combined influence of radiation absorption and hall

Ain Shams Engineering Journal (2016) 7, 383–397

Ain Shams University

Ain Shams Engineering Journal

www.elsevier.com/locate/asejwww.sciencedirect.com

ENGINEERING PHYSICS AND MATHEMATICS

Combined influence of radiation absorption and

Hall current effects on MHD double-diffusive free

convective flow past a stretching sheet

* Corresponding author.

Peer review under responsibility of Ain Shams University.

Production and hosting by Elsevier

http://dx.doi.org/10.1016/j.asej.2015.11.0242090-4479 � 2016 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V.This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

G. Sreedevia,*, R. Raghavendra Rao

a, D.R.V. Prasada Rao

b, A.J. Chamkha

c

aDepartment of Mathematics, K.L. University, Green Fields, Vaddeswaram, Guntur 522502, AP, IndiabDepartment of Mathematics, S.K. University, Anantapur 515003, AP, IndiacMechanical Engineering Department, Prince Mohammad Bin Fahd University (PMU), Al-Khobar 31952, Saudi Arabia

Received 26 July 2015; revised 14 November 2015; accepted 29 November 2015Available online 13 February 2016

KEYWORDS

Radiation absorption;

Variable viscosity;

Soret effect;

MHD;

Hall current;

Heat and mass transfer

Abstract An analysis has been carried out on the influence of radiation absorption, variable vis-

cosity, Hall current of a magnetohydrodynamic free-convective flow and heat and mass transfer

over a stretching sheet in the presence of heat generation/absorption. The fluid viscosity is assumed

to vary as an inverse linear function of temperature. The boundary-layer equations governing the

fluid flow, heat and mass transfer under consideration have been reduced to a system of nonlinear

ordinary differential equations by employing a similarity transformation. Using the finite difference

scheme, numerical solutions to the transform ordinary differential equations have been obtained

and the results are presented graphically. The numerical results obtained are in good agreement

with the existing scientific literature.� 2016 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an

open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Industries based on engineering disciplines, preferably chemi-

cal engineering sectors encounter fluid, heat, and mass flowinduced by continuous stretching heated surfaces. Some exam-ples are the extrusion process, wire and fiber coating, polymer

processing, foodstuff processing, and design of various heatexchangers. As stretching brings a unidirectional orientation

to the extrudate, the quality of the final product considerablydepends on the flow and heat and mass transfer mechanism.

Steady heat flow on a moving continuous flat surface was firstconsidered by Sakiadis [1] who developed a numerical solutionusing a similarity transformation. Crane [2] motivated by the

processes of polymer extrusion, in which the extrudate emergesfrom a narrow slit, examined semi-infinite fluid flow driven bya linearly stretching surface. Many researchers and academi-cians have advanced their studies relating to problems involv-

ing MHD in stretching sheet considering various parameters:Sharidan [3], Carraagher et al. [4], Gupta and Gupta [5], Ishaket al. [6–8], Aziz et al. [9], Abel et al. [10], Mukhopadhyay and

Mondal [11], Krishnendu [12], Swati Mukhopadhyay et al.[13], Laha et al. [14], Afzal [15], Prasad et al. [16], Wang [17]among others. Rashad et al. [18] considered MHD free

http://crossmark.crossref.org/dialog/?doi=10.1016/j.asej.2015.11.024&domain=pdf

http://creativecommons.org/licenses/by-nc-nd/4.0/

http://dx.doi.org/10.1016/j.asej.2015.11.024

http://www.sciencedirect.com/science/journal/20904479

http://dx.doi.org/10.1016/j.asej.2015.11.024

http://creativecommons.org/licenses/by-nc-nd/4.0/

Nomenclature

~B magnetic induction vector

Bo strength of applied magnetic fieldC dimensionless concentration variableCf local skin-friction coefficientCs concentration susceptibility factor

Cp specific heat at constant pressureD thermal molecular diffusivityDm solution diffusivity of the medium~E intensity vector of the electric fieldE charge of the electronG Grashof number

go acceleration due to gravity~J electric current density vectorK mean absorption coefficientKf thermal conductivity

k0 measures the rate of reactionKT thermal diffusion ratioM Hartmann number

m Hall current parameterN Buoyancy ratioNr thermal radiation parameter

Nu local Nusselt numberne number of density of the electronPr Prandtl number

Q1 radiation absorptionSc Schmidt numberSr Soret parameterSh local Sherwood number

T dimensional temperature variable

Tw fluid temperature at wallsT1 free stream temperatureu,v,w velocity componentsx,y,z direction components

le magnetic permeability~V velocity vector

Greek symbols

a thermal diffusivitybc coefficient of species concentrationbT coefficient of thermal expansiong similarity variable

q1 fluid densityl co-efficient of dynamic viscosityr electrical conductivity of the fluid

h non-dimensional temperaturehr viscosity parameter/ non-dimensional concentration

k Heat generation/absorption parameterc non-dimensional chemical reaction parametersx, sz shear stress componentsm kinematic viscosity

Subscriptsc cold wallw warm side wall

384 G. Sreedevi et al.

convective heat and mass transfer of a chemically-reactingfluid from radiate stretching surface embedded in a saturated

porous medium. Theuri et al. [19] studied unsteady double dif-fusive magnetohydrodynamic boundary layer flow of a chem-ically reacting fluid over a flat permeable surface.

The influence of a uniform transverse magnetic field on themotion of an electrically conducting fluid past a stretchingsheet was investigated by Pavlov [20]. The effect of chemical

reaction on free-convective flow and mass transfer of a viscous,incompressible and electrically conducting fluid over a stretch-ing sheet was investigated by Afify [21] in the presence of trans-verse magnetic field. Das et al. [22] investigated MHD

Boundary layer slip flow and heat transfer of nanofluid pasta vertical stretching sheet with non-uniform heat generation/absorption. Ibrahim and Makinde [23] studied the double-

diffusive mixed convection and MHD Stagnation point flowof nanofluid over a stretching sheet. Rudraswamy et al. [24]considered the effects of magnetic field and chemical reaction

on stagnation-point flow and heat transfer of a nanofluid overan inclined stretching sheet.

In most of the investigations, the electrical conductivity ofthe fluid was assumed to be uniform and the magnetic field

intensity was assumed to be low. However, in an ionized fluidwhere the density is low and thereby magnetic field intensity isvery strong, the conductivity normal to the magnetic field is

reduced due to the spiraling of electrons and ions about themagnetic lines of force before collisions take place and acurrent induced in a direction normal to both the electric

and magnetic fields. This phenomenon is known as Hall Effect.Hall currents are important and they have a marked effect on

the magnitude and direction of the current density and conse-quently on the magnetic force term. The problem of MHD freeconvection flow with Hall currents has many important engi-

neering applications such as in power generators, MHD accel-erators, refrigeration coils, transmission lines, electrictransformers, and heating elements, and Watanabe and Pop

[25], Abo-Eldahab and Salem [26], Rana et al. [27], Singhand Gorla [28], and Shit [29] among others have advancedstudies on Hall effect on MHD past stretching sheet. Shateyiand Motsa [30] considered boundary layer flow and double dif-

fusion over an unsteady stretching surface with Hall effect.Chamkha et al. [31] analyzed the unsteady MHD free convec-tive heat and mass transfer from a vertical porous plate with

Hall current, thermal radiation and chemical reaction effects.The effect of radiation on MHD flow and heat transfer

problem has become more important industrially. At high

operating temperature, radiation effect can be quite significant.Many processes in engineering areas occur at high temperatureand knowledge of radiation heat transfer becomes very impor-tant for the design of the pertinent equipment. Nuclear power

plants, gas turbines and the various propulsion devices for air-craft, missiles, satellites and space vehicles are examples ofsuch engineering areas. Several authors [32–38] are among var-

ious names who studied the problems relating to MHD withradiation effect. Makinde [39] analyzed chemically reactinghydromagnetic unsteady flow of a radiating fluid past a

Fig. 1 Physical Sketch of the Problem.

MHD double-diffusive free convective flow 385

vertical plate with constant heat flux. Chamkha et al. [40] fur-ther analyzed the unsteady coupled heat and mass transfer bymixed convection flow of a micropolar fluid near the stagna-

tion point on a vertical surface in the presence of radiationand chemical reaction.

To accurately predict the flow and heat transfer rates, it is

necessary to take into account the temperature-dependentviscosity of the fluid. The effect of temperature-dependent vis-cosity on heat and mass transfer laminar boundary layer flow

has been discussed by many authors [41–46] in various situa-tions. They showed that when this effect was included, the flowcharacteristics might change substantially compared with theconstant viscosity assumption. Salem [47] investigated variable

viscosity and thermal conductivity effects on MHD flow andheat transfer in viscoelastic fluid over a stretching sheet. Deviand Ganga [48] have considered the viscous dissipation effects

on MHD flows past stretching porous surfaces in porousmedia. Recently, Mohamed El-Aziz [49] analyzed unsteadymixed convection heat transfer along a vertical stretching sur-

face with variable viscosity and viscous dissipation. Tian et al.[50] investigated the 2D boundary layer flow and heat transferin variable viscosity MHD flow over a stretching plate.

Studies related to double diffusive hydromagnetic bound-ary layer flow with heat and mass transfer over flat surfacesare extremely important and have many applications in engi-neering and industrial processes [51,52]. For instance, heat

and mass transfer occur in processes, such as drying, evapora-tion at the surface of a water body, and energy transfer in a wetcooling tower, cooling of nuclear reactors, MHD power gener-

ators, MHD pump, chemical vapor deposition on surfaces,formation and dispersion of fog, and distribution of tempera-ture and moisture over agriculture fields. Moreover, when heat

and mass transfer occur simultaneously between the fluxes, thedriving potentials are of more intricate nature [53–55]. Anenergy flux can be generated not only by temperature gradients

but by composition gradients. The energy flux caused by acomposition is called Dufour or diffusion-thermo effect. Tem-perature gradients can also create mass fluxes, and this is theSoret or thermal-diffusion effect. The concept of Soret and

Dufour effects on heat and mass transfer has been developedfrom the kinetic theory of gases by Chapman and Cowling[56] and Hirshfelder et al. [57]. They explained the phenomena

and derived the necessary formulae to calculate the thermaldiffusion coefficient and the thermal-diffusion factor formonatomic gases or for polyatomic gas mixtures. The

thermal-diffusion effect, for instance, has been utilized for iso-tope separation and in mixture between gases with very lightmolecular weight (Hydrogen–Helium) and of medium molecu-lar weight (Nitrogen–air) and the diffusion-thermo effect was

found to be of a magnitude such that it cannot be neglectedKafoussias and Williams [58]. Alam et al. [59] studied Dufourand Soret effects on steady free convection and mass transfer

flow past a semi-infinite vertical porous plate in a porous med-ium. Makinde [60] studied numerically the influence of a mag-netic field on heat and mass transfer by mixed convection from

vertical surfaces in porous media considering Soret andDufour effects. Makinde and Olanrewaju [61] have investi-gated the Dufour and Soret effects on unsteady mixed convec-

tion flow past a porous plate moving through a binary mixtureof chemically reacting fluid. The thermal diffusion and diffu-sion thermo effects on chemically reacting hydromagneticboundary layer flow of heat and mass transfer past a moving

vertical plate with suction/injection have been addressed byOlanrewaju and Makinde [62]. Zhang et al. [63] 2013 consid-ered the unsteady heat and mass transfer in MHD over an

oscillatory stretching surface with Soret and Dufour effects.Recently, Rashidi et al. [64] analyzed heat and mass transferfor MHD viscoelastic fluid flow over a vertical stretching sheet

considering Soret and Dufour effects.Motivated by the above-mentioned researchers, this paper

aims at studying the combined influence of radiation absorp-

tion, variable viscosity effect, thermo-diffusion and Hall cur-rent on the hydromagnetic free-convective flow, heat andmass transfer flow past a stretching surface in the presenceof heat generation/absorption source. Since the governing

differential equations are highly nonlinear, the governingequations have been solved by employing fifth-orderRunge–Kutta–Fehlberg method along with shooting

technique. The effects of various parameters on the velocity,temperature and concentration as well as on the local skin-friction coefficient, local Nusselt number and local Sherwood

number are presented graphically and discussed quantitatively.

2. Problem formulation

We consider the steady free-convective flow, heat and masstransfer of an incompressible, viscous and electrically conduct-ing fluid past a stretching sheet and the sheet is stretched with a

velocity proportional to the distance from a fixed origin O(cf. Fig. 1).

A uniform strong magnetic field of strength B0 is imposedalong the y-axis and the effect of Hall current is taken into

account. Taking Hall effects into account the generalizedOhm’s law provided by Cowling [65] may be put in the follow-ing form

~J ¼ r1þm2

~Eþ ~V � þ~B� 1

ene~f � ~B

� �

where m ¼ rBo

encis defined as the Hall current parameter. A very

interesting fact that the effect of Hall current gives rise to aforce in the z-direction which in turn produces a cross-flow

velocity in this direction and then the flow becomes three-dimensional. The temperature and the species concentrationare maintained at prescribed constant values Tw, Cw at the


sheet and T1 and C1 are the fixed values far away from the

sheet.Following Lai and Kulacki [66], the fluid viscosity l is

assumed to vary as a reciprocal of a linear function of temper-

ature given by

1

l¼ 1

l1½1þ c0ðT� T1Þ� ð1Þ

1

l1¼ aðT� T1Þ ð2Þ

where a ¼ c0l1

and Tr ¼ T1 � 1c0.

In the above equation both a and Tr are constants, and

their values depend on the thermal property of the fluid, i.e.,g0. In general a > 0 represent for liquids, whereas for gasesa < 0.

Owing to the above-mentioned assumptions, the boundarylayer free-convection flow with mass transfer and generalizedOhm’s law with Hall current effect are governed by the follow-

ing system of equations:

@u

@xþ @v

@y¼ 0 ð3Þ

qm u @u@xþ v @u

@y

� �¼ @

@yl @u

@y

� �þ qmg0bs T� T1ð Þ

þqmg0bc C� C1ð Þ � rB20

1þm2 ðuþmwÞ

9=; ð4Þ

qm u@w

@xþ v

@w

@y

� �¼ @

@yl@w

@y

� �þ rB2

0

1þm2ðmu� wÞ ð5Þ

qmCp u@T

@xþ v

@T

@y

� �¼ kf

@2T

@y2�Q T� T1ð Þ þQ 0

1 ðC� C1Þ

ð6Þ

u@C

@xþ v

@C

@y¼ Dm

@2C

@y2� k0 C� C1ð Þ þDmKT

Tm

@2T

@y2ð7Þ

where (u; v;w) are the velocity components along the (x; y; z)directions respectively. Dm is the solution diffusivity of the

medium, KT is the thermal diffusion ratio, Cs is the concentra-tion susceptibility, Cp is the specific heat at constant pressure

and Tm is the mean fluid temperature.The boundary conditions for the present problem can be

written as

u ¼ bx; v ¼ w ¼ 0;T ¼ Tw;C ¼ Cw at y ¼ 0 ð8Þ

u ! 0;w ! 0;T ! T1;C ! C1 at y ! 1 ð9Þwhere bð> 0Þ being stretching rate of the sheet. The boundaryconditions on velocity in Eq. (8) are the no-slip condition at

the surface y = 0, while the boundary conditions on velocityat y ! 1 follow from the fact that there is no flow far awayfrom the stretching surface.

To examine the flow regime adjacent to the sheet, the fol-lowing transformations are invoked

u ¼ bxf 0ðgÞ; v ¼ �ffiffiffiffiffibv

pfðgÞ;w ¼ bxgðgÞ;

g ¼ffiffiffib

v

ry; hðgÞ ¼ T� T1

Tw � T1;/ ¼ C� C1

Cw � C1ð10Þ

where f is a dimensionless stream function, h is the similarityspace variable, h and / are the dimensionless temperature

and concentration respectively. Clearly, the continuity Eq.

(3) is satisfied by u and v defined in Eq. (9). Substituting Eq.(10) the Eqs. (4)–(7) reduce to

h� hrhr

� �f 0 � ff 00ð Þ þ f 000 � h 0

h� hr

� �f 00�

h� hrhr

� �GðhþN/Þ þM2 h 0 � hr

hr

� �f 0 þmg

1þm2

� �¼ 0

ð11Þ

h� hrhr

� �f 0g� fg0ð Þ þ g00 � h 0

h� hr

� �g 0

�M2 h� hrhr

� �mf 0 þ g

1þm2

� �¼ 0 ð12Þ

1þ 4Nrr3

� �h00 þ Prkh�Q/ ¼ 0 ð13Þ

/00 � Scðf/ 0 � c/Þ ¼ �ScSrh00 ð14ÞSimilarly, the transformed boundary conditions are given by

f 0ðgÞ ¼ 1; fðgÞ ¼ 0; gðgÞ ¼ 0; hðgÞ ¼ 1;/ðgÞ ¼ 1 at g ¼ 0 ð15Þ

f 0ðgÞ ! 0; gðgÞ ! 0; hðgÞ ! 0;/ðgÞ ! 0 at g ! 1 ð16Þwhere a prime denotes the differentiation with respect to g onlyand the dimensionless parameters appearing in the Eqs. (11)–

(16) are respectively defined as hr ¼ Tr�T1Tw�T1

¼ � 1c0ðTw�T1Þ

h ithe

viscosity parameter, M ¼ rB20

P1bthe magnetic parameter,

Pr ¼ qCpv

kthe Prandtl number, c ¼ k0

bðCw � C1Þ the non-

dimensional chemical reaction parameter, G ¼ g0bTðTw�T1Þb2x

the

local Grashof number, N ¼ bCðCw�C1ÞbTðTw�T1Þ the Buoyancy ratio,

Nr ¼ kk04T31r� the thermal radiation parameter, Sr ¼ DmKT

CsCpthe

Soret parameter, Q1 ¼ Q 01ðCw�C1Þ

kfb2ðTw�T1Þ the radiation absorption

parameter, Sc ¼ lq1D

the Schmidt number and k ¼ Qq1Cpb

is

defined as the heat generation/absorption parameter.

3. Numerical procedure

The coupled ordinary differential Eqs. (11)–(14) are of third-

order in f, and second-order in g, h and / which have beenreduced to a system of nine simultaneous equations offirst-order for nine unknowns. In order to solve this systemof equations numerically we require nine initial conditions

but two initial conditions on f and one initial condition eachon g, h and / are known. However the values of f1, g, h and/ are known at g ! 1. These four end conditions are utilized

to produce four unknown initial conditions at g = 0 by usingshooting technique. The most crucial factor of this scheme is tochoose the appropriate finite value of g1. In order to estimate

the value of g1, we start with some initial guess value and solvethe boundary value problem consisting of Eqs. (11)–(14) to

obtain f11(0), g1(0), h1(0) and /1(0). The solution process is

repeated with another large value of g1 until two successive

values of f11(0), g1(0), h1(0) and /1(0) differ only after desiredsignificant digit. The last value of g1is taken as the final value

of g1 for a particular set of physical parameters for determin-

ing velocity components f1ðgÞ, g(gÞ, temperature hðgÞ andconcentration /ðgÞ in the boundary layer. After knowing all


the nine initial conditions, we solve this system of simultaneousequations using fifth-order Runge–Kutta–Fehlberg integrationscheme with automatic grid generation scheme which ensures

convergence at a faster rate. The value of g1 greatly dependsalso on the set of the physical parameters such as Magneticparameter, Hall parameter, buoyancy ratio, Heat source

parameter, Prandtl number, thermal radiation parameter, por-ous parameter, Schmidt number, Soret number and chemicalreaction parameter so that no numerical oscillations would

occur. During the computation, the shooting error was con-trolled by keeping it to be less than 10�6. Thus, the couplednonlinear boundary value problem of third-order in f,second-order in g, h and / has been reduced to a system of

nine simultaneous equations of first-order for nine unknownsas follows:

f1 ¼ f;f11 ¼ f2;f12 ¼ f3;

f13 ¼f7

f6�hr

� �f3�

f6�hrhr

� �f22� f1f3� �þ f6�hr

hr

� �Gðf6þ f8Þ

�M2 f6�hrhr

� �f2þmf41þm2

� �f4 ¼ g;f14 ¼ f5;

f15 ¼f7

f6�hr

� �f5�

f6�hrhr

� �f2f4� f1f5ð Þ

þM2 f6�hrhr

� �mf2� f41þm2

� �f6 ¼ h; f16 ¼ f7;

f17 ¼ ½�3Prkf6=ð3þ4NrÞ�þ3Qf8=ð3þ4NrÞf8 ¼/; f 08 ¼ f9;

f 09 ¼ Scðcf8� f1f9Þ�3ScSrðPrkf6þQ1f8=ð3þ4NrÞh i

ð17ÞWhere

f1 ¼ f; f2 ¼ f 1; f3 ¼ f 11; f4 ¼ h; f5 ¼ g1; f6 ¼ h; f7 ¼ h1;

f8 ¼ /; f 18 ¼ /1 ¼ f9 ð18Þand a prime denotes differentiation with respect to g. The

boundary conditions now become

f1 ¼ 0; f2 ¼ 1; f4 ¼ 0; f6 ¼ 1; f8 ¼ 1 at g ¼ 0

f2 ! 0; f4 ! 0; f6 ! 0; f8 ! 0 at g ! 1 ð19Þ

Since f3(0), f5(0), f7(0) and f9(0) are not prescribed so wehave to start with the initial approximations as f3(0) = s10,

f5(0) = s20, f7(0) = s30 and f9(0) = s40. Let c1; c2; c3 and c4be the correct values of f3(0), f5(0) and f7(0) respectively. Theresultant system of nine ordinary differential equations is inte-

grated using fifth-order Runge–Kutta–Fehlberg method anddenotes the values of f3, f5, f7 and f9 at g ¼ g1 by f3(s10, s20,s30, s40;g1), f5(s10, s20, s30, s40;g1), f7(s10, s20, s30, s40; g1) andf9(s10, s20, s30, s40; g1) respectively. Since f3, f5, f7 and f9 atg ¼ g1 are clearly function of c1; c2; c3 and c4, they areexpanded in Taylor series around c1 � s10; c2 � s20; c3 � s30and c4 � s40 respectively by retaining only the linear terms.The use of difference quotients is made for the derivativesappeared in these Taylor series expansions.

Thus, after solving the system of Taylor series expansionsfor dc1 ¼ c1 � s10; dc2 ¼ c2 � s20; dc3 ¼ c3 � s30, and dc4 ¼c4 � s40 we obtain the new estimates s11 ¼ s10 þ ds10,s21 = s20 þ ds20, s31 ¼ s30 þ ds30 and s41 ¼ s40 þ ds40. Thereafterthe entire process is repeated starting with f1(0), f2(0), s11, f4(0),s21, s31 and s41 as initial conditions. Iteration of the whole

outlined process is repeated with the latest estimates ofc1; c2; c3 and c4 until prescribed boundary conditions aresatisfied.

Finally, s1n = s1ðn�1Þ þ ds1ðn�1Þ, s2n ¼ s2ðn�1Þ þ ds2ðn�1Þ,s3n = s3ðn�1Þ þ ds3ðn�1Þ and s4n = s4ðn�1Þ þ ds4ðn�1Þ for

n= 1,2,3, . . . are obtained which seemed to be the mostdesired approximate initial values of f3(0), f5(0), f7(0) and

f9(0). In this way all the six initial conditions are determined.Now it is possible to solve the resultant system of seven simul-taneous equations by fifth-order Runge–Kutta–Fehlberg inte-

gration scheme so that velocity, temperature fields andconcentration for a particular set of physical parameters caneasily be obtained. The results are provided in several tablesand graphs.

3.1. Skin friction, Nusselt number and Sherwood number

The important characteristics of the present investigation are

the local skin-friction coefficient Cf, the local Nusselt number

Nu and the local Sherwood number Sh defined by

Cf ¼ sw

lbxffiffibv

q ¼ f 00ð0Þ ð20Þ

where

sw ¼ l@T

@y

� �y¼0

¼ lbx

ffiffiffib

v

rf 00ð0Þ;Nu ¼ _qw

kffiffibv

qðTw � T1Þ

ð21Þ

where

qw ¼ �k@T

@y

� �y¼0

¼ �k

ffiffiffib

v

rðTw � T1Þh 0ð0Þ and

Sh ¼ mw

Dffiffibv

qðCw � C1Þ

¼ �/ 0ð0Þ ð22Þ

where mw ¼ �D @C@y

� �y¼0

¼ �Dffiffibv

qðCw � C1Þ/ 0ð0Þ.

If we consider M ¼ m ¼ 0, hr ! 1 andSr ¼ Sc ¼ hr ¼ N ¼ 0, the present flow problem becomes

hydrodynamic boundary-layer flow past a stretching sheetwhose analytical solution was put forwarded by Crane (2) asfollows

fðgÞ ¼ 1� e�n i:e:; f 0ðgÞ ¼ e�n ð23ÞAn attempt has been made to validate our result for the axial

velocity f 0ðgÞ. We compared our results with this analytical

solution and found to be in good agreement.

4. Comparison

In this analysis, it should be mentioned that the resultsobtained herein are compared with the results of Shit et al.[67] and Rahman et al. [68] as shown in Tables 1 and 2 and

the results are found to be in good agreement.

5. Results and discussion

Boundary value problem for momentum, heat and mass trans-fer in flow of Newtonian fluid over a stretching sheet was

Table 1 Comparison of Nu and Sh at g= 0 with Shit et al. [67] with Sr = 0, Q1 = 0.

M Nr c k hr Shit et al. [67] results Present results

Nu(0) Sh(0) Nu(0) Sh(0)

0.5 1 0.5 0.5 �2 �0.6912 0.6265 �0.69045 0.62431

1.5 1 0.5 0.5 �2 �0.6977 0.6543 �0.69452 0.65234

0.5 3 0.5 0.5 �2 �12.3751 0.9278 �12.3698 0.91998

0.5 1 1.5 0.5 �2 �0.6956 1.0959 �0.6921 1.091243

0.5 1 �0.5 0.5 �2 �0.6966 0.4898 �0.69345 0.4876

0.5 1 0.5 1.5 �2 �0.6968 0.4245 �0.6923 0.4143

0.5 1 0.5 1.5 �2 �0.5974 0.4071 �0.5946 0.4052

0.5 1 0.5 1.5 �4 �0.6969 0.6253 �0.6956 0.6214

Table 2 Comparison of Nu and Sh at g= 0 with Rahman

et al. [68] with Pr = 0.71, m= Sr = Q1 = 0, Nr = k ¼ c= 0,

for different values of hr.

Different values of

hrRahman et al. [68]

results

Present results

Nu(0) Sh(0) Nu(0) Sh(0)

�2 �0.37865 4.805362 �0.37756 4.80543

�4 �0.30536 4.54266 �0.30493 4.54265

2 0.248266 4.485639 0.24233 4.48562

4 0.238773 4.33562 0.23877 4.33561


solved numerically using Runge–Kutta–Fehlberg method withshooting technique. In order to check the accuracy the numer-

ical solution procedure used a comparison of skin friction withthat of Shit et al. [67] for Sr = Q1 = 0 for different values ofM, Nr, c; k; hr and Rahman et al. [68] for m= Sr = Q1 =

Nr = c ¼ k = 0 for different values of hr and the results aretabulated in Tables 1 and 2. From these Tables 1 and 2 thecomparison in the above case is found to be in good agreement

and thus verifies the accuracy of numerical method used in thepresent work. The slight deviation in the value may be due tothe use of Runge–Kutta–Fehlberg method which has fifth-order accuracy; thus, the present results are more accurate.

Hence this confirms that the numerical method adopted inthe present study gives very accurate results. The system ofcoupled nonlinear Eqs. (14)–(17) together with the boundary

conditions (18) and (19) has been solved numerically.

Fig. 2 Effect of Hall current (m) on axial velocity (f0Þ.

In our present study the numerical values of the physicalparameters have been chosen so that M, m, Nr, c, Sr and hrare varied over a range which is listed in the following figures(cf. [25,21,67,69]).

Figs. 2–5 represent the axial velocity f 0 and the g 0 compo-nent of velocity (g), which is induced due to the presence of theHall effects. All these figures show that for any particular val-ues of the physical parameters g(gÞ reaches a maximum value

at a certain height g above the sheet and beyond which g(g)decreases gradually in asymptotic nature for different valuesof m, c, Q1, and Sr.

5.1. Velocity profiles

From Figs. 2 and 3 it is found that the axial velocity f 0ðgÞincreases with increase in Hall parameter. This increase isdue to the fact that as m increases the Lorentz force opposesthe flow and leads to the degeneration of the fluid motion.

In this study, in the absence of magnetic field parameterM = 0 there is no transverse velocity g(gÞ= 0 and as magneticfield increases a cross flow velocity in the transverse directiongreatly induces to the Hall effect. With increase in Hall param-

eter m, the effective conductivity decreases which results in the

reduction of a magnetic damping force on f 0ðgÞ and the reduc-

tion in damping force is coupled with the fact that magnetic

field has propelling effect on f 0ðgÞ. For increase in the Hallparameter m, we notice an enhancement in a cross flow veloc-

ity. Figs. 4 and 5 represent f 0ðgÞ and g(gÞ with chemical

Fig. 3 Effect of Hall current (m) on cross flow velocity (g).

Fig. 4 Effect of chemical reaction parameter (cÞ on axial velocity

(f 0Þ.

Fig. 5 Effect of chemical reaction parameter (cÞ on cross flow

velocity (g).

Fig. 6 Effect of radiation absorption parameter (Q1) on axial

velocity (f 0Þ.

Fig. 7 Effect of radiation absorption parameter (Q1) on cross

flow velocity (g).

Fig. 8 Effect of Soret parameter (Sr) on axial velocity (f 0Þ.


reaction parameter c. It is pertinent to mention that c > 0 cor-responds to a destructive chemical reaction while c < 0 indi-cates a generative chemical reaction. It can be seen from the

profiles that the axial velocity reduces in the degeneratingchemical reaction case and enhances in generating chemicalreaction case in the boundary layer. This is due to the fact that

increase in the rate of chemical reaction rate leads to thinningof a momentum in a boundary layer in degenerating chemicalreaction and thickening of the boundary layer in generating

chemical reaction case. It can be seen from the profiles thatthe cross flow velocity reduces in the degenerating chemicalreaction case and enhances in the generation chemical reaction

case. The effect of radiation absorption parameter Q1, on f 0ðgÞand g(gÞ is shown in Figs. 6 and 7. It is found that higher theradiative absorption larger the cross flow velocity and axial

velocity in the flow region. This is due to the fact that whenheat is absorbed the buoyancy forces correct the flow.

Figs. 8 and 9 represent f 0ðgÞ and g(gÞ with Soret parameter

Sr. It can be observed from the profiles that the axial velocityreduces with increase in Sr in the boundary layer. We noticedepreciation in g with increase in the Soret parameter Sr. It

is due to the fact that an increase in the Soret parameter Sr,results in the decrease in thermal boundary layer thickness

and species boundary layer, which leads to a depreciation.

Figs. 10 and 11 represent the axial velocity f 0ðgÞ, and cross

flow velocity g(gÞ with viscous parameter hr. The axial velocityincreases with increase hr and the cross flow velocity g(gÞreduces with the increase in viscosity parameter. It is also clear

Fig. 9 Effect of Soret parameter (Sr) on cross flow velocity (g).

Fig. 10 Effect of variable viscosity (hr) on axial velocity (f 0Þ.

Fig. 11 Effect of variable viscosity (hr) on cross flow velocity (g).

Fig. 12 Effect of Hall current (m) on temperature (hÞ.

Fig. 13 Effect of chemical reaction parameter (cÞ on tempera-

ture (hÞ.


velocity profile occurs near the surface of the plate. This is dueto fact that increase in viscosity parameter hr enhances thethickness of the momentum boundary layer.

5.2. Temperature profiles

Fig. 12 shows that the transverse flow in the z-directiondecreases gradually with Hall parameter m, reaching maxi-

mum values for every profile and then decreases the thicknessof thermal boundary layer. An increase in the Hall parameterm, results in deprecation in the temperature. The temperature

reduces in the degeneration chemical reaction case andenhances in the generating case in the boundary layer as shownin Fig. 13. It can be easily seen that the thermal boundary layer

releases the energy which causes the temperature of fluid todecrease with increase in degenerating chemical reaction casec > 0. In the case of generating chemical reaction c < 0 the

thermal boundary layer absorbs the energy which causes thetemperature of the fluid to increase with increase in chemicalreaction parameter. It is due to the fact that the thermalboundary layer thickness decreases with increase of c > 0

and enhances c < 0. We find that an increase in the radiationabsorption Q1 leads to an enhancement in the temperatureas illustrated in Fig. 14. This is because larger the values of

Q1 correspond to an increased dominance of conduction over

absorption radiation thereby increasing buoyancy force (thus,vertical velocity) and thickness of the thermal and momentumboundary layers. An increase in the Soret parameter Sr, results

in a depreciation in the temperature. Thus the effect of the

Fig. 14 Effect of radiation absorption parameter (Q1) on

temperature (hÞ.

Fig. 15 Effect of Soret parameter (Sr) on temperature (hÞ.

Fig. 16 Effect of variable viscosity (hr) on temperature (hÞ.

Fig. 17 Effect of Hall current (m) on concentration (/Þ.

Fig. 18 Effect of chemical reaction parameter (cÞ on concentra-

tion (/Þ.

Fig. 19 Effect of radiation absorption parameter (Q1) on

concentration (/Þ.


thermo-diffusion is to reduce the temperature of the fluid flowas shown in Fig. 15. Fig. 16 represents temperature with vis-

cosity parameter hr. It is seen that the effect of hr is to increasethe temperature profile in the flow region. This is due to thefact that the thickness of the boundary layer increases withthe increase of hr.

5.3. Concentration profiles

From Fig. 17 we notice that an increase in the Hall parameterm, results in the deprecation in concentration. Fig. 18 showsthat we obtain a destructive type chemical reaction because

Fig. 20 Effect of Soret parameter (Sr) on concentration (/Þ.

Fig. 21 Effect of variable viscosity (hr) on concentration (/Þ. Fig. 23 Variation of Skin friction (sxÞ on c.

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

00 1 2 3 4 5

i

ii

iii

iv

v

vi

M

x

Fig. 22 Variation of Skin friction (sxÞ on m, Q1, and Sr.


the concentration decreases for increasing chemical reactionparameter which indicates that the diffusion rates can be

tremendously changed by chemical reaction. This is due tothe fact that an increase in the chemical reaction c causes theconcentration at the boundary layer to become thinner which

decreases the concentration of the diffusing species. Thisdecrease in the concentration of the diffusing species dimin-ishes the mass diffusion. The concentration reduces in the

degeneration chemical reaction case and enhances in the gener-ating case in the boundary layer as shown in Fig. 18. It is dueto the fact that the thermal boundary layer and solutal bound-ary layer thickness decreases with increase of c > 0 and

enhances c < 0. We find that an increase in the radiation

Table 3 Skin friction (sx) at g= 0.

M I II III IV V V

1 �0.224087 0.10169 �0.2735 �0.3211 0.2119 0

2 �0.81729 �0.30814 �0.8761 �0.9172 0.43328 0

3 �1.15584 �0.51255 �1.1933 �1.2286 0.57359 0

m 0.5 1.5 0.5 0.5 0.5 0

Sr 0.5 0.5 1.5 2.5 0.5 0

c 0.5 0.5 0.5 0.5 1.5 2

Q1 0.5 0.5 0.5 0.5 0.5 0

absorption Q1 leads to an enhancement in the concentrationas illustrated in Fig. 19. This is due to the fact that the thermal

and solutal boundary layer thickness increases with radiationabsorption parameter Q1. An increase in the Soret parameterSr, results in depreciation in the concentration. Thus the effect

of the thermo-diffusion is to reduce the concentration asshown Fig. 20. Fig. 21 represents concentration with viscosityparameter hr. It is found that an increase in hr, reduces the

concentration in the boundary layer. This is due to the factthat the increase in hr, results in reducing the thickness ofthe solutal boundary layer.

The skin friction component sx at g = 0 is shown in Table 3

and in Figs. 22 and 23. We find that sx enhances with increase

I VII VIII IX X

.4697 0.3349 0.3386 �0.2051 �0.1877

.1668 0.49562 0.5039 �0.8084 �0.7993

.48331 0.61406 0.61898 �1.1394 �1.1245

.5 0.5 0.5 0.5 0.5

.5 0.5 0.5 0.5 0.5

.5 �0.5 �1.5 0.5 0.5

.5 0.5 0.5 1.5 2.5

Table 5 Nusselt number (Nu) g= 0.

M I II III IV V VI VII VIII IX X

1 �0.0440326 �0.04255 �0.03504 �0.02651 �0.02088 �0.1188 �0.0546 �0.06049 �0.1340 �0.2195

2 �0.0298597 �0.02911 �0.02816 �0.02685 �0.00849 �0.1994 �0.04542 �0.05385 �0.08645 �0.1377

3 �0.0138387 �0.01259 �0.013134 �0.01217 �0.00739 �0.1539 �0.03395 �0.04903 �0.05374 �0.0677

m 0.5 1.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

Sr 0.5 0.5 1.5 2.5 0.5 0.5 0.5 0.5 0.5 0.5

c 0.5 0.5 0.5 0.5 1.5 2.5 �0.5 �1.5 0.5 0.5

Q1 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 1.5 2.5

Fig. 27 Variation of Nusselt number (Nu) on c.

Table 4 Skin friction (sz) at g= 0.


1 0.327984 0.45713 0.32178 0.31555 0.31193 0.349701 0.33493 0.33863 0.32980 0.33141

2 0.479339 0.75467 0.47617 0.46233 0.43328 0.56688 0.49562 0.50394 0.49875 0.50828

3 0.603959 1.01283 0.59265 0.58189 0.57359 0.68331 0.61406 0.61898 0.60705 0.60980

m 0.5 1.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

Sr 0.5 0.5 1.5 2.5 0.5 0.5 0.5 0.5 0.5 0.5

c 0.5 0.5 0.5 0.5 1.5 2.5 �0.5 �1.5 0.5 0.5

Q1 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 1.5 2.5

Fig. 24 Variation of Skin friction (sZÞ on m, Q1, and Sr. Fig. 26 Variation of Nusselt number (Nu) on m, Q1, and Sr.

Fig. 25 Variation of Skin friction (sZÞ on c.


Table 6 Sherwood Number (Sh) g= 0.


1 0.670855 0.695265 0.839894 1.000084 1.24235 �0.113321 0.515931 0.443561 0.70129 0.729954

2 0.973886 1.00175 1.007028 1.073886 1.99504 �0.261495 0.711571 0.589033 1.06146 1.14085

3 0.599217 0.639085 0.78174 0.953674 1.14356 �0.21619 0.465302 0.405146 0.63384 0.66154

m 0.5 1.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

Sr 0.5 0.5 1.5 2.5 0.5 0.5 0.5 0.5 0.5 0.5

c 0.5 0.5 0.5 0.5 1.5 2.5 �0.5 �1.5 0.5 0.5

Q1 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 1.5 2.5

Fig. 28 Variation of Sherwood (Sh) number on m, Q1, and Sr. Fig. 29 Variation of Sherwood number (Sh) on c.


in the Hartmann number M and reduces with Hall parameterm. This is due to the fact that the effective conductivity

enhances with increase in M and m, which enhances the mag-netic damping force. With respect to chemical reaction param-eter c, we find that jsxj reduces c in the degenerating chemical

reaction case and enhances in the generating chemical reactioncase. It enhances with increase in Soret parameter Sr andreduces with Q1.

The skin friction component sz is depicted in Table 4 and inFigs. 24 and 25 for different parameter values. It is found thatwith an increase in M or m sz enhances. With respect to chem-ical reaction parameter c, we notice a depreciation in jszj withc 6 1:5 and enhancement with c P 2:5 in degeneration chem-ical reaction case, while in the generating chemical reactioncase, jszj enhances with c < 0 at g = 0. An increase in Q1

enhances jszj and reduces with Sr.The rate of heat transfer (Nusselt number) Nu at g = 0 is

shown in Table 5 for different parametric values. It can be seen

from the tabular values that the rate of heat transfer reduceswith increase in Hartmann number M and Hall parameterm. This is because the applied magnetic field tends to impleadthe flow motion and thus reduce the surface friction force.

With respect to increase in chemical reaction parameter c, wefound that Nu reduces with increase in c 6 1:5 and enhanceswith higher c P 2:5 in the degenerating chemical reaction case,

while in the generating chemical reaction case, jNuj enhancesat g = 0. An increase in radiation absorption parameter Q1

results in an enhancement in jNuj at g= 0, while it reduces

with increase in Sr as shown in Figs. 26 and 27. It is due tothe fact that the thermal boundary layer absorbs energy whichcauses the temperature rise considerably with increase in the

flow field due to the buoyancy effect. Therefore the desiredtemperature can be maintained by controlling the heat absorp-

tion effect in practical chemical engineering applications.The rate of mass transfer (Sherwood number) Sh at g= 0 is

exhibited in Table 6 and also in graphical representation in

Figs. 28 and 29 for different parametric values. An increasein Hall parameter m leads to an enhancement jShj at g= 0.We observe a reversed effect in the behavior of jShj with M.

With reference to chemical reaction parameter c < 0, we findthat the rate of mass transfer enhances with c 6 1:5 andreduces with higher c P 2:5 in the degenerating chemical reac-tion case, while in the generating chemical reaction case jShjreduces with c. An increase in Sr results in an enhancementin jShj at g= 0. Thus higher the thermo-diffusion effect, largerthe jShj at the sheet. Also jShj experiences an enhancement

with increase in Q1.

6. Conclusions

The problem of combined influence of radiation absorption,variable viscosity effect, Hall current of a magneto hydrody-namic free-convective flow and heat and mass transfer over a

stretching sheet in the presence of heat generation/absorptionhas been analyzed. The fluid viscosity is assumed to vary asan inverse linear function of temperature. The numerical

results were obtained and compared with previously reportedcases available in the literature and they were found to be ingood agreement. Graphical results for various parametric con-ditions were presented and discussed for different values. The

main findings are summarized below:


� In the presence of Hall current, both axial and cross flow

velocity distribution enhances. Increase in Hall parameterm results into a depreciation in the temperature and concen-tration. With increase in skin friction parameter sx, Hall

current reduces at g= 0 and enhances sz at g = 0. The rateof heat transfer reduces with increase in Hall parameter,whereas the rate of mass transfer enhances with increasein Hall parameter.

� The presence of degenerating chemical reaction cases, theaxial velocity and cross flow velocity reduces and both thesevelocities enhance in the generating chemical reaction case.

The temperature and concentration reduce in thedegeneration chemical reaction case and enhance in the gen-eration chemical reaction case. The skin friction parameter

reduces with chemical reaction parameter c in the degener-ating case and enhances with the generating case. The rateof heat transfer reduces with lower values of c and enhanceswith higher values of c. The rate of mass transfer enhances

with lower values of c and reduces with higher values of c.� With increase in the radiation absorption the velocity, tem-perature and concentration increase. An increase in skin

friction parameter sx, radiation absorption reduces atg = 0, whereas radiation absorption parameter enhanceswith skin friction parameter sz at g = 0. The rate of heat

transfer and mass transfer enhances with an increase inradiation absorption.

� An increase in thermo-diffusion Sr, the axial velocity

reduces and cross flow velocity decreases and reduces thetemperature and concentration. The skin friction parametersx increases with increase in Sr at g = 0. The skin frictionparameter sz reduces with increase in Sr at g = 0. The rate

of heat transfer reduces with increase in thermo-diffusionSr. Thus higher the thermo-diffusion Sr, larger the rate ofmass transfer at the sheet.

� The axial velocity increases the boundary layer thicknesswith the increase in the viscosity parameter hr. The crossflow velocity g(gÞ thinning the momentum boundary layer

with increase in the viscosity. With increase in the viscosityparameter the temperature of the fluid flow also increasesand reduces the concentration of the boundary layer.

Acknowledgments

The authors express their gratitude to the referees for theirconstructive review of the paper and for valuable comments.

References

[1] Sakiadis BC. Boundary layer behaviour on continuous solid

surfaces. Am Inst Chem Eng J 1961;7(2):221–5.

[2] Crane LJ. Flow past a stretching plate. Z Angew Math Phys

1970;21:645–7.

[3] Sharidan S, Mahmood T, Pop I. Similarity solutions for the

unsteady boundary layer flow and heat transfer due to a stretching

sheet. Int J Appl Mech Eng 2006;11(3):647–54.

[4] Carragher P, Crane LJ. Heat transfer on continuous stretching

surface. ZAMM 1982;62:564–77.

[5] Gupta PS, Gupta AS. Heat and mass transfer on a stretching

sheet with suction or blowing. Can J Chem Eng 2009;55:744–6.

[6] Ishak A, Nazar R, Pop I. Mixed convection on the stagnation

point flow toward a vertical continuously stretching sheet. ASME

J Heat Transfer 2007;129:1087–90.

[7] Ishak A, Nazar R, Pop I. Mixed convection on the stagnation

point flow of a micro fluid towards s a stretching sheet. Meccanica

2008;43:411–8.

[8] Ishak A, Nazar R, Pop I. Boundary layer flow and heat transfer

over an unsteady stretching vertical surface. Meccanica

2009;44:369–75.

[9] Aziz RC, Hashim I, Alomari AK. Thin film flow and heat transfer

on an unsteady stretching sheet with internal heating. Meccanica

2011;46:349–57.

[10] Abel MS, Sanjayanand E, Nandeppa navar MM. Viscoelastic

MHD flow and heat transfer over a stretching sheet with viscous

and ohmic dissipations. Commun Nonlinear Sci Numer Simul

2008;13(9):1808–21.

[11] Mukhopadhyay S, Mondal IC. Heat analysis for fluid flow over

an exponentially stretching porous sheet with surface heat flux in

porous medium. Ain Shams Eng J 2013;4:103–10.

[12] Krishnendu B. Heat transfer analysis in unsteady boundary layer

stagnation-point flow towards a shrinking/stretching. Ain Shams

Eng J 2013;4:259–64.

[13] Mukhopadhyay S, Ranjan P, Krishnnedu B, layek GC. Casson

fluid flow over an unsteady stretching surface. Ain Shams Eng J

2013;4:933–8.

[14] Laha MK, Gupta PS, Gupta AS. Heat transfer characteristics of

the flow of an incompressible viscous fluid over a stretching sheet.

Warme Sto¨ffubertrag 1989;24:151–3.

[15] Afzal N. Heat transfer from a stretching surface. Int J Heat Mass

Transfer 1993;36(4):1128–31.

[16] Prasad KV, Abel AS, Datti PS. Diffusion of chemically reactive

species of non-Newtonian fluid immersed in a porous medium

over a stretching sheet. Int J Non-Linear Mech 2003;38

(5):651–7.

[17] Wang C. Analytic solutions for a liquid film on an unsteady

stretching surface. Heat Mass Transfer 2006;42:43–8.

[18] Rashad AM, Modather M, Chamkha AJ. MHD free convective

heat and mass transfer of a chemically-reacting fluid from radiate

stretching surface embedded in a saturated porous medium. Int J

Chem React Eng 2011;9:A66.

[19] David Theuri, Makinde Oluwole Daniel, Kabuki Nancy Wan-

gare. Unsteady double diffusive magneto hydrodynamic bound-

ary layer flow of a chemically reacting fluid over a flat permeable

surface. Austr J Basic Appl Sci 2013;7(4):78–89.

[20] Pavlov KB. Magnetohydrodynamic flow of an incompressible

viscous fluid caused by deformation of plane surface. Magnit

Gidrodinam 1974;4:146–7.

[21] Afify AA. MHD free-convective flow and mass transfer over a

stretching sheet with chemical reaction. Heat Mass Transfer

2004;40:495–500.

[22] Das S, Jana RN, Makinde OD. MHD boundary layer slip flow

and heat transfer of nanofluid past a vertical stretching sheet with

non-uniform heat generation/absorption. Int J Nanosci 2014;13

(3):1450019.

[23] IbrahimW, Makinde OD. Double-diffusive mixed convection and

MHD stagnation point flow of nanofluid over a stretching sheet. J

Nanofluids 2015;4:28–37.

[24] Rudraswamy NG, Gireesha BJ, Chamkha AJ. Effects of magnetic

field and chemical reaction on stagnation-point flow and heat

transfer of a nanofluid over an inclined stretching sheet. J

Nanofluids 2015;4:239–46.

[25] Watanabe T, Pop I. Hall effects on magneto-hydrodynamic

boundary layer flow over a continuous moving flat plate. Acta

Mech 1995;108:35–47.

[26] Abo-Eldahab EM, Salem AM. Hall effects on MHD free

convection flow of a non-Newtonian power-law fluid at a

stretching surface. Int Commun Heat Mass Transfer

2004;31:343–54.

[27] Rana MA, Siddiqui AM, Ahmed N. Hall effect on Hartmann

flow and heat transfer of a Burger’s fluid. Phys Lett A 2008;

372:562–8.

http://refhub.elsevier.com/S2090-4479(16)00020-4/h0005



















































































[28] Singh AK, Gorla SR. Free convective heat and mass transfer with

Hall current, Joule heating and thermal diffusion. Heat Mass

Transfer 2009;45:1341–9.

[29] Shit GC. Hall effects on MHD free convective flow and mass

transfer over a stretching sheet. Int J Appl Math Mech 2009;5

(8):22–38.

[30] Shateyi S, Motsa SS. Boundary layer flow and double diffusion

over an unsteady stretching surface with hall effect. Chem Eng

Commun 2011;198:1545–65.

[31] Chamkha AJ, Mansour MA, Aly AM. Unsteady MHD free

convective heat and mass transfer from a vertical porous plate

with hall current, thermal radiation and chemical reaction effects.

Int J Numer Methods Fluids 2011;65:432–47.

[32] Abdus Sattar MD, Hamid KMD. Unsteady free-convection

interaction with thermal radiation in a boundary layer flow past

a vertical porous plate. J Math Phys Sci 1996;30:25–37.

[33] Muthucumaraswamy R, Senthil Kumar G. Heat and mass

transfer effects on moving vertical plate in the presence of thermal

radiation. Theor Appl Mech 2004;31(1):35–46.

[34] Makinde OD. Free convection flow with thermal radiation and

mass transfer past a moving vertical plate. Int Commun Heat

Mass Transfer 2005;32(10):1411–9.

[35] Cortell R. Viscoelastic fluid flow and heat transfer over a

stretching sheet under the effects of a non-uniform heat source,

viscous dissipation and thermal radiation. Int J Heat Mass

Transfer 2007;50(15–16):3152–62.

[36] Abel MS, Mahesha N. Heat transfer in MHD viscoelastic fluid

over a stretching sheet with variable thermal conductivity, non-

uniform heat source and radiation. Appl Math Model 2008;32

(10):1965–83.

[37] Pal D. Heat and mass transfer in stagnation-point flow towards a

stretching surface in the presence of buoyancy force and thermal

radiation. Meccanica 2009;44:145–58.

[38] Pal D, Hiremath PS. Computational modeling of heat transfer

over an unsteady stretching surface embedded in a porous

medium. Meccanica 2010;45:415–24.

[39] Makinde OD. Chemically reacting hydromagnetic unsteady flow

of a radiating fluid past a vertical plate with constant heat flux. Z

Naturforsch A 2012;67(5):239–47.

[40] Chamkha AJ, El-Kabeir SMM, Rashad AM. Unsteady coupled

heat and mass transfer by mixed convection flow of a micropolar

fluid near the stagnation point on a vertical surface in the presence

of radiation and chemical reaction. Prog Appl Fluid Dyn

2015;15:186–96.

[41] Mukhopadhyay S, Layek GC, Samad SA. Study of MHD

boundary layer flow over a heated stretching sheet with variable

viscosity. Int J Heat MassTransfer 2005;48(21–22):4460–6.

[42] Mukhopadhyay S, Layek GC. Effects of thermal radiation and

variable fluid viscosity on free convective flow and heat transfer

past a porous stretching surface. Int J Heat Mass Transfer 2008;51

(9–10):2167–78.

[43] Ali ME. The effect of variable viscosity on mixed convection heat

transfer along a vertical moving surface. Int J Therm Sci 2006;45

(1):60–9.

[44] Makinde OD. Laminar falling liquid film with variable viscosity

along an inclined heated plate. Appl Math Comput 2006;175

(1):80–8.

[45] Prasad KV, Vajravelu K, Datti PS. The effects of variable fluid

properties on the hydro-magnetic flow and heat transfer over a

non-linearly stretching sheet. Int J Therm Sci 2010;49(3):603–10.

[46] Alam MS, Rahman MM, Sattar MA. Transient magnetohydro-

dynamic free convective heat and mass transfer flow with

thermophoresis past a radiate inclined permeable plate in the

presence of variable chemical reaction and temperature dependent

viscosity. Nonlinear Anal Modell Control 2009;14(1):3–20.

[47] Salem AM. Variable viscosity and thermal conductivity effects on

MHD flow and heat transfer in viscoelastic fluid over a stretching

sheet. Phys Lett A 2007;369(4):315–22.

[48] Devi SP Anjali, Ganga B. Effects of viscous and Joules dissipation

on MHD flow, heat and mass transfer past a stretching porous

media. Nonlinear Anal Model Control 2009;14(3):303–14.

[49] El Aziz Mohamed Abd. Unsteady mixed convection heat transfer

along a vertical stretching surface with variable viscosity and

viscous dissipation. J Egypt Math Soc 2014;22(3):529–37.

[50] Tian Xi-Yan, Li Ben-Wen, Wu Ya-Shuai, Zhang Jing-Kui.

Chebyshev collocation spectral method simulation for the 2D

boundary layer flow and heat transfer in variable viscosity MHD

fluid over a stretching plate. Int J Heat Mass Transfer

2015;89:829–37.

[51] Mc Dougall TJ. Double-diffusive convection caused by coupled

molecular diffusion. J Fluid Mech 1982;126:379–97.

[52] Chamkha AJ. Hydromagnetic natural convection from an

isothermal inclined surface adjacent to a thermally stratified

porous medium. Int J Eng Sci 1997;35:975–86.

[53] Jaluria Y. Natural convection heat and mass transfer. Oxford

(UK): Pergamon Press; 1980.

[54] Abd El-Aziz M. Thermal-diffusion and diffusion-thermo effects

on combined heat and mass transfer by hydromagnetic three-

dimensional free convection over a permeable stretching surface

with radiation. Phys Lett 2009;372(3):263–72.

[55] Mohamed RA, Abo-Dahab SM. Influence of chemical reaction

and thermal radiation on the heat and mass transfer in MHD

micropolar flow over a vertical moving porous plate in a porous

medium with heat generation. Int J Therm Sci 2009;48:1800–13.

[56] Chapman S, Cowling TG. The mathematical theory of non-

uniform gases. Cambridge (UK): Cambridge Univ. Press; 1952.

[57] Hirshfelder JO, Curtis CF, Bird RB. Molecular theory of gases

and liquids. NewYork: Wiley; 1954.

[58] Kafoussias NG, Williams EW. Thermal-diffusion and diffusion-

thermo effects on mixed free-forced convective and mass transfer

boundary layer flow with temperature dependent viscosity. Int J

Eng Sci 1995;33(9):1369–84.

[59] Alam MS, Ferdows M, Ota M, Maleque MA. Dufour and Soret

effects on steady free convection and mass transfer flow past a

semi-infinite vertical porous plate in a porous medium. Int J Appl

Mech Eng 2006;11(3):535–45.

[60] Makinde OD. On MHD mixed convection with Soret and Dufour

effects past a vertical plate embedded in a porous medium. Latin

Am Appl Res 2011;41:63–8.

[61] Makinde OD, Olanrewaju PO. Unsteady mixed convection with

Soret and Dufour effects past a porous plate moving through a

binary mixture of chemically reacting fluid. Chem Eng Commun

2011;198(7):920–38.

[62] Olanrewaju PO, Makinde OD. Effects of thermal diffusion and

diffusion thermo on chemically reacting MHD boundary layer

flow of heat and mass transfer past a moving vertical plate with

suction/injection. Arab J Sci Eng 2011;36:1607–19.

[63] Zheng X, Jin LC, Zhang XX, Zhang JH. Unsteady heat and mass

transfer in MHD over an oscillatory stretching surface with Soret

and Dufour effects. Acta Mech Sin 2013;29(5):667–75.

[64] Rashidi Mohammad Mehdi, Ali Mohamed, Rostami Behnam,

Rostami Peyman, Xie Gong-Nan. Heat and mass transfer for

MHD viscoelastic fluid flow over a vertical stretching sheet with

considering soret and dufour effects. Math Probl Eng 2015,

861065.

[65] Cowling TG. Magneto hydrodynamics. New York: Interscience

Publishers; 1957.

[66] Lai FC, Kulacki FA. The effect of variable viscosity on convective

heat transfer along a vertical surface in a saturated porous

medium. Int J Heat Mass Transfer 1990;33:1028–31.

[67] Shit GC, Haldar R. Combined effects of thermal radiation and

hall current on MHD free-convective flow and mass transfer over

a stretching sheet with variable viscosity. J Appl Fluid Mech

2012;5:113–21.

[68] Rahman M, Salahuddin KM. Study of hydromagnetic heat and

mass transfer flow over an inclined heated surface with variable










































































































































viscosity and electric conductivity. Commun Nonliner Sci Numer

Simul 2010;15:2073–85.

[69] Salem AM. Variable viscosity and thermal conductivity effects on

MHD flow and heat transfer in viscoelastic fluid over a stretching

sheet. Phys Lett A 2007;369(4):315–22.

G. Sreedevi did her Master of Computer

Applications (MCA) from IGNOU, New

Delhi, India, and earned her M.Sc in Applied

Mathematics from Sri Krishnadevaraya

University, Anantapur, India, in 2004. She

has obtained her M.Phil in Fluid Dynamics

from Sri Krishnadevaraya University, Anan-

tapur, India, in 2010. She is currently a third-

year Ph.D. scholar from Koneru Lakshmaiah

University, Guntur, India. As part of her

research, she has submitted papers to Inter-

national Journals and presented papers in the International Confer-

ences at Gulf International Conference on Applied Mathematics,

Kuwait (GICAM-2013) and International Conference on Computa-

tional Heat and Mass Transfer, NIT, Warangal, India (ICCHMT-

2015) which got published from the Springer and Elsevier publishing

houses respectively. She has overall teaching experience of eight years

in various institutions. Her current areas of interest and research are in

the field of Fluid Dynamics, Heat and Mass Transfer Magnetohy-

drodynamics and Nanofluids.

R. Raghavendra Rao has graduated from

Acharya Nagarjuna University, Nagarjuna

Nagar, India, in 1990 and obtained M.Sc in

Applied Mathematics from Andhra Univer-

sity, Waltair, India, in 1993. He was awarded

Ph.D. in Mathematics by Sri Venkateswara

University, Tirupathi, India, in 2000 with the

research title ‘‘Hydrodynamic Lubrication

Theory: Effects of Velocity-Slip and Viscosity

Variation on Different Bearing Systems”. He

assisted in teaching undergraduate and post-

graduate courses for the Department of Mathematics during the time

of his research. After that, he worked as a Lecturer in Koneru Lak-

shmaiah College of Engineering, Guntur, India, between 2000 and

2005. He also worked as an Assistant Professor in Department of

Mathematics, Eritrea Institute of Technology, Eritrea (Northeast

Africa), from 2006 to 2010. He has been working as an Associate

Professor in Mathematics in Koneru Lakshmaiah University, Guntur,

India, since 2010. His main areas of research interest are Fluid

Mechanics, Hydrodynamic lubrication theory and Tribology of bear-

ings. He attended and presented ten papers at different seminars/-

conferences. He has published thirteen research papers in national/

international journals.

D.R.V. Prasada Rao is a Professor & UGC-

BSC Faculty Fellow, Department of Mathe-

matics, Sri Krishnadevaraya University,

Anantapur, India. He did his M.Sc from Sri

Venkateswara University, Tirupathi, India, in

1973 and earned his Ph.D. from the same

University in 1979. He has over 40 years of

teaching experience and groomed many

graduate and postgraduate students and

research scholars. His areas of interest are

Fluid Dynamics, Bio-mechanics, Heat and

Mass Transfer in Porous Medium, Chemical Reactions in Fluid Flows

and Nano fluids. He has authored and co-authored over 110 papers in

archival journals and conferences.

Ali J. Chamkha is Professor and Chairman of

the Mechanical Engineering Department at

the Prince Mohammad Bin Fahd University

(PMU) in the Kingdom of Saudi Arabia. He

earned his Ph.D. in Mechanical Engineering

from Tennessee Technological University,

USA, in 1989. His research interests include

multiphase fluid-particle dynamics, nanofluids

dynamics, fluid flow in porous media, heat

and mass transfer, magnetohydrodynamics

and fluid-particle separation. He has served as

an Associate Editor for many journals such as International Journal of

Numerical Method for Heat and Fluid Flow, Journal of Applied Fluid

Mechanics, International Journal for Microscale and Nanoscale

Thermal and Fluid Transport Phenomena and he is currently the

Deputy Editor-in-Chief for the International Journal of Energy and

Technology. He has authored and co-authored over 450 publications

in archival international journals and conferences.






combined influence of radiation absorption and hall

Documents