hydromagnetic natural convection flow with heat and mass ... · convection ﬂow of micro-polar...

Journal of the Egyptian Mathematical Society (2015) 23, 197–207

Egyptian Mathematical Society

Journal of the Egyptian Mathematical Society

www.etms-eg.orgwww.elsevier.com/locate/joems

ORIGINAL ARTICLE

Hydromagnetic natural convection flow with heat

and mass transfer of a chemically reacting and heat

absorbing fluid past an accelerated moving vertical

plate with ramped temperature and ramped surface

concentration through a porous medium

* Corresponding author.E-mail addresses: [email protected] (G.S. Seth), hussain.modassir@

yahoo.com (S.M. Hussain), [email protected] (S. Sarkar).

Peer review under responsibility of Egyptian Mathematical Society.

Production and hosting by Elsevier

1110-256X ª 2014 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society.

http://dx.doi.org/10.1016/j.joems.2014.03.006

G.S. Seth a,*, S.M. Hussain b, S. Sarkar a

a Department of Applied Mathematics, Indian School of Mines, Dhanbad 826004, Indiab Department of Mathematics, O. P. Jindal Institute of Technology, Raigarh 496109, India

Received 11 January 2014; revised 21 March 2014; accepted 26 March 2014Available online 29 April 2014

KEYWORDS

Ramped temperature;

Ramped surface concentra-

tion;

Chemical reaction;

Heat absorption;

Nusselt number;

Sherwood number

Abstract Unsteady hydromagnetic natural convection flow with heat and mass transfer of an elec-

trically conducting, viscous, incompressible, chemically reacting and heat absorbing fluid past an

accelerated moving vertical plate with ramped temperature and ramped surface concentration

through a porous medium in the presence of thermal and mass diffusions is studied. The exact solu-

tions of momentum, energy and concentration equations, under the Boussinesq approximation, are

obtained in closed form by Laplace Transform technique. The expressions for skin friction, Nusselt

number and Sherwood number are also derived. The variations in fluid velocity, fluid temperature

and species concentration are displayed graphically whereas numerical values of skin friction, Nus-

selt number and Sherwood number are presented in tabular form for various values of pertinent

flow parameters. Natural convection flow near a ramped temperature plate with ramped surface

concentration is also compared with the flow near an isothermal plate with uniform surface concen-

tration.

2010 MATHEMATICS SUBJECT CLASSIFICATION: 76W05; 76R10; 80A20; 80A32

ª 2014 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society.

1. Introduction

The problems of hydromagnetic convective flow in a porousmedium have drawn considerable attentions of severalresearchers owing to its importance in various scientific andtechnological applications viz. problems of boundary layer

http://crossmark.crossref.org/dialog/?doi=10.1016/j.joems.2014.03.006&domain=pdf

mailto:[email protected]





http://www.sciencedirect.com/science/journal/1110256X


Nomenclature

B0 uniform magnetic field

C0 species concentrationD chemical molecular diffusivityGc solutal Grashof numberK1 permeability parameter

K01 permeability of porous mediumPr Prandtl numberR dimensionless constant

T dimensionless fluid temperatureu dimensionless fluid velocity in g directionC dimensionless species concentration

Cp specific heat at constant pressureg acceleration due to gravityGr thermal Grashof numberK2 chemical reaction parameter

M magnetic parameter

Q0 heat absorption coefficient

Sc Schmidt numberT0 fluid temperatureu0 fluid velocity in g direction

Greek symbols

a0 thermal diffusivityb0 volumetric coefficient of thermal expansion for

species concentration

t kinematic coefficient of viscosity/ dimensionless heat absorption coefficientq fluid densityb* volumetric coefficient of expansion

r electrical conductivity

198 G.S. Seth et al.

flow control, plasma studies, geothermal energy extraction,metallurgy, chemical, mineral and petroleum engineering,

etc. and on the performance of so many engineering devicesusing electrically conducting fluids, namely, MHD generators,MHD pumps, MHD accelerators, MHD flow-meters, nuclear

reactors, plasma jet engines, etc. Raptis and Kafousias [1]investigated steady hydromagnetic free convection flowthrough a porous medium bounded by an infinite vertical platewith constant suction velocity. Raptis [2] discussed unsteady

two-dimensional natural convection flow of an electrically con-ducting, viscous and incompressible fluid along an infinite ver-tical plate embedded in a porous medium. Chamkha [3]

studied unsteady MHD free convection flow through a porousmedium supported by a surface. Chamkha [4] also studiedMHD natural convection flow near an isothermal inclined sur-

face adjacent to a thermally stratified porous medium. Aldosset al. [5] investigated combined free and forced convection flowfrom a vertical plate embedded in a porous medium in the

presence of a magnetic field. Kim [6] analyzed unsteadyMHD free convection flow past a moving semi-infinite verticalporous plate embedded in a porous medium with variable suc-tion. Theoretical/experimental investigations of convective

boundary layer flow with heat and mass transfer induced dueto a moving surface with a uniform or non-uniform velocityplay an important role in several manufacturing processes in

industry which include the boundary layer flow along materialhandling conveyers, extrusion of plastic sheets, cooling of aninfinite metallic plate in cooling bath, glass blowing, continu-

ous casting and levitation, design of chemical processingequipment, formation and dispersion of fog, distribution oftemperature and moisture over agricultural fields and grovesof trees, damage of crops due to freezing, common industrial

sight especially in power plants, etc. Keeping in view theimportance of such study, Jha [7] considered hydromagneticfree convection and mass transfer flow past a uniformly accel-

erated moving vertical plate through a porous medium. Ibra-him et al. [8] investigated unsteady hydromagnetic freeconvection flow of micro-polar fluid and heat transfer past a

vertical porous plate through a porous medium in the presence

of thermal and mass diffusions with a constant heat source.Makinde and Sibanda [9] studied MHD mixed convective flow

with heat and mass transfer past a vertical plate embedded in aporous medium with constant wall suction. Makinde [10] ana-lyzed hydromagnetic mixed convection flow and mass transfer

over a vertical porous plate with constant heat flux embeddedin a porous medium. Makinde [11] also investigated MHDboundary layer flow with heat and mass transfer over a mov-ing vertical plate with a convective surface boundary

condition.It is noticed that there may be an appreciable temperature

difference between the surface of the solid body and ambient

fluid in so many fluid flow problems of practical interests. Thisprompted many researchers to consider temperature depen-dent heat sources and/or sinks, which may have strong influ-

ence on heat transfer characteristics [12]. The researchstudies related to heat generating and/or heat absorbing fluidflow are of considerable importance in several physical prob-

lems viz. fluids undergoing exothermic and/or endothermicchemical reaction [12], its applications in the field of nuclearenergy [13], convection in Earth’s mantle [14], post accidentheat removal [15], fire and combustion modeling [16], develop-

ment of metal waste from spent nuclear fuel [17], etc. Exactmathematical modeling of internal heat generation/absorptionis very much complicated. It is found that some simple

mathematical models yet idealized may present their averagebehavior for most of the physical situations. Taking into con-sideration of this fact Sparrow and Cess [18] discussed the

effects of temperature dependent heat absorption in theirresearch study on steady stagnation point flow and heat trans-fer. Moalem [19] investigated steady heat transfer in a porousmedium with temperature dependent heat source. Chamkha

and Khaled [20] considered hydromagnetic combined heatand mass transfer by natural convection from a permeable ver-tical plate embedded in a fluid saturated porous medium in the

presence of heat generation or absorption. Kamel [21] investi-gated unsteady hydromagnetic convection flow due to heatand mass transfer through a porous medium bounded by an

infinite vertical porous plate with temperature dependent heat

Hydromagnetic natural convection flow with heat and mass transfer of a chemically reacting and heat absorbing fluid 199

sources and sinks. Chamkha [22] studied unsteady hydromag-netic two dimensional convective laminar boundary layer flowwith heat and mass transfer of a viscous, incompressible, elec-

trically conducting and temperature dependent heat absorbingfluid along a semi infinite vertical permeable moving plate inthe presence of a uniform transverse magnetic field. Makinde

[23] investigated heat and mass transfer by MHD mixed con-vection stagnation point flow toward a vertical plate embeddedin a highly porous medium with radiation and internal heat

generation. In all these investigations, numerical/analyticalsolution is obtained by assuming conditions for the velocityand temperature at the plate as continuous and well defined.However, there are several problems of practical interests

which may require non-uniform or arbitrary conditions atthe plates. Keeping in view this fact, several researchers,namely, Hayday et al. [24], Kelleher [25], Kao [26], Lee and

Yovanovich [27] and Chandran et al. [28] studied natural con-vection flow from a vertical plate with step discontinuities inthe surface temperature considering different aspects of the

problem. Patra et al. [29] investigated the effects of radiationon natural convection flow of a viscous and incompressiblefluid near a vertical flat plate with ramped temperature. They

compared the effects of radiative heat transfer on natural con-vection flow near a ramped temperature plate with the flownear an isothermal plate. Seth and Ansari [30] investigatedunsteady hydromagnetic natural convection flow of a viscous,

incompressible, electrically conducting and temperaturedependent heat absorbing fluid past an impulsively movingvertical plate with ramped temperature in a porous medium

taking into account the effects of thermal diffusion. Subse-quently, Seth et al. [31] extended the problem studied by Sethand Ansari [30] to consider the effects of rotation on flow-field.

In many chemical engineering processes, there does occurthe chemical reaction between a foreign mass and the fluid.Chemical reactions can be classified as either heterogeneous

or homogeneous processes. This depends on whether theyoccur at an interface or as a single phase volume reaction.These processes take place in numerous industrial applicationsviz. polymer production, manufacturing of ceramics or glass-

ware, food processing, etc. Afify [32] studied the effect of radi-ation on free convective flow and mass transfer past a verticalisothermal cone surface with chemical reaction in the presence

of a transverse magnetic field. Muthucumaraswamy and Chan-drakala [33] investigated radiative heat and mass transfereffects on moving isothermal vertical plate in the presence of

chemical reaction. Ibrahim et al. [34] analyzed the effect ofchemical reaction and radiation absorption on the unsteadyMHD free convection flow past a semi-infinite vertical perme-able moving plate with heat source. Bakr [35] discussed the

effects of chemical reaction on MHD free convection and masstransfer flow of a micro-polar fluid with oscillatory plate veloc-ity and constant heat source in a rotating frame of reference.

Chamkha [36] studied MHD flow of a uniformly stretched ver-tical permeable surface in the presence of heat generation/absorption and chemical reaction. Chamkha et al. [37] dis-

cussed the effects of Joule heating, chemical reaction and ther-mal radiation on unsteady hydromagnetic natural convectionboundary layer flow with heat and mass transfer of a micro-

polar fluid from a semi-infinite heated vertical porous platein the presence of a uniform transverse magnetic field. Bhat-tacharya and Layek [38] obtained similarity solution ofMHD boundary layer flow with mass diffusion and chemical

reaction over a porous flat plate with suction/blowing.Recently, Mohamed et al. [39] investigated unsteady MHDfree convection heat and mass transfer boundary layer flow

of viscous, incompressible, optically thick and electrically con-ducting fluid through a porous medium along an impulsivelymoving hot vertical plate in the presence of homogeneous

chemical reaction of first order and temperature dependentheat sink. They obtained analytical solution of the governingequations in closed form by Laplace transform technique.

Objective of present investigation is to study unsteadyhydromagnetic natural convection flow with heat and masstransfer of a chemically reacting and heat absorbing fluid pastan accelerated moving vertical plate with ramped temperature

and ramped surface concentration through a porous medium.Such study may find application in solar collection systems,fire dynamics in insulations, geothermal energy systems, cata-

lytic reactors, nuclear waste repositories, recovery of petro-leum products and gases (e.g. CBM: Coal Bed Methane andUCG: Underground Coal Gasification), etc.

2. Formulation of the problem and its solution

Consider unsteady hydromagnetic natural convection flow

with heat and mass transfer of an electrically conducting, vis-cous, incompressible, chemically reacting and temperaturedependent heat absorbing fluid past an accelerated moving infi-

nite vertical plate embedded in a porous medium in the pres-ence of thermal and mass diffusions. Choose the co-ordinatesystem in such a way that x0-axis is along the plate in upwarddirection, y0-axis normal to the plane of the plate and z0-axis

perpendicular to x0y0-plane. The fluid is permeated by uniformtransverse magnetic field B0 applied parallel to y0-axis. Initially,i.e. at time t0 6 0, both the fluid and plate are at rest and main-

tained at uniform temperature T01 and uniform surface concen-tration C01. At time t0 > 0, plate starts moving in x0-directionagainst the gravitational field with time dependent velocity

U(t0). Temperature of the plate is raised or lowered toT01 þ ðT0w � T01Þt0=t0 and the level of concentration at the sur-face of the plate is raised or lowered to C01 þ ðC0w � C01Þt0=t0when 0 < t0 6 t0. Thereafter, i.e. at t

0 > t0, plate is maintainedat the uniform temperature T0w and the level of concentration atthe surface of the plate is preserved at uniform concentrationC0w. Here t0 is characteristic time. It is assumed that there exists

a homogeneous chemical reaction of first order with constantrate K02 between the diffusing species and the fluid. The sche-matic diagram of the physical problem is shown in Fig. 1. Since

the plate is of infinite extent along x0 and z0-directions and iselectrically non-conducting, all physical quantities except pres-sure depend on y0 and t0 only. The induced magnetic field pro-

duced by fluid motion is neglected in comparison with appliedone. This statement is justified because magnetic Reynoldsnumber is very small for liquid metals and partially ionizedfluid [40]. Also no external electric field is applied so the effect

of polarization of fluid is negligible. This corresponds to thecase where no energy is added or extracted from the fluid byelectrical means [40].

Taking into consideration the assumptions made above, thegoverning equations for natural convection flow with heat andmass transfer of an electrically conducting, viscous, incom-

pressible, chemically reacting and temperature dependent heatabsorbing fluid through a porous medium in the presence of

Figure 1 Schematic diagram of the physical problem.


thermal and mass diffusions, under Boussinesq approxima-tion, are given by

@u0

@t0¼ t

@2u0

@y02� rB2

0

qu0 � t

K01u0 þ gb0ðT0 �T01Þ þ gb�ðC0 �C01Þ; ð2:1Þ

@T0

@t0¼ a0

@2T0

@y02� Q0

qCp

ðT0 � T01Þ; ð2:2Þ

@C0

@t0¼ D

@2C0

@y02� K02ðC0 � C01Þ; ð2:3Þ

Appropriate initial and boundary conditions for fluid velocity,fluid temperature and species concentration are given by

u0 ¼ 0;T0 ¼ T01;C0 ¼ C01 for y0 P 0 and t0 6 0; ð2:4aÞ

u0 ¼ Uðt0Þ at y0 ¼ 0 for t0 > 0; ð2:4bÞ

T0 ¼ T01 þ ðT0w � T01Þt0=t0;C0 ¼ C01 þ ðC0w � C01Þt0=t0;

�at y0 ¼ 0 for 0 < t0 6 t0;

ð2:4cÞ

T0 ¼ T0w;C0 ¼ C0w at y0 ¼ 0 for t0 > t0; ð2:4dÞ

u0 ! 0; T0 ! T01; C0 ! C01 as y0 ! 1 for t0 > 0: ð2:4eÞ

In order to represent Eqs. (2.1)–(2.3) along with initial and

boundary conditions Eq. (2.4) in dimensionless form, follow-ing dimensionless quantities and parameters are introduced.

y¼y0=U0t0; u¼u0=U0; t¼ t0=t0; T¼ðT0 �T01Þ=ðT0w�T01Þ;C¼ðC0 �C01Þ=ðC0w�C01Þ; Gr¼ tgb0ðT0w�T01Þ=U3

0;

Gc¼ tgb�ðC0w�C01Þ=U30; M¼rB2

0t=qU20; K1¼K01U

20=t

2;

Pr¼ t=a0; Sc¼ t=D; K2¼ tK02=U20 and/¼ tQ0=qCpU

20;

9>>>>=>>>>;ð2:5Þ

Using dimensionless quantities and parameters defined in(2.5), Eqs. (2.1)–(2.3), in dimensionless form, become

@u

@t¼ @

2u

@y2�Mu� u

K1

þ GrTþ GcC; ð2:6Þ

@T

@t¼ 1

Pr

@2T

@y2� /T; ð2:7Þ

@C

@t¼ 1

Sc

@2C

@y2� K2C: ð2:8Þ

According to above non-dimensionalization process, charac-teristic time t0 may be defined as

t0 ¼ t=U20: ð2:9Þ

Appropriate initial and boundary conditions (2.4), in dimen-sionless form, assume the following form

u ¼ 0; T ¼ 0; C ¼ 0 for y P 0 and t 6 0; ð2:10aÞ

u ¼ FðtÞ at y ¼ 0 for t > 0; ð2:10bÞ

T ¼ t; C ¼ t at y ¼ 0 for 0 < t 6 1; ð2:10cÞ

T ¼ 1; C ¼ 1 at y ¼ 0 for t > 1; ð2:10dÞ

u! 0; T! 0; C! 0 as y!1 for t > 0; ð2:10eÞ

where F(t) = U(t0)/U0.The fluid flow described by the Eqs. (2.6)–(2.8) subject to

initial and boundary conditions (2.10) is quite general. In orderto analyze the flow features of the fluid flow we now consider a

particular case of interest, namely, uniformly acceleratedmovement of the plate, i.e. F(t) = Rt where R is dimensionlessconstant.

Eqs. (2.6)–(2.8) are solved analytically with the help ofLaplace transform technique subject to the initial and bound-ary conditions (2.10) and the exact solutions for fluid velocity

F(y, t), fluid temperature T(y, t) and species concentrationC(y, t) are obtained and are presented after simplification inthe following form

uðy; tÞ ¼ R

2tþ y

2ffiffiffikp

� �d1 þ t� y

2ffiffiffikp

� �d2

� �

� d1

d23

fF1ðy; tÞ �Hðt� 1ÞF1ðy; t� 1Þg

� d2

d24

fF2ðy; tÞ �Hðt� 1ÞF2ðy; t� 1Þg; ð2:11Þ


Tðy; tÞ ¼ Pðy; tÞ �Hðt� 1ÞPðy; t� 1Þ; ð2:12Þ

Cðy; tÞ ¼ Qðy; tÞ �Hðt� 1ÞQðy; t� 1Þ; ð2:13Þ

where

F1ðy; tÞ ¼ed3t

2ey

ffiffiffiffiffiffiffiffiffiffiðkþd3Þp

erfcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkþ d3Þt

pþ y

2ffiffitp

� ��

þe�yffiffiffiffiffiffiffiffiffiffiðkþd3Þp

erfc �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkþ d3Þt

pþ y

2ffiffitp

� �

�eyffiffiffiffiffiffiffiffiffiffiffiffiffiffiPrð/þd3Þp

erfcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið/þ d3Þt

pþ y

2

ffiffiffiffiffiPr

t

r !

�e�yffiffiffiffiffiffiffiffiffiffiffiffiffiffiPrð/þd3Þp

erfc �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið/þ d3Þt

pþ y

2

ffiffiffiffiffiPr

t

r !)

� d3

2tþ 1

d3

þ y

2ffiffiffikp

� �d1

�þ tþ 1

d3

� y

2ffiffiffikp

� �d2

� d3

2tþ 1

d3

þ y

2ffiffiffikp

� �d1

�þ tþ 1

d3

� y

2ffiffiffikp

� �

d2� tþ 1

d3

þ y

2

ffiffiffiffiffiPr

/

s !d3 � tþ 1

d3

� y

2

ffiffiffiffiffiPr

/

s !d4

);

F2ðy; tÞ ¼ed4t

2ey

ffiffiffiffiffiffiffiffiffiffiðkþd4Þp


pþ y

2ffiffitp

� ��



pþ y

2ffiffitp

� �

�eyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiScðK2þd4Þp

erfcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðK2 þ d4Þt

pþ y

2

ffiffiffiffiffiSc

t

r !

�e�yffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiScðK2þd4Þp

erfc �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðK2 þ d4Þt

pþ y

2

ffiffiffiffiffiSc

t

r !)

� d4

2tþ 1

d4

þ y

2ffiffiffikp

� �d1

�þ tþ 1

d4

� y

2ffiffiffikp

� �

d2� tþ 1

d4

þ y

2

ffiffiffiffiffiffiSc

K2

r� �d5 � tþ 1

d4

� y

2


K2

r� �d6

�;

Pðy; tÞ ¼ 1

2tþ y

2

ffiffiffiffiffiPr

/

s !d3 þ t� y

2

ffiffiffiffiffiPr

/

s !d4

( );

and

Cðy; tÞ ¼ 1

2tþ y

2


K2

r� �d5 þ t� y

2


K2

r� �d6

��;

where

k ¼ Mþ 1

K1

� �; d1 ¼ Gr=ð1� PrÞ; d2 ¼ Gc=ð1� ScÞ;

d3 ¼ ðPr/� kÞ=ð1� PrÞ; d4 ¼ ðScK2 � kÞ=ð1� ScÞ;

d1 ¼ eyffiffikperfc

ffiffiffiffiktpþ y

2ffiffitp

� �;

d2 ¼ e�yffiffikperfc �

ffiffiffiffiktpþ y

2ffiffitp

� �;

d3 ¼ eyffiffiffiffiffiffiPr/p

erfcffiffiffiffiffi/t

pþ y

2

ffiffiffiffiffiPr

t

r !;

d4 ¼ e�yffiffiffiffiffiffiPr/p

erfc �ffiffiffiffiffi/t

pþ y

2

ffiffiffiffiffiPr

t

r !;

d5 ¼ eyffiffiffiffiffiffiffiScK2

perfc

ffiffiffiffiffiffiffiK2t

pþ y

2

ffiffiffiffiffiSc

t

r !;

d6 ¼ e�yffiffiffiffiffiffiffiScK2

perfc �


pþ y

2

ffiffiffiffiffiSc

t

r !:

3. Solution in the case of isothermal plate with uniform surface

concentration

In order to highlight the effects of ramped temperature andramped surface concentration on fluid flow, it may be worth-while to compare such flow with the one near an accelerated

moving vertical plate with uniform temperature and uniformsurface concentration. Keeping in view the assumptions madein this paper, the solution for fluid velocity, fluid temperature

and species concentration for natural convection flow past anaccelerated moving vertical isothermal plate with uniform sur-face concentration is obtained and is expressed in the followingform

uðy; tÞ ¼ 1

2R tþ y

2ffiffiffikp

� �þ d1

d3

þ d2

d4

� �d1

�þ R t� y

2ffiffiffikp

� �þ d1

d3

þ d2

d4

� �d2

�

� d1

2d3

ed3t eyffiffiffiffiffiffiffiffiffiffiðkþd3Þp


pþ y

2ffiffitp

� ��



pþ y

2ffiffitp

� �

� eyffiffiffiffiffiffiffiffiffiffiffiffiffiffiPrð/þd3Þp

erfcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið/þ d3Þt

pþ y

2

ffiffiffiffiffiPr

t

r !

� e�yffiffiffiffiffiffiffiffiffiffiffiffiffiffiPrð/þd3Þp

erfcð�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið/þ d3Þt

pþ y

2

ffiffiffiffiffiPr

t

rÞ#

� d1

2d3

ðd3 þ d4Þ �d2

2d4

ed4t eyffiffiffiffiffiffiffiffiffiffiðkþd4Þp


pþ y

2ffiffitp

� ��


� erfc �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkþ d4Þt

pþ y

2ffiffitp

� �

� eyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiScðK2þd4Þp

erfcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðK2 þ d4Þt

pþ y

2

ffiffiffiffiffiSc

t

r !

� e�yffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiScðK2þd4Þp

�erfc �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðK2 þ d4Þt

pþ y

2

ffiffiffiffiffiSc

t

r !#� d2

2d4

ðd5 þ d6Þ;

ð3:1Þ

Tðy; tÞ ¼ 1

2ðd3 þ d4Þ; ð3:2Þ

Cðy; tÞ ¼ 1

2ðd5 þ d6Þ: ð3:3Þ

4. Skin friction, Nusselt number and Sherwood number

The expressions for the skin friction s, Nusselt number Nu andSherwood number Sh, which are measures of the shear stress,rate of heat transfer and rate of mass transfer at the plate

respectively, are presented in the following form for the platewith ramped temperature and ramped surface concentrationand isothermal plate with uniform surface concentration:

Figure 2 Velocity profiles when K2 = 0.2, Gr = 4, Gc = 5, /= 1 and t= 0.5.


For the plate with ramped temperature and ramped surfaceconcentration

s ¼ R1

2ffiffiffikp þ t

ffiffiffikp� �

erfcffiffiffiffiktp

� 1n o

�ffiffiffit

p

re�kt

" #

� d1

d23

fF5ð0; tÞ �Hðt� 1ÞF5ð0; t� 1Þg

� d1

d23

fF6ð0; tÞ �Hðt� 1ÞF6ð0; t� 1Þg; ð4:1Þ

Nu ¼ P2ð0; tÞ �Hðt� 1ÞP2ð0; t� 1Þ; ð4:2Þ

Sh ¼ Q2ð0; tÞ �Hðt� 1ÞQ2ð0; t� 1Þ; ð4:3Þ

where

F5ð0; tÞ ¼ ed3tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkþ d3Þ

perfc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkþ d3Þt

p � 1

n o� 1ffiffiffiffiffi

ptp e�ðkþd3Þt

�

þffiffiffiffiffiPr

pt

re�ð/þd3Þt�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPrð/þ d3Þ

perfc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið/þ d3Þt

p � 1

n o#

� d3

1

2ffiffiffikp þ

ffiffiffikp

tþ 1

d3

� �� erfc

ffiffiffiffiktp

� 1n o

� 1

2

ffiffiffiffiffiPr

/

sþ

ffiffiffiffiffiffiffiffiPr/

ptþ 1

d3

� �( )erfc

ffiffiffiffiffi/t

p � 1

n o

� tþ 1

d3

� �� 1ffiffiffiffiffi

ptp e�kt þ tþ 1

d3

� � ffiffiffiffiffiPr

pt

re�/t

#;

F6ð0; tÞ ¼ ed4tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkþ d4Þ

perfc


p � 1

n o� 1ffiffiffiffiffi

ptp e�ðkþd4Þt

�

þffiffiffiffiffiSc

pt

re�ðK2þd4Þt�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiScðK2þ d4Þ

perfc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðK2þ d4Þt

p � 1

n o#

� d4

1

2ffiffiffikp þ

ffiffiffikp

tþ 1

d4

� �� erfc

ffiffiffiffiktp

� 1n o

� 1

2


K2

rþ

ffiffiffiffiffiffiffiffiffiffiScK2

ptþ 1

d4

� �� erfc


p � 1

n o

� tþ 1

d4

� �� 1ffiffiffiffiffi

ptp e�ktþ tþ 1

d4

� � ffiffiffiffiffiSc

pt

re�K2t

#;

P2ð0;tÞ¼1

2

ffiffiffiffiffiPr

/

sþ2t


p !erfc

ffiffiffiffiffi/t

p �1

n o�2t

ffiffiffiffiffiPr

pt

re�/t

" #;

Q2ð0; tÞ¼1

2


K2

rþ2t


p� �erfcð

ffiffiffiffiffiffik2t

pÞ�1

n o�2t

ffiffiffiffiffiSc

pt

re�K2t

" #;

and for isothermal plate with uniform surface concentration

s¼ R

2ffiffiffikp þ

ffiffiffikp

Rtþ d1

d3

þ d2

d4

� �� erfc

ffiffiffiffiffiktp

� 1n o

� Rtþ d1

d3

þ d2

d4

� �1ffiffiffiffiffiptp e�kt � d1

d3

ed3tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkþ d3Þ

perfc


p � 1

n oh� 1ffiffiffiffiffi

ptp e�ðkþd3Þt �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPrð/þ d3Þ

perfc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið/þ d3Þt

p � 1

n o

þffiffiffiffiffiPr

pt

re�ð/þd3Þt

#� d1

d3


perfcð

ffiffiffiffiffi/t

pÞ � 1

n o�

ffiffiffiffiffiPr

pt

re�/t

" #

� d2

d4

ed4t

ffiffiffiffiffiSc

pt

re�ðK2þd4Þt

"þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkþ d4Þ

p� erfc


p � 1

n o

� 1ffiffiffiffiffiptp e�ðkþd4Þt �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiScðK2 þ d4Þ

perfc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðK2 þ d4Þt

p � 1

n o�

� d2

d4


perfc


p � 1

n o�

ffiffiffiffiffiSc

pt

re�K2t

" #; ð4:4Þ

Nu ¼ffiffiffiffiffiffiffiffiPr/

perfc

ffiffiffiffiffi/t

p � 1

n o�

ffiffiffiffiffiPr

pt

re�/t; ð4:5Þ

Sh ¼ffiffiffiffiffiffiffiffiffiffiScK2

perfc


p � 1

n o�

ffiffiffiffiffiSc

pt

re�K2t: ð4:6Þ

5. Results and discussion

In order to study the influence of magnetic field, thermal buoy-

ancy force, species buoyancy force, heat absorption, chemicalreaction and time on flow-field in the boundary layer region,the numerical values of fluid velocity, computed from the ana-lytical solutions reported in Sections 2 and 3, are displayed

graphically versus boundary layer co-ordinate y in Figs. 2–7for various values of magnetic parameter M, thermal Grashofnumber Gr, solutal Grashof number Gc, heat absorption co-

efficient /, chemical reaction parameter K2 and time t takingK1 = 0.5, Pr = 0.71 (ionized air), Sc = 0.22 and R = 1. It isevident from Figs. 2–7 that, for both ramped temperature plate

with ramped surface concentration and isothermal plate withuniform surface concentration, fluid velocity u increasesrapidly in the region near the surface of the plate, attains a

maximum distinctive value and then decreases properly onincreasing boundary layer coordinate y to approach freestream value. Fig. 2 illustrates the influence of magnetic fieldon the fluid velocity. For both ramped temperature plate with

ramped surface concentration and isothermal plate with uni-form surface concentration, fluid velocity decreases on increas-ing magnetic parameterM. This implies that magnetic field has

retarding influence on fluid flow for both ramped temperatureplate with ramped surface concentration and isothermal platewith uniform surface concentration. This is due to the fact that

the application of a magnetic field to an electrically conductingfluid gives rise to a resistive force called Lorentz force whichhas a tendency to slow down the motion of fluid in the bound-

ary layer region. Figs. 3 and 4 depict the influence of thermalbuoyancy force and species buoyancy force on fluid velocity.The thermal Grashof number Gr signifies the relative effectof the thermal buoyancy force to the viscous hydrodynamic

force. The solutal Grashof number Gc characterizes the ratioof the species buoyancy force and viscous hydrodynamic force.As expected, it is observed that, for both ramped temperature

Figure 3 Velocity profiles when M = 4, K2 = 0.2, Gc = 5, /= 1 and t= 0.5.

Figure 4 Velocity profiles when M= 4, K2 = 0.2, Gr = 4, /= 1 and t= 0.5.

Figure 5 Velocity profiles when M= 4, K2 = 0.2, Gr = 4,

Gc = 5 and t = 0.5.

Figure 6 Velocity profiles when M= 4, Gr = 4, Gc = 5, / = 1

and t= 0.5.

Figure 7 Velocity profiles when M= 4, K2 = 0.2, Gr = 4,

Gc = 5 and / = 1.


plate with ramped surface concentration and isothermal plate

with uniform surface concentration, fluid velocity increases onincreasing either Gr or Gc. This implies that fluid velocity is get-ting accelerated due to enrichment in either thermal buoyancy

force or species buoyancy force.Fig. 5 demonstrates the effects of heat absorption on fluid

velocity. It is noticed that, for both ramped temperature platewith ramped surface concentration and isothermal plate with

uniform surface concentration, fluid velocity decreases onincreasing /. This implies that, both for ramped temperatureplate with ramped surface concentration and isothermal plate

with uniform surface concentration, heat absorption tends toretard fluid velocity throughout the boundary layer region.This may be attributed to the fact that the tendency of heat

absorption (thermal sink) is to reduce the fluid temperaturewhich causes the strength of thermal buoyancy force todecrease resulting in a net reduction in the fluid velocity.

Fig. 6 displays the effect of chemical reaction on fluid veloc-ity. It is observed that fluid velocity decreases on increasing K2

for both ramped temperature plate with ramped surface con-centration and isothermal plate with uniform surface concen-

tration. This implies that chemical reaction has a retardinginfluence on fluid flow for both ramped temperature plate withramped surface concentration and isothermal plate with uni-

form surface concentration. Fig. 7 depicts the effects of timeon fluid velocity. It is observed that fluid velocity increaseson increasing time t. This implies that there is an enhancement

in fluid velocity for both ramped temperature plate withramped surface concentration and isothermal plate with uni-form surface concentration with the progress of time. It is

noted from Figs. 2–7 that fluid velocity is faster in the caseof isothermal plate with uniform surface concentration thanthat in the case of ramped temperature plate with ramped sur-face concentration.

Figure 8 Temperature profiles when Pr = 0.71 and t = 0.5.

Figure 9 Temperature profiles when / = 1 and Pr = 0.71.

Figure 10 Concentration profiles when Sc = 0.22 and t= 0.5.

Figure 11 Concentration profiles when K2 = 0.2 and Sc = 0.22.


The numerical solutions for fluid temperature and speciesconcentration, computed from analytical solutions reportedin Sections 2 and 3, are depicted graphically in Figs. 8–11

for different values of heat absorption coefficient /, chemicalreaction parameter K2 and time t. It is observed fromFigs. 8–11 that, fluid temperature and species concentration

are maximum at the surface of the plate and decrease properlyon increasing boundary layer coordinate y to approach freestream value. Figs. 8 and 9 illustrate the effects of heat absorp-tion coefficient / and time t on fluid temperature. It is evident

that, for both ramped temperature plate with ramped surface

concentration and isothermal plate with uniform surface con-centration, fluid temperature decreases on increasing /whereas it increases on increasing time t. This implies that,

for both ramped temperature plate with ramped surface con-centration and isothermal plate with uniform surface concen-tration, heat absorption has a tendency to reduce the fluidtemperature and there is a rise in fluid temperature with the

progress of time in the boundary layer region. Figs. 10 and11 demonstrate the effects of chemical reaction parameter K2

and time t on species concentration. It is noticed that, for both

ramped temperature plate with ramped surface concentrationand isothermal plate with uniform surface concentration, spe-cies concentration decreases on increasing K2 whereas it

increases on increasing time t. This implies that, for bothramped temperature plate with ramped surface concentrationand isothermal plate with uniform surface concentration,

chemical reaction tends to reduce species concentration andthere is enrichment in species concentration with the progressof time. It may be noted from Figs. 8–11 that the fluid temper-ature and species concentration are lower in the case of

ramped temperature plate with ramped surface concentrationthan that in the case of isothermal plate with uniform surfaceconcentration.

The numerical values of skin friction s, computed from theanalytical expressions reported in Section 4, are presented intabular form in Tables 1–3 for various values of M, Gr, Gc,

K2, / and t taking R= 1, K1 = 0.5, Sc = 0.22 andPr = 0.71 whereas those of Nusselt number Nu and Sherwoodnumber Sh, calculated from the analytical expressions reportedin Section 4, are exhibited in tabular form in Tables 4 and 5 for

different values of /, K2 and t.It is found from Table 1 that, for ramped temperature plate

with ramped surface concentration, skin friction �s increases

on increasing time t whereas, for isothermal plate with uniformsurface concentration, it decreases in magnitude on increasingt whenM P 4. This implies that, for ramped temperature plate

with ramped surface concentration, skin friction is gettingenhanced with the progress of time whereas, for isothermalplate with uniform surface concentration, skin friction is

getting reduced with the progress of time when M P 4. It isworthy to note from Table 1 that there exists flow separationat the isothermal plate on increasing time t when M = 6 andon increasing M when t= 0.7. It is observed from Tables 2

Table 1 Skin friction �s when K1 = 0.5, Gr = 4, Gc = 5, K2 = 0.2 and / = 1.

Ramped Isothermal

Mfltfi 0.3 0.5 0.7 0.3 0.5 0.7

2 0.361179 0.298291 0.186799 �0.52415 �0.94968 �0.880144 0.484248 0.554626 0.594741 �0.79709 �0.70894 �0.359606 0.595863 0.777144 0.937804 �0.71756 �0.38275 0.10460

Table 2 Skin friction �s when M = 4, Gc = 5, K1 = 0.5, K2 = 0.2 and t= 0.5.

Ramped Isothermal

Grfl/fi 1 3 5 1 3 5

2 0.554626 0.568566 0.579712 �0.708944 �0.644484 �0.5830824 0.339164 0.367043 0.389335 �1.22794 �1.09902 �0.9762146 0.123702 0.165521 0.198959 �1.74693 �1.55355 �1.36935

Table 3 Skin friction �s when M = 4, Gr = 4, K1 = 0.5, / = 1 and t= 0.5.

Ramped Isothermal

GcflK2fi 0.2 2 5 0.2 2 5

3 0.817907 0.834995 0.857408 �0.066691 0.003292 0.087601

5 0.554626 0.583106 0.620462 �0.708944 �0.592306 �0.4517907 0.291346 0.331217 0.383515 �1.35120 �1.18790 �0.991182

Table 4 Nusselt number �Nu when Pr = 0.71.

Ramped Isothermal

/fltfi 0.3 0.5 0.7 0.3 0.5 0.7

1 0.571348 0.779133 0.969291 1.11605 0.983021 0.925311

3 0.664544 0.966961 1.26243 1.55005 1.48794 1.47004

5 0.749004 1.12943 1.50704 1.92093 1.89157 1.88595

Table 5 Sherwood number �Sh when Sc = 0.22.

Ramped Isothermal

K2fltfi 0.3 0.5 0.7 0.3 0.5 0.7

0.2 0.295649 0.386593 0.463189 0.525702 0.428415 0.379505

2 0.344659 0.488076 0.625355 0.839945 0.785973 0.757863

5 0.416933 0.628694 0.838894 1.1897 1.12945 1.09522


and 3 that, for ramped temperature plate with ramped surfaceconcentration, skin friction �s increases on increasing either /or K2 and it decreases on increasing either Gr or Gc. For

isothermal plate with uniform surface concentration, �sdecreases in magnitude, attains a minimum and then increaseson increasing K2 when Gc = 3 and it decreases in magnitude

on increasing K2 when Gc = 5and7. �s increases in magnitudeon increasing either Gr or Gc for isothermal plate with uniformsurface concentration. This implies that, for ramped tempera-

ture plate with ramped surface concentration, heat absorptionand chemical reaction have tendency to enhance skin frictionwhereas thermal and species buoyancy forces have reverseeffect on it. For isothermal plate with uniform surface

concentration, heat absorption has a tendency to reduce skinfriction whereas chemical reaction tends to reduce skin frictionwhen Gc P 5 and thermal and species buoyancy forces have

tendency to enhance skin friction. It is evident from Table 3that there exists flow separation at the isothermal plate onincreasing K2 when Gc = 3 and on increasing Gc when

K2 = 5. It is perceived from Table 4 that, for ramped temper-ature plate with ramped surface concentration, Nusselt num-ber �Nu increases on increasing / or t. For isothermal plate

with uniform surface concentration, �Nu increases on increas-ing / and decreases on increasing t. This implies that, for bothramped temperature plate with ramped surface concentrationand isothermal plate with uniform surface concentration, heat


absorption tends to enhance rate of heat transfer at the plate.Rate of heat transfer at the plate is getting enhanced forramped temperature plate with ramped surface concentration

whereas rate of heat transfer at the plate is getting reducedfor isothermal plate with uniform surface concentration withthe progress of time.

It is observed from Table 5 that, for both ramped temper-ature plate with ramped surface concentration and isothermalplate with uniform surface concentration, Sherwood number

�Sh increases on increasing K2. On increasing time t, �Sh

increases for ramped temperature plate with ramped surfaceconcentration whereas it decreases for isothermal plate withuniform surface concentration. This implies that, for both

ramped temperature plate with ramped surface concentrationand isothermal plate with uniform surface concentration,chemical reaction tends to enhance rate of mass transfer at

the plate. For ramped temperature plate with ramped surfaceconcentration, rate of mass transfer at the plate is gettingenhanced whereas for isothermal plate with uniform surface

concentration rate of mass transfer at the plate is gettingreduced with the progress of time.

6. Conclusions

The present study brings out the following significant findings:

(i) For both ramped temperature plate with ramped surfaceconcentration and isothermal plate with uniform surface

concentration:Magnetic field, heat absorption and chemical reactionhave retarding influence on fluid flow. Fluid velocity isgetting accelerated with the progress of time. Heat

absorption has tendency to reduce fluid temperatureand there is a rise in fluid temperature with the progressof time. Chemical reaction tends to reduce species con-

centration and there is enrichment in species concentra-tion with the progress of time.

(ii) For ramped temperature plate with ramped surface con-

centration: skin friction is getting enhanced with theprogress of time when M P 4 whereas heat absorptionand chemical reaction have an accelerating influence

on fluid flow whereas thermal and species buoyancyforces have reverse effect on it.

(iii) For isothermal plate with uniform surface concentra-tion: Skin friction is getting reduced with the progress

of time when M P 4. Heat absorption has a tendencyto reduce skin friction whereas chemical reaction tendsto reduce skin friction when Gc P 5 and thermal and

species buoyancy forces have tendency to enhance skinfriction. There exists flow separation at the plate onincreasing time t when M = 6 and on increasing M

when t = 0.7. Also, there exists flow separation at theplate on increasing K2 when Gc = 3 and on increasingGc when K2 = 5.

(iv) For both ramped temperature plate with ramped surface

concentration and isothermal plate with uniform surfaceconcentration, heat absorption tends to enhance rate ofheat transfer at the plate. Rate of heat transfer at the

plate is getting enhanced for ramped temperature platewith ramped surface concentration whereas rate of heattransfer at the plate is getting reduced for isothermal

plate with uniform surface concentration with the pro-

gress of time.(v) For both ramped temperature plate with ramped surface

concentration and isothermal plate with uniform surface

concentration, chemical reaction tends to enhance rateof mass transfer at the plate. For ramped temperatureplate with ramped surface concentration, rate of masstransfer at the plate is getting enhanced whereas for iso-

thermal plate with uniform surface concentration rate ofmass transfer at the plate is getting reduced with the pro-gress of time.

Acknowledgement

Authors are grateful to the reviewers for their comments andsuggestions which helped them to improve the quality of the

research paper.

References

[1] A. Raptis, N. Kafousias, Heat transfer in flow through a porous

medium bounded by an infinite vertical plate under the action of

a magnetic field, Int. J. Energy Res. 6 (1982) 241.

[2] A. Raptis, Flow through a porous medium in the presence of a

magnetic field, Int. J. Energy Res. 10 (1986) 97.

[3] A.J. Chamkha, Transient MHD free convection from a porous

medium supported by a surface, Fluid/Part. Sep. J. 10 (1997) 101.

[4] A.J. Chamkha, Hydromagnetic natural convection from an

isothermal inclined surface adjacent to a thermally stratified

porous medium, Int. J. Eng. Sci. 35 (1997) 975.

[5] T.K. Aldoss, M.A. Al-Nimr, M.A. Jarrah, B.J. Al-Shaer,

Magnetohydrodynamic mixed convection from a vertical plate

embedded in a porous medium, Numer. Heat Transf. 28 (1995)

635.

[6] Y.J. Kim, Unsteady MHD convective heat transfer past a semi-

infinite vertical porous moving plate with variable suction, Int. J.

Eng. Sci. 38 (2000) 833.

[7] B.K. Jha, MHD free convective and mass-transfer flow through

a porous medium, Astrophys. Space Sci. 175 (1991) 283.

[8] F.S. Ibrahim, I.A. Hassanien, A.A. Bakr, Unsteady

Magnetohydrodynamic micropolar fluid flow and heat transfer

over a vertical porousmedium in the presence of thermal andmass

diffusion with constant heat source, Can. J. Phys. 82 (2004) 775.

[9] O.D. Makinde, P. Sibanda, Magnetohydrodynamic mixed

convective flow and heat and mass transfer past a vertical

plate in a porous medium with constant wall suction, J. Heat

Transf. 130 (2008) 112602-1–112602-8.

[10] O.D. Makinde, OnMHD boundary layer flow and mass transfer

past a vertical plate in a porous medium with constant heat flux,

Int. J. Numer. Meth. Heat Fluid Flow 19 (2009) 546.

[11] O.D. Makinde, On MHD heat and mass transfer over a moving

vertical plate with a convective surface boundary condition,

Can. J. Chem. Eng. 88 (2010) 983.

[12] K. Vajravelu, J. Nayfeh, Hydromagnetic convection at a cone

and a wedge, Int. Comm. Heat Mass Transf. 19 (1992) 701.

[13] J.C. Crepeau, R. Clarksean, Similarity solutions of natural

convection with internal heat generation, ASME J. Heat Transf.

119 (1997) 183.

[14] D.P. McKenzie, J.M. Roberts, N.O. Weiss, Convection in

Earth’s mantle: towards a numerical simulation, J. Fluid Mech.

62 (1974) 465.

[15] I. Baker, R. Faw, F.A. Kulacki, Post accident heat removal

part-1: heat transfer within an internally heated non boiling

liquid layer, Nucl. Sci. Eng. 61 (1976) 222.

http://refhub.elsevier.com/S1110-256X(14)00043-1/h0005













































[16] M.A. Delichatsios, Air entrainment in to buoyant jet flames and

pool fires, in: P.J. DiNenno et al. (Eds.), The SFPA Handbook

of Fire Protection Engineering, NFPA publications, M. A.

Quincy, 1988, p. 306.

[17] B.R. Westphal, D.D. Keiser, R.H. Rigg, D.V. Laug, Production

of metal waste forms from spent nuclear fuel treatment, in: DOE

Spent Nuclear Fuel Conference, Salt Lake City, UT, 1994, p.

288.

[18] E.M. Sparrow, R.D. Cess, The effect of magnetic field on free

convection heat transfer, Int. J. Heat Mass Transf. 3 (1961) 267.

[19] D. Moalem, Steady state heat transfer with porous medium with

temperature dependent heat generation, Int. J. Heat Mass

Transf. 19 (1976) 529.

[20] A.J. Chamkha, A.A. Khaled, Hydromagnetic combined heat

and mass transfer by natural convection from a permeable

surface embedded in a fluid saturated porous medium, Int. J.

Numer. Meth. Heat Fluid Flow 10 (2000) 455.

[21] M.H. Kamel, Unsteady MHD convection through porous

medium with combined heat and mass transfer with heat

source/sink, Energy Convers. Manage. 42 (2001) 393.

[22] A.J. Chamkha, Unsteady MHD convective heat and mass

transfer past a semi-infinite vertical permeable moving plate with

heat absorption, Int. J. Eng. Sci. 42 (2004) 217.

[23] O.D. Makinde, Heat and mass transfer by MHD mixed

convection stagnation point flow toward a vertical plate

embedded in a highly porous medium with radiation and

internal heat generation, Meccanica 47 (2012) 1173.

[24] A.A. Hayday, D.A. Bowlus, R.A. McGraw, Free convection

from a vertical plate with step discontinuities in surface

temperature, ASME J. Heat Transf. 89 (1967) 244.

[25] M. Kelleher, Free convection from a vertical plate with

discontinuous wall temperature, ASME J. Heat Transf. 93

(1971) 349.

[26] T.T. Kao, Laminar free convective heat transfer response along

a vertical flat plate with step jump in surface temperature, Lett.

Heat Mass Transf. 2 (1975) 419.

[27] S. Lee, M.M. Yovanovich, Laminar natural convection from a

vertical plate with a step change in wall temperature, ASME J.

Heat Transf. 113 (1991) 501.

[28] P. Chandran, N.C. Sacheti, A.K. Singh, Natural convection

near a vertical plate with ramped wall temperature, Heat Mass

Transf. 41 (2005) 459.

[29] R.R. Patra, S. Das, R.N. Jana, S.K. Ghosh, Transient approach

to radiative heat transfer free convection flow with ramped wall

temperature, J. Appl. Fluid Mech. 5 (2012) 9.

[30] G.S. Seth, Md.S. Ansari, MHD natural convection flow past an

impulsively moving vertical plate with ramped wall temperature

in the presence of thermal diffusion with heat absorption, Int. J.

Appl. Mech. Eng. 15 (2010) 199.

[31] G.S. Seth, R. Nandkeolyar, Md.S. Ansari, Effect of rotation on

unsteady hydromagnetic natural convection flow past an

impulsively moving vertical plate with ramped temperature in

a porous medium with thermal diffusion and heat absorption,

Int. J. Appl. Math Mech. 7 (2011) 52.

[32] A.A. Afify, Effect of radiation on free convective flow and mass

transfer past a vertical isothermal cone surface with chemical

reaction in the presence of a transverse magnetic field, Can. J.

Phys. 82 (2004) 447.

[33] R. Muthucumaraswamy, P. Chandrakala, Radiative heat and

mass transfer effects on moving isothermal vertical plate in the

presence of chemical reaction, Int. J. Appl. Mech. Eng. 11 (2006)

639.

[34] F.S. Ibrahim, A.M. Elaiw, A. Bakr, Effect of the chemical

reaction and radiation absorption on the unsteady MHD free

convection flow past a semi-infinite vertical permeable moving

plate with heat source and suction, Comm. Nonlin. Sci. Numer.

Simul. 13 (2008) 1056.

[35] A.A. Bakr, Effects of chemical reaction on MHD free

convection and mass transfer flow of a micropolar fluid with

oscillatory plate velocity and constant heat source in a rotating

frame of reference, Comm. Nonlin. Sci. Numer. Simul. 16 (2011)

698.

[36] A.J. Chamkha, MHD flow of a uniformly stretched vertical

permeable surface in the presence of heat generation/absorption

and a chemical reaction, Int. Comm. Heat Mass Transf. 30

(2003) 413.

[37] A.J. Chamkha, R.A. Mohamed, S.E. Ahmed, Unsteady MHD

natural convection from a heated vertical porous plate in a

micropolar fluid with Joule heating, chemical reaction and

radiation effects, Meccanica 46 (2011) 399.

[38] K. Bhattacharyya, G.C. Layek, Similarity solution of MHD

boundary layer flow with diffusion and chemical reaction over a

porous flat plate with suction/blowing, Meccanica 47 (2012)

1043.

[39] R.A. Mohamed, A-N.A. Osman, S.M. Abo-Dahab,

Unsteady MHD double-diffusive convection boundary-layer

flow past a radiate hot vertical surface in porous media in the

presence of chemical reaction and heat sink, Meccanica 48

(2013) 931.

[40] K.R. Cramer, S.I. Pai, Magnetofluid Dynamics for Engineers

and Applied Physicists, McGraw Hill Book Company, New

York, USA, 1973.




























































































hydromagnetic natural convection flow with heat and mass ... · convection ﬂow of micro-polar...

Documents