashof number on velocityprofile

0

02

04

06

08

1

12

14

16

Pr = 072 1 3 71

0 2 4 6 8120578

f998400

Figure 4 Effects of Prandtl number on velocity profile

0

025

05

075

1

125

15

R = 05 1 2 25

0 2 4 6 8120578

f998400

Figure 5 Effects of radiation parameter on velocity profile


Q = 05 07 1 15

0 2 4 6 8120578

0

05

1

15

2

f998400

Figure 6 Effects of internal heat generation on velocity profile

Ec = 01 03 05 1

0 2 4 6 8120578

0

05

1

15

2

f998400

Figure 7 Effects of Eckert number on velocity profile

0

02

04

06

08

1

12

14

16

18

Sc = 024 06 078 1

0 2 4 6 8120578

f998400

Figure 8 Effects of Schmidt number on velocity profile

Kr = 05 1 15 2

0

02

04

06

08

1

12

14

16

0 2 4 6 8120578

f998400

Figure 9 Effects of chemical reaction parameter on velocity profile

0

025

05

075

1

125

15

Bi = 01 05 1 10

0 2 4 6 8120578

f998400

Figure 10 Effects of local Biot number on velocity profile

Pr = 072 1 3 71

0

02

04

06

08

1

12

14

16

0 2 4 6 8120578

120579

Figure 11 Effects of Prandtl number on temperature profile


0

02

04

06

08

1

0 2 4 6 8120578

R = 05 1 2 25120579

Figure 12 Effects of radiation parameter on temperature profile

2

Q = 05 07 1 15

0

02

04

06

08

1

12

14

16

18

0 2 4 6 8120578

120579

Figure 13 Effects of internal heat generation parameter on temper-ature profile

0

05

1

15

2

Ec = 01 03 05 1

0 2 4 6 8120578

120579

Figure 14 Effects of Eckert number on temperature profile

0

02

04

06

08

1

Bi = 01 05 1 10

0 2 4 6 8120578

120579

Figure 15 Effects of local Biot number on temperature profile

0

02

04

06

08

1

Sc = 024 06 078 1

0 2 4 6 8120578

120601

Figure 16 Effects of Schmidt number on concentration profile

Kr = 05 1 15 2

0

02

04

06

08

1

0 2 4 6 8120578

120601

Figure 17 Effects of chemical reaction parameter on concentrationprofile


A set of non-linear coupled differential equations governingthe fluid velocity temperature and concentration is solvednumerically for various material parameters A comprehen-sive set of graphical results for the velocity temperature andconcentration is presented and discussed Our results revealamong others that the internal heat generation thermalradiation and the Eckert number prevent the flow of heatfrom the left surface to the right surface of the plate unlessthe local Grashof number is strong enough to convert awayboth the internally generated heat in the fluid Generally thefluid velocity increases gradually away from the plate attainsits peak value within the boundary layer and decreases to thefree stream zero value satisfying the boundary conditions It isinteresting to note that the fluid velocity within the boundarylayer increases with increasing values of the exponentiallydecaying internal heat generation thermal radiation and theEckert number little away from the wall plate and attains itspeak before obeying the boundary conditions The velocityand concentration both decrease with an in increase in theSchmidt number and the chemical reaction parameter

References

[1] E L Cussler Diffusion Mass Transfer in Fluid Systems Cam-bridge University Press London UK 1998

[2] U N Das R Deka and V M Soundalgekar ldquoEffects of masstransfer on flowpast an impulsively started infinite vertical platewith constant heat flux and chemical reactionrdquo Forschung imIngenieurwesenEngineering Research vol 60 no 10 pp 284ndash287 1994

[3] S P Anjalidevi andRKandasamy ldquoEffects of chemical reactionheat and mass transfer on laminar flow along a semi infinitehorizontal platerdquoHeat andMass Transfer vol 35 no 6 pp 465ndash467 1999

[4] M A Seddeek A A Darwish andM S Abdelmeguid ldquoEffectsof chemical reaction and variable viscosity on hydromagneticmixed convection heat and mass transfer for Hiemenz flowthrough porous media with radiationrdquo Communications inNonlinear Science and Numerical Simulation vol 12 no 2 pp195ndash213 2007

[5] A M Salem and M Abd El-Aziz ldquoEffect of Hall currents andchemical reaction on hydromagnetic flow of a stretching ver-tical surface with internal heat generationabsorptionrdquo AppliedMathematical Modelling vol 32 no 7 pp 1236ndash1254 2008

[6] R A Mohamed ldquoDouble-diffusive convection-radiation inter-action on unsteady MHD flow over a vertical moving porousplate with heat generation and Soret effectsrdquoAppliedMathemat-ical Sciences vol 3 no 13-16 pp 629ndash651 2009

[7] F S IbrahimAM Elaiw andAA Bakr ldquoEffect of the chemicalreaction and radiation absorption on the unsteady MHDfree convection flow past a semi infinite vertical permeablemoving plate with heat source and suctionrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 6 pp1056ndash1066 2008

[8] J C Crepeau and R Clarksean ldquoSimilarity solutions of naturalconvection with internal heat generationrdquo Journal of HeatTransfer vol 119 no 1 pp 183ndash185 1997

[9] P M Patil and P S Kulkarni ldquoEffects of chemical reaction onfree convective flowof a polar fluid through a porousmedium inthe presence of internal heat generationrdquo International Journalof Thermal Sciences vol 47 no 8 pp 1043ndash1054 2008

[10] A Mahdy ldquoEffect of chemical reaction and heat generationor absorption on double-diffusive convection from a verticaltruncated cone in porous media with variable viscosityrdquo Inter-national Communications in Heat andMass Transfer vol 37 no5 pp 548ndash554 2010

[11] A Ishak ldquoSimilarity solutions for flow and heat transfer overa permeable surface with convective boundary conditionrdquoApplied Mathematics and Computation vol 217 no 2 pp 837ndash842 2010

[12] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009

[13] O D Makinde and P O Olanrewaju ldquoBuoyancy effects onthermal boundary layer over a vertical plate with a convectivesurface boundary conditionrdquo Journal of Fluids Engineering vol132 no 4 Article ID 044502 4 pages 2010

[14] O DMakinde ldquoSimilarity solution for natural convection froma moving vertical plate with internal heat generation and aconvective boundary conditionrdquo Thermal Science vol 15 ppS137ndashS143 2011

[15] P O Olanrewaju J A Gbadeyan T Hayat and A A HendildquoEffects of internal heat generation thermal radiation andbuoyancy force on a boundary layer over a vertical plate with aconvective surface boundary conditionrdquo South African Journalof Science vol 107 no 9-10 article 476 6 pages 2011

[16] O D Makinde and P O Olanrewaju ldquoCombined effects ofinternal heat generation and buoyancy force on boundary layerflow over a vertical plate with a convective surface boundaryconditionrdquoThe Canadian Journal of Chemical Engineering vol90 no 5 pp 1289ndash1294 2011

[17] RM Sonth S K KhanM S Abel and K V Prasad ldquoHeat andmass transfer in a visco-elastic fluid flow over an acceleratingsurface with heat sourcesink and viscous dissipationrdquoHeat andMass Transfer vol 38 no 3 pp 213ndash220 2002

[18] C-H Chen ldquoCombined heat and mass transfer in MHDfree convection from a vertical surface with Ohmic heatingand viscous dissipationrdquo International Journal of EngineeringScience vol 42 no 7 pp 699ndash713 2004

[19] E M Abo-Eldahab andM A El Aziz ldquoViscous dissipation andJoule heating effects on MHD-free convection from a verticalplate with power-law variation in surface temperature in thepresence of Hall and ion-slip currentsrdquo Applied MathematicalModelling vol 29 no 6 pp 579ndash595 2005

[20] 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

[21] I J Uwanta ldquoEffects of chemical reaction and radiation onheat and mass transfer past a semi-infinite vertical porous platewith constant mass flux and dissipationrdquo European Journal ofScientific Research vol 87 no 2 pp 190ndash200 2012

[22] P O Olanrewaju J A Gbadeyan T Hayat and A A HendildquoEffects of internal heat generation thermal radiation andbuoyancy force on a boundary layer over a vertical plate with aconvective surface boundary conditionrdquo South African Journalof Science vol 107 no 9-10 pp 1ndash6 2011

[23] M A Hossain M A Alim and D A S Rees ldquoThe effectof radiation on free convection from a porous vertical platerdquoInternational Journal of Heat and Mass Transfer vol 42 no 1pp 181ndash191 1998


[24] A Raptis ldquoFlow of a micropolar fluid past a continuouslymoving plate by the presence of radiationrdquo International Journalof Heat and Mass Transfer vol 41 no 18 pp 2865ndash2866 1998

[25] A J Chamkha ldquoHydromagnetic natural convection from anisothermal inclined surface adjacent to a thermally stratifiedporous mediumrdquo International Journal of Engineering Sciencevol 35 no 10-11 pp 975ndash986 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


Physics Research International


Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


first order if the rate of reaction is directly proportional to theconcentration itself (Cussler [1]) For example the formationof smog is a first-order homogeneous reaction Consider theemission of nitrogen dioxide from automobiles and othersmog stacks This nitrogen dioxide reacts chemically in theatmosphere with unburned hydrocarbons (aided by sunlight)and produces peroxyacetylnitrate which forms an envelopewhich is turned photochemical smog The study of heat andmass transfer with chemical reaction is of great practicalimportance in many branches of science and engineeringDas et al [2] studied the effects of mass transfer flow past animpulsively started infinite vertical plate with constant heatflux and chemical reaction Anjalidevi and Kandasamy [3]found the effects of chemical reaction heat andmass transferon laminar flow along a semi-infinite horizontal plate Morerecently intensive studies have been carried out to investigatethe effects of chemical reaction and different flow types (seeSeddeek et al [4] Salem and Abd El-Aziz [5] Mohamed [6]and Ibrahim et al [7])

The study of heat generation or absorption inmoving flu-ids is important in problems dealing with chemical reactionsand those concernedwith dissociating fluidsHeat generationeffects may alter the temperature distribution and these inturn can affect the particle deposition rate in nuclear reactorselectronic chips and semiconductorrsquos wafers Although exactmodeling of internal heat generation or absorption is quitedifficult some simple mathematical models can be used toexpress its general behavior formost physical situations Heatgeneration or absorption can be assumed to be constantspace dependent or temperature dependent Crepeau andClarksean [8] have used a space-dependent exponentiallydecaying heat generation or absorption in their study onflow and heat transfer from vertical plate Several interestingcomputational studies of reactiveMHD boundary layer flowswith heat andmass transfer in the presence of heat generationor absorption have appeared in recent years (see Patil andKulkarni [9] Salem and Abd El-Aziz [5] Mohamed [6] andMahdy [10])

Convective heat transfer studies are very important inprocesses involving high temperatures such as gas turbinesnuclear plants and thermal energy storage Ishak [11] exam-ined the similarity solutions for flow and heat transfer overa permeable surface with convective boundary conditionMoreover Aziz [12] studied a similarity solution for laminarthermal boundary layer over a flat plate with a convectivesurface boundary condition and also studied hydrodynamicand thermal slip flow boundary layers over a flat plate with aconstant heat flux boundary condition Makinde and Olan-rewaju [13] investigated the buoyancy effects on a thermalboundary layer over a vertical plate with a convective surfaceboundary condition More recently Makinde [14] studiedsimilarity solution for natural convection from a movingvertical plate with internal heat generation and a convectiveboundary condition Olanrewaju et al [15] examined theeffects of internal heat generation thermal radiation andbuoyancy force on a boundary layer over a vertical platewith a convective surface boundary condition Makinde andOlanrewaju [16] investigated the combined effects of internalheat generation and buoyancy force on boundary layer flow

over a vertical plate with a convective surface boundarycondition

Viscous dissipation changes the temperature distribu-tions by playing a role like an energy source which leadsto the affected heat transfer rates The merit of the effect ofviscous dissipation depends on whether the plate is beingcooled or heated Heat transfer analysis over porous surfaceis of much practical interest due to its abundant applicationsTo be more specific heat-treated materials traveling betweena feed roll and wind-up roll or materials manufacturedby extrusion glass-fiber and paper production cooling ofmetallic sheets or electronic chips and crystal growingjust to name a few In these cases the final product ofdesired characteristics depends on the rate of cooling in theprocess and the process of stretching The work of Sonthet al [17] deals with the effect of the viscous dissipationterm along with temperature-dependent heat sourcesink onmomentum and heat and mass transfer in a viscoelastic fluidflow over an accelerating surface Chen [18] examined theeffect of combined heat and mass transfer on MHD-freeconvection from a vertical surface with ohmic heating andviscous dissipationThe effect of viscous dissipation and Jouleheating on MHD-free convection flow past a semi-infinitevertical flat plate in the presence of the combined effect ofHall and nonslip currents for the case of power-law variationof the wall temperature is analyzed by Abo-Eldahab and ElAziz [19]

In many new engineering areas processes such as fossilfuel combustion energy processes solar power technologyastrophysical flows gas turbines and the various propulsiondevices for aircraft missiles satellites and space vehiclereentry occur at high temperatures so knowledge of radiationheat transfer beside the convective heat transfer plays a veryimportant role and hence its effect cannot be neglected Alsothermal radiation is of major importance in many processesin engineering areas which occur at a very high temperaturefor the design of many advanced energy conversion systemsand pertinent equipment The Rosseland approximation isused to describe the radiative heat flux in the energy equa-tion Pal and Mondal [20] investigate the unsteady two-dimensional MHD non-Darcian mixed convection heat andmass transfer past a semi-infinite vertical permeable plateembedded in a porous medium by taking into account Soretand Dufour effects in the presence of suction or injectionthermal radiation and first-order chemical reaction Uwanta[21] studied the effects of chemical reaction and radiation onheat and mass transfer past a semi-infinite vertical porousplate with constant mass flux and dissipation Olanrewaju etal [22] found the effects of internal heat generation thermalradiation and buoyancy force on a boundary layer over avertical plate with a convective surface boundary condition

The objective of this paper was to explore the effectsof thermal radiation heat generation viscous dissipationand chemical reaction on the similarity solution for naturalconvection from a moving vertical plate under a convectiveboundary condition which is an extension of Makinde[14] with the addition of thermal radiation viscous dissi-pation and chemical reaction parameter for more physicalimplications Using a similarity approach the governing





0in






119891


120597119906

120597119909+120597V

120597119910= 0 (1)

119906120597119906

120597119909+ V

120597119906

120597119910= ]

1205972119906

1205971199102+ 119892120573 (119879 minus 119879

infin) + 119892120573


infin) (2)

120588119862119901(119906

120597119906

120597119909+ V

120597119906

120597119910) = 119896

1205972119879

1205971199102+ 119902 minus

120597119902119903

120597119910+ 120583(

120597119906

120597119910)

2

(3)

119906120597119862

120597119909+ V

120597119862

120597119910= 119863

1205972119862

1205971199102minus 1198701015840

119903(119862 minus 119862

infin) (4)







119906 (119909 0) = 1198800 V (119909 0) = 0

minus119870120597119879

120597119910(119909 0) = ℎ

119891[119879119891minus 119879 (119909 0)]

119862119891(119909 0) = 119860119909

120582+ 119862infin


infin

(5)

g

y

x

Tinfin

Cinfin

u T C


minusk120597T


u = U0 = 0

q



119903is given by

119902119903= minus

4120590lowast

3119870119904

1205971198794

120597119910 (6)






1198794asymp 41198793


4

infin (7)


120597119902119903

120597119910= minus

16120590lowast

3119870

12059721198794

1205971199102 (8)


120578 = 119910radic1198800

V119909=119910

119909radicRe119909

119906

1198800

= 1198911015840 V =

1

2119909radicRe119909(1205781198911015840minus 119891)

120579 (120578) =119879 minus 119879infin

119879119891minus 119879infin

120601 (120578) =119862 minus 119862

infin

119862119891minus 119862infin

(9)






119891101584010158401015840+1

21198911198911015840+ Gr 120579 + Gc120601 = 0

12057910158401015840[1 +

4

3119877] +

1

2Pr1198911205791015840 + Pr119876119890minus120578 + Ec Pr(11989110158401015840)

2

= 0

12060110158401015840+1

2Sc1198911206011015840 minus Kr Sc120601 = 0

119891 (0) = 0 1198911015840(0) = 1

1205791015840(0) = minusBi [1 minus 120579 (0)] 120601 (0) = 1


(10)

where

Bi =ℎ119891

119896radic]119909

1198800

Pr = ]

120572

Gr =119909119892120573 (119879


1198802

0


1198802

0

Ra =4120572

lowast

1205901198793

infin

119896119870 119876 =

1199092119902119890120578

119896Re119909(119879119891minus 119879infin)

Ec =1198802

0

119896 (119879119891minus 119879infin)

Sc = ]

119863 Kr = Kr1015840119909

1198800

(11)



119909must




ℎ119891= 119888119909minus12

120573 = 119898119909minus1119902 = 119897119909

minus1119890minus120578 (12)


Bi = 119888

119896radic

]

1198800

Gr =

119898119892 (119879119891minus 119879infin)

1198802

0

Gc =

119898119892 (119862119891minus 119862infin)

1198802

0

119876 =119897]

1198961198800(119879119891minus 119879infin)

(13)




10158401015840(0) minus120579

1015840(0) and minus120601











Bi Gr Pr Q 11989110158401015840(0)Makinde [14] 120579

1015840(0)Makinde [14] 120579(0)Makinde [14] 119891

10158401015840(0) present 120579

1015840(0) present 120579(0) present




11989110158401015840(0) 120579

1015840(0) 120579(0) 120601

1015840(0)

01 10 10 10 05 01 01 06 05 0617935 00690068 0309932 069090410 10 10 10 05 01 01 06 05 0995463 024145 075855 0705335100 10 10 10 05 01 01 06 05 116294 0325667 0967433 07113701 20 10 10 05 01 01 06 05 0914867 00688259 0311741 070301701 30 10 10 05 01 01 06 05 121966 00680544 0319456 071474701 10 20 10 05 01 01 06 05 127296 0067557 032443 071175501 10 30 10 05 01 01 06 05 18969 0064555 035445 072982301 10 10 10 05 01 01 06 05 0598557 00696409 0303591 068967901 10 10 30 05 01 01 06 05 0537444 0070888 029112 068573601 10 10 10 10 01 01 06 05 0635229 00684205 0315795 069199501 10 10 10 15 01 01 06 05 0645997 00680532 0319468 069267601 10 10 10 05 02 01 06 05 0694807 00621171 0378829 069427101 10 10 10 05 03 01 06 05 0769222 00553443 0446557 069747101 10 10 10 05 01 02 06 05 0654059 00662619 0337381 069265301 10 10 10 05 01 03 06 05 0693151 00632311 0367689 069452101 10 10 10 05 01 01 078 05 0562746 00687271 0312729 078661601 10 10 10 05 01 01 10 05 0511741 0068638 0311362 088992801 10 10 10 05 01 01 06 10 0514718 00688634 0311366 087701101 10 10 10 05 01 01 06 15 0488687 00687467 0312533 103195





0 1 2 3 4 5 6 70

05

1

15

2

25

3

Gr = 05 1 15 2

120578

f998400





1015840(0) respec-


4 Conclusions


0

05

1

15

2

0 2 4 6 8

Gc = 05 1 15 2

120578

f998400


0

02

04

06

08

1

12

14

16

Pr = 072 1 3 71

0 2 4 6 8120578

f998400


0

025

05

075

1

125

15

R = 05 1 2 25

0 2 4 6 8120578

f998400



Q = 05 07 1 15

0 2 4 6 8120578

0

05

1

15

2

f998400


Ec = 01 03 05 1

0 2 4 6 8120578

0

05

1

15

2

f998400


0

02

04

06

08

1

12

14

16

18

Sc = 024 06 078 1

0 2 4 6 8120578

f998400


Kr = 05 1 15 2

0

02

04

06

08

1

12

14

16

0 2 4 6 8120578

f998400


0

025

05

075

1

125

15

Bi = 01 05 1 10

0 2 4 6 8120578

f998400


Pr = 072 1 3 71

0

02

04

06

08

1

12

14

16

0 2 4 6 8120578

120579



0

02

04

06

08

1

0 2 4 6 8120578

R = 05 1 2 25120579


2

Q = 05 07 1 15

0

02

04

06

08

1

12

14

16

18

0 2 4 6 8120578

120579


0

05

1

15

2

Ec = 01 03 05 1

0 2 4 6 8120578

120579


0

02

04

06

08

1

Bi = 01 05 1 10

0 2 4 6 8120578

120579


0

02

04

06

08

1

Sc = 024 06 078 1

0 2 4 6 8120578

120601


Kr = 05 1 15 2

0

02

04

06

08

1

0 2 4 6 8120578

120601




References
































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







0in






119891


120597119906

120597119909+120597V

120597119910= 0 (1)

119906120597119906

120597119909+ V

120597119906

120597119910= ]

1205972119906

1205971199102+ 119892120573 (119879 minus 119879

infin) + 119892120573


infin) (2)

120588119862119901(119906

120597119906

120597119909+ V

120597119906

120597119910) = 119896

1205972119879

1205971199102+ 119902 minus

120597119902119903

120597119910+ 120583(

120597119906

120597119910)

2

(3)

119906120597119862

120597119909+ V

120597119862

120597119910= 119863

1205972119862

1205971199102minus 1198701015840

119903(119862 minus 119862

infin) (4)







119906 (119909 0) = 1198800 V (119909 0) = 0

minus119870120597119879

120597119910(119909 0) = ℎ

119891[119879119891minus 119879 (119909 0)]

119862119891(119909 0) = 119860119909

120582+ 119862infin


infin

(5)

g

y

x

Tinfin

Cinfin

u T C


minusk120597T


u = U0 = 0

q



119903is given by

119902119903= minus

4120590lowast

3119870119904

1205971198794

120597119910 (6)






1198794asymp 41198793


4

infin (7)


120597119902119903

120597119910= minus

16120590lowast

3119870

12059721198794

1205971199102 (8)


120578 = 119910radic1198800

V119909=119910

119909radicRe119909

119906

1198800

= 1198911015840 V =

1

2119909radicRe119909(1205781198911015840minus 119891)

120579 (120578) =119879 minus 119879infin

119879119891minus 119879infin

120601 (120578) =119862 minus 119862

infin

119862119891minus 119862infin

(9)






119891101584010158401015840+1

21198911198911015840+ Gr 120579 + Gc120601 = 0

12057910158401015840[1 +

4

3119877] +

1

2Pr1198911205791015840 + Pr119876119890minus120578 + Ec Pr(11989110158401015840)

2

= 0

12060110158401015840+1

2Sc1198911206011015840 minus Kr Sc120601 = 0

119891 (0) = 0 1198911015840(0) = 1

1205791015840(0) = minusBi [1 minus 120579 (0)] 120601 (0) = 1


(10)

where

Bi =ℎ119891

119896radic]119909

1198800

Pr = ]

120572

Gr =119909119892120573 (119879


1198802

0


1198802

0

Ra =4120572

lowast

1205901198793

infin

119896119870 119876 =

1199092119902119890120578

119896Re119909(119879119891minus 119879infin)

Ec =1198802

0

119896 (119879119891minus 119879infin)

Sc = ]

119863 Kr = Kr1015840119909

1198800

(11)



119909must




ℎ119891= 119888119909minus12

120573 = 119898119909minus1119902 = 119897119909

minus1119890minus120578 (12)


Bi = 119888

119896radic

]

1198800

Gr =

119898119892 (119879119891minus 119879infin)

1198802

0

Gc =

119898119892 (119862119891minus 119862infin)

1198802

0

119876 =119897]

1198961198800(119879119891minus 119879infin)

(13)




10158401015840(0) minus120579

1015840(0) and minus120601











Bi Gr Pr Q 11989110158401015840(0)Makinde [14] 120579

1015840(0)Makinde [14] 120579(0)Makinde [14] 119891

10158401015840(0) present 120579

1015840(0) present 120579(0) present




11989110158401015840(0) 120579

1015840(0) 120579(0) 120601

1015840(0)

01 10 10 10 05 01 01 06 05 0617935 00690068 0309932 069090410 10 10 10 05 01 01 06 05 0995463 024145 075855 0705335100 10 10 10 05 01 01 06 05 116294 0325667 0967433 07113701 20 10 10 05 01 01 06 05 0914867 00688259 0311741 070301701 30 10 10 05 01 01 06 05 121966 00680544 0319456 071474701 10 20 10 05 01 01 06 05 127296 0067557 032443 071175501 10 30 10 05 01 01 06 05 18969 0064555 035445 072982301 10 10 10 05 01 01 06 05 0598557 00696409 0303591 068967901 10 10 30 05 01 01 06 05 0537444 0070888 029112 068573601 10 10 10 10 01 01 06 05 0635229 00684205 0315795 069199501 10 10 10 15 01 01 06 05 0645997 00680532 0319468 069267601 10 10 10 05 02 01 06 05 0694807 00621171 0378829 069427101 10 10 10 05 03 01 06 05 0769222 00553443 0446557 069747101 10 10 10 05 01 02 06 05 0654059 00662619 0337381 069265301 10 10 10 05 01 03 06 05 0693151 00632311 0367689 069452101 10 10 10 05 01 01 078 05 0562746 00687271 0312729 078661601 10 10 10 05 01 01 10 05 0511741 0068638 0311362 088992801 10 10 10 05 01 01 06 10 0514718 00688634 0311366 087701101 10 10 10 05 01 01 06 15 0488687 00687467 0312533 103195





0 1 2 3 4 5 6 70

05

1

15

2

25

3

Gr = 05 1 15 2

120578

f998400





1015840(0) respec-


4 Conclusions


0

05

1

15

2

0 2 4 6 8

Gc = 05 1 15 2

120578

f998400


0

02

04

06

08

1

12

14

16

Pr = 072 1 3 71

0 2 4 6 8120578

f998400


0

025

05

075

1

125

15

R = 05 1 2 25

0 2 4 6 8120578

f998400



Q = 05 07 1 15

0 2 4 6 8120578

0

05

1

15

2

f998400


Ec = 01 03 05 1

0 2 4 6 8120578

0

05

1

15

2

f998400


0

02

04

06

08

1

12

14

16

18

Sc = 024 06 078 1

0 2 4 6 8120578

f998400


Kr = 05 1 15 2

0

02

04

06

08

1

12

14

16

0 2 4 6 8120578

f998400


0

025

05

075

1

125

15

Bi = 01 05 1 10

0 2 4 6 8120578

f998400


Pr = 072 1 3 71

0

02

04

06

08

1

12

14

16

0 2 4 6 8120578

120579



0

02

04

06

08

1

0 2 4 6 8120578

R = 05 1 2 25120579


2

Q = 05 07 1 15

0

02

04

06

08

1

12

14

16

18

0 2 4 6 8120578

120579


0

05

1

15

2

Ec = 01 03 05 1

0 2 4 6 8120578

120579


0

02

04

06

08

1

Bi = 01 05 1 10

0 2 4 6 8120578

120579


0

02

04

06

08

1

Sc = 024 06 078 1

0 2 4 6 8120578

120601


Kr = 05 1 15 2

0

02

04

06

08

1

0 2 4 6 8120578

120601




References
































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119891101584010158401015840+1

21198911198911015840+ Gr 120579 + Gc120601 = 0

12057910158401015840[1 +

4

3119877] +

1

2Pr1198911205791015840 + Pr119876119890minus120578 + Ec Pr(11989110158401015840)

2

= 0

12060110158401015840+1

2Sc1198911206011015840 minus Kr Sc120601 = 0

119891 (0) = 0 1198911015840(0) = 1

1205791015840(0) = minusBi [1 minus 120579 (0)] 120601 (0) = 1


(10)

where

Bi =ℎ119891

119896radic]119909

1198800

Pr = ]

120572

Gr =119909119892120573 (119879


1198802

0


1198802

0

Ra =4120572

lowast

1205901198793

infin

119896119870 119876 =

1199092119902119890120578

119896Re119909(119879119891minus 119879infin)

Ec =1198802

0

119896 (119879119891minus 119879infin)

Sc = ]

119863 Kr = Kr1015840119909

1198800

(11)



119909must




ℎ119891= 119888119909minus12

120573 = 119898119909minus1119902 = 119897119909

minus1119890minus120578 (12)


Bi = 119888

119896radic

]

1198800

Gr =

119898119892 (119879119891minus 119879infin)

1198802

0

Gc =

119898119892 (119862119891minus 119862infin)

1198802

0

119876 =119897]

1198961198800(119879119891minus 119879infin)

(13)




10158401015840(0) minus120579

1015840(0) and minus120601











Bi Gr Pr Q 11989110158401015840(0)Makinde [14] 120579

1015840(0)Makinde [14] 120579(0)Makinde [14] 119891

10158401015840(0) present 120579

1015840(0) present 120579(0) present




11989110158401015840(0) 120579

1015840(0) 120579(0) 120601

1015840(0)

01 10 10 10 05 01 01 06 05 0617935 00690068 0309932 069090410 10 10 10 05 01 01 06 05 0995463 024145 075855 0705335100 10 10 10 05 01 01 06 05 116294 0325667 0967433 07113701 20 10 10 05 01 01 06 05 0914867 00688259 0311741 070301701 30 10 10 05 01 01 06 05 121966 00680544 0319456 071474701 10 20 10 05 01 01 06 05 127296 0067557 032443 071175501 10 30 10 05 01 01 06 05 18969 0064555 035445 072982301 10 10 10 05 01 01 06 05 0598557 00696409 0303591 068967901 10 10 30 05 01 01 06 05 0537444 0070888 029112 068573601 10 10 10 10 01 01 06 05 0635229 00684205 0315795 069199501 10 10 10 15 01 01 06 05 0645997 00680532 0319468 069267601 10 10 10 05 02 01 06 05 0694807 00621171 0378829 069427101 10 10 10 05 03 01 06 05 0769222 00553443 0446557 069747101 10 10 10 05 01 02 06 05 0654059 00662619 0337381 069265301 10 10 10 05 01 03 06 05 0693151 00632311 0367689 069452101 10 10 10 05 01 01 078 05 0562746 00687271 0312729 078661601 10 10 10 05 01 01 10 05 0511741 0068638 0311362 088992801 10 10 10 05 01 01 06 10 0514718 00688634 0311366 087701101 10 10 10 05 01 01 06 15 0488687 00687467 0312533 103195





0 1 2 3 4 5 6 70

05

1

15

2

25

3

Gr = 05 1 15 2

120578

f998400





1015840(0) respec-


4 Conclusions


0

05

1

15

2

0 2 4 6 8

Gc = 05 1 15 2

120578

f998400


0

02

04

06

08

1

12

14

16

Pr = 072 1 3 71

0 2 4 6 8120578

f998400


0

025

05

075

1

125

15

R = 05 1 2 25

0 2 4 6 8120578

f998400



Q = 05 07 1 15

0 2 4 6 8120578

0

05

1

15

2

f998400


Ec = 01 03 05 1

0 2 4 6 8120578

0

05

1

15

2

f998400


0

02

04

06

08

1

12

14

16

18

Sc = 024 06 078 1

0 2 4 6 8120578

f998400


Kr = 05 1 15 2

0

02

04

06

08

1

12

14

16

0 2 4 6 8120578

f998400


0

025

05

075

1

125

15

Bi = 01 05 1 10

0 2 4 6 8120578

f998400


Pr = 072 1 3 71

0

02

04

06

08

1

12

14

16

0 2 4 6 8120578

120579



0

02

04

06

08

1

0 2 4 6 8120578

R = 05 1 2 25120579


2

Q = 05 07 1 15

0

02

04

06

08

1

12

14

16

18

0 2 4 6 8120578

120579


0

05

1

15

2

Ec = 01 03 05 1

0 2 4 6 8120578

120579


0

02

04

06

08

1

Bi = 01 05 1 10

0 2 4 6 8120578

120579


0

02

04

06

08

1

Sc = 024 06 078 1

0 2 4 6 8120578

120601


Kr = 05 1 15 2

0

02

04

06

08

1

0 2 4 6 8120578

120601




References
































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





Bi Gr Pr Q 11989110158401015840(0)Makinde [14] 120579

1015840(0)Makinde [14] 120579(0)Makinde [14] 119891

10158401015840(0) present 120579

1015840(0) present 120579(0) present




11989110158401015840(0) 120579

1015840(0) 120579(0) 120601

1015840(0)

01 10 10 10 05 01 01 06 05 0617935 00690068 0309932 069090410 10 10 10 05 01 01 06 05 0995463 024145 075855 0705335100 10 10 10 05 01 01 06 05 116294 0325667 0967433 07113701 20 10 10 05 01 01 06 05 0914867 00688259 0311741 070301701 30 10 10 05 01 01 06 05 121966 00680544 0319456 071474701 10 20 10 05 01 01 06 05 127296 0067557 032443 071175501 10 30 10 05 01 01 06 05 18969 0064555 035445 072982301 10 10 10 05 01 01 06 05 0598557 00696409 0303591 068967901 10 10 30 05 01 01 06 05 0537444 0070888 029112 068573601 10 10 10 10 01 01 06 05 0635229 00684205 0315795 069199501 10 10 10 15 01 01 06 05 0645997 00680532 0319468 069267601 10 10 10 05 02 01 06 05 0694807 00621171 0378829 069427101 10 10 10 05 03 01 06 05 0769222 00553443 0446557 069747101 10 10 10 05 01 02 06 05 0654059 00662619 0337381 069265301 10 10 10 05 01 03 06 05 0693151 00632311 0367689 069452101 10 10 10 05 01 01 078 05 0562746 00687271 0312729 078661601 10 10 10 05 01 01 10 05 0511741 0068638 0311362 088992801 10 10 10 05 01 01 06 10 0514718 00688634 0311366 087701101 10 10 10 05 01 01 06 15 0488687 00687467 0312533 103195





0 1 2 3 4 5 6 70

05

1

15

2

25

3

Gr = 05 1 15 2

120578

f998400





1015840(0) respec-


4 Conclusions


0

05

1

15

2

0 2 4 6 8

Gc = 05 1 15 2

120578

f998400


0

02

04

06

08

1

12

14

16

Pr = 072 1 3 71

0 2 4 6 8120578

f998400


0

025

05

075

1

125

15

R = 05 1 2 25

0 2 4 6 8120578

f998400



Q = 05 07 1 15

0 2 4 6 8120578

0

05

1

15

2

f998400


Ec = 01 03 05 1

0 2 4 6 8120578

0

05

1

15

2

f998400


0

02

04

06

08

1

12

14

16

18

Sc = 024 06 078 1

0 2 4 6 8120578

f998400


Kr = 05 1 15 2

0

02

04

06

08

1

12

14

16

0 2 4 6 8120578

f998400


0

025

05

075

1

125

15

Bi = 01 05 1 10

0 2 4 6 8120578

f998400


Pr = 072 1 3 71

0

02

04

06

08

1

12

14

16

0 2 4 6 8120578

120579



0

02

04

06

08

1

0 2 4 6 8120578

R = 05 1 2 25120579


2

Q = 05 07 1 15

0

02

04

06

08

1

12

14

16

18

0 2 4 6 8120578

120579


0

05

1

15

2

Ec = 01 03 05 1

0 2 4 6 8120578

120579


0

02

04

06

08

1

Bi = 01 05 1 10

0 2 4 6 8120578

120579


0

02

04

06

08

1

Sc = 024 06 078 1

0 2 4 6 8120578

120601


Kr = 05 1 15 2

0

02

04

06

08

1

0 2 4 6 8120578

120601




References
































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0 1 2 3 4 5 6 70

05

1

15

2

25

3

Gr = 05 1 15 2

120578

f998400





1015840(0) respec-


4 Conclusions


0

05

1

15

2

0 2 4 6 8

Gc = 05 1 15 2

120578

f998400


0

02

04

06

08

1

12

14

16

Pr = 072 1 3 71

0 2 4 6 8120578

f998400


0

025

05

075

1

125

15

R = 05 1 2 25

0 2 4 6 8120578

f998400



Q = 05 07 1 15

0 2 4 6 8120578

0

05

1

15

2

f998400


Ec = 01 03 05 1

0 2 4 6 8120578

0

05

1

15

2

f998400


0

02

04

06

08

1

12

14

16

18

Sc = 024 06 078 1

0 2 4 6 8120578

f998400


Kr = 05 1 15 2

0

02

04

06

08

1

12

14

16

0 2 4 6 8120578

f998400


0

025

05

075

1

125

15

Bi = 01 05 1 10

0 2 4 6 8120578

f998400


Pr = 072 1 3 71

0

02

04

06

08

1

12

14

16

0 2 4 6 8120578

120579



0

02

04

06

08

1

0 2 4 6 8120578

R = 05 1 2 25120579


2

Q = 05 07 1 15

0

02

04

06

08

1

12

14

16

18

0 2 4 6 8120578

120579


0

05

1

15

2

Ec = 01 03 05 1

0 2 4 6 8120578

120579


0

02

04

06

08

1

Bi = 01 05 1 10

0 2 4 6 8120578

120579


0

02

04

06

08

1

Sc = 024 06 078 1

0 2 4 6 8120578

120601


Kr = 05 1 15 2

0

02

04

06

08

1

0 2 4 6 8120578

120601




References
































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Q = 05 07 1 15

0 2 4 6 8120578

0

05

1

15

2

f998400


Ec = 01 03 05 1

0 2 4 6 8120578

0

05

1

15

2

f998400


0

02

04

06

08

1

12

14

16

18

Sc = 024 06 078 1

0 2 4 6 8120578

f998400


Kr = 05 1 15 2

0

02

04

06

08

1

12

14

16

0 2 4 6 8120578

f998400


0

025

05

075

1

125

15

Bi = 01 05 1 10

0 2 4 6 8120578

f998400


Pr = 072 1 3 71

0

02

04

06

08

1

12

14

16

0 2 4 6 8120578

120579



0

02

04

06

08

1

0 2 4 6 8120578

R = 05 1 2 25120579


2

Q = 05 07 1 15

0

02

04

06

08

1

12

14

16

18

0 2 4 6 8120578

120579


0

05

1

15

2

Ec = 01 03 05 1

0 2 4 6 8120578

120579


0

02

04

06

08

1

Bi = 01 05 1 10

0 2 4 6 8120578

120579


0

02

04

06

08

1

Sc = 024 06 078 1

0 2 4 6 8120578

120601


Kr = 05 1 15 2

0

02

04

06

08

1

0 2 4 6 8120578

120601




References
































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0

02

04

06

08

1

0 2 4 6 8120578

R = 05 1 2 25120579


2

Q = 05 07 1 15

0

02

04

06

08

1

12

14

16

18

0 2 4 6 8120578

120579


0

05

1

15

2

Ec = 01 03 05 1

0 2 4 6 8120578

120579


0

02

04

06

08

1

Bi = 01 05 1 10

0 2 4 6 8120578

120579


0

02

04

06

08

1

Sc = 024 06 078 1

0 2 4 6 8120578

120601


Kr = 05 1 15 2

0

02

04

06

08

1

0 2 4 6 8120578

120601




References
































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





References
































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics











FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



research article similarity solution of heat and mass...

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