Effects of Hall current, rotation and Soret effects on MHD ... · heat and mass transfer in MHD free convection ﬂow from a vertical surface with Ohmic heating and viscous dissipation

Ain Shams Engineering Journal (2016) xxx, xxx–xxx

Ain Shams University

Ain Shams Engineering Journal

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

ENGINEERING PHYSICS AND MATHEMATICS

Effects of Hall current, rotation and Soret effects

on MHD free convection heat and mass transfer

flow past an accelerated vertical plate through a

porous medium

* Corresponding author.E-mail addresses: [email protected] (D. Sarma), kamalesh.

[email protected] (K.K. Pandit).

Peer review under responsibility of Ain Shams University.

Production and hosting by Elsevier

http://dx.doi.org/10.1016/j.asej.2016.03.0052090-4479 � 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/).

Please cite this article in press as: Sarma D, Pandit KK, Effects of Hall current, rotation and Soret effects on MHD free convection heat and mass transfer flowaccelerated vertical plate through a porous medium, Ain Shams Eng J (2016), http://dx.doi.org/10.1016/j.asej.2016.03.005

D. Sarma, K.K. Pandit *

Department of Mathematics, Cotton College, Guwahati 781001, Assam, India

Received 30 July 2015; revised 24 February 2016; accepted 2 March 2016

KEYWORDS

Hall current;

Rotation;

MHD;

Soret effect;

Porous medium;

Free convection

Abstract The effects of Hall current, rotation and Soret effects on an unsteady MHD free convec-

tion heat and mass transfer flow of a viscous, incompressible and electrically conducting fluid past

an infinite vertical plate embedded in a porous medium are investigated. It is assumed that the entire

system rotates with a uniform angular velocity X0 about the normal to the plate and a uniform

transverse magnetic field is applied along the normal to the plate directed into the fluid region.

The magnetic Reynolds number is considered to be so small that the induced magnetic field can

be neglected. Exact solution of the governing equations is obtained in closed form by Laplace trans-

form technique. Exact solution is also obtained in case of unit Schmidt number. The expressions for

primary and secondary fluid velocity, fluid temperature, species concentration, skin friction due to

primary and secondary velocity fields at the plate are obtained.� 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

The study of natural convection flow induced by the simulta-neous action of thermal and solutal buoyancy forces acting

over bodies with different geometries in a fluid with porous

medium is prevalent in many natural phenomena and has

varied a wide range of industrial applications. For example,the presence of pure air or water is impossible because someforeign mass may be present either naturally or mixed with

air or water due to industrial emissions, in atmospheric flows.Natural processes such as attenuation of toxic waste in waterbodies, vaporization of mist and fog, photosynthesis, transpi-

ration, sea-wind formation, drying of porous solids, and for-mation of ocean currents [1] occur due to thermal andsolutal buoyancy forces developed as a result of difference in

temperature or concentration or a combination of these two.Such configuration is also encountered in several practicalsystems for industry based applications viz. cooling of moltenmetals, heat exchanger devices, petroleum reservoirs, insulation

past an

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2 D. Sarma, K.K. Pandit

systems, filtration, nuclear waste repositories, chemical cat-alytic reactors and processes, desert coolers, frost formationin vertical channels, wet bulb thermometers, etc. Considering

the importance of such fluid flow problems, extensive andin-depth research works have been carried out by severalresearchers [2–10] in the past.

Investigation of hydromagnetic natural convection flowwith heat and mass transfer in porous and non-porous mediahas drawn considerable attentions of several researchers owing

to its applications in geophysics, astrophysics, aeronautics,meteorology, electronics, chemical, and metallurgy and petro-leum industries. Magnetohydrodynamic (MHD) natural con-vection flow of an electrically conducting fluid with porous

medium has also been successfully exploited in crystal forma-tion. Oreper and Szekely [11] have found that the presenceof a magnetic field can suppress natural convection currents

and the strength of magnetic field is one of the important fac-tors in reducing non-uniform composition thereby enhancingquality of the crystal. In addition to it, the thermal physics

of hydromagnetic problems with mass transfer is of muchsignificance in MHD flow-meters, MHD energy generators,MHD pumps, controlled thermo-nuclear reactors, MHD

accelerators, etc. Keeping in view the importance of suchstudy, Hossain and Mandal [12] investigated mass transfereffects on unsteady hydromagnetic free convection flow pastan accelerated vertical porous plate. Jha [13] studied hydro-

magnetic free convection and mass transfer flow past a uni-formly accelerated vertical plate through a porous mediumwhen magnetic field is fixed with the moving plate. Elbashbe-

shy [14] discussed heat and mass transfer along a vertical platein the presence of magnetic field. Chen [15] analyzed combinedheat and mass transfer in MHD free convection flow from a

vertical surface with Ohmic heating and viscous dissipation.Ibhrahim et al. [16] considered unsteady MHD micropolarfluid flow and heat transfer past a vertical porous plate

through a porous medium in the presence of thermal and massdiffusions with a constant heat source. Chamkha [17] investi-gated unsteady MHD convective flow with heat and masstransfer past a semi-infinite vertical permeable moving plate

in a uniform porous medium with heat absorption. Makindeand Sibanda [18] investigated MHD mixed convection flowwith heat and mass transfer past a vertical plate embedded

in a uniform porous medium with constant wall suction inthe presence of uniform transverse magnetic field. Makinde[19] studied MHD mixed convection flow and mass transfer

past a vertical porous plate embedded in a porous mediumwith constant heat flux. Eldabe et al. [20] discussed unsteadyMHD flow of a viscous and incompressible fluid with heatand mass transfer in a porous medium near a moving vertical

plate with time-dependent velocity.Investigation of hydromagnetic natural convection flow in

a rotating medium is of considerable importance due to its

application in various areas of astrophysics, geophysics andfluid engineering viz. maintenance and secular variations inEarth’s magnetic field due to motion of Earth’s liquid core,

structure of the magnetic stars, internal rotation rate of theSun, turbo machines, solar and planetary dynamo problems,rotating drum separators for liquid metal MHD applications,

rotating MHD generators, etc. Taking into consideration theimportance of such study, unsteady hydromagnetic naturalconvection flow past a moving plate in a rotating medium isstudied by a number of researchers such as the studies of Singh

Please cite this article in press as: Sarma D, Pandit KK, Effects of Hall current, rotatioaccelerated vertical plate through a porous medium, Ain Shams Eng J (2016), http:

[21,22], Raptis and Singh [23], Kythe and Puri [24], Tokis [25],Nanousis [26] and Singh et al. [27].

In all these investigations, the effects of thermal radiation

are not taken into account. However, thermal radiation effectson hydromagnetic natural convection flow with heat and masstransfer play a vital role in manufacturing processes viz. glass

production, design of fins, steel rolling, furnace design, castingand levitation, etc. Moreover, several engineering processesoccur at very high temperatures where the knowledge of radia-

tive heat transfer becomes indispensible for the design of per-tinent equipment. Nuclear power plants, gas turbines andvarious propulsion devices for missiles, aircraft, satellites andspace vehicles are examples of such engineering areas [28]. It

is worthy to note that unlike convection/conduction the gov-erning equations taking into account the effects of thermalradiation become quite complicated. Hence many difficulties

arise while solving such equations. However, some reasonableapproximations are proposed to solve the governing equationswith radiative heat transfer. Viskanta and Grosh [29] were one

of the initial investigators to study the effects of thermal radi-ation on temperature distribution and heat transfer in anabsorbing and emitting media flowing over a wedge. They used

Rosseland approximation for the radiative heat flux vector tosimplify the energy equation. Cess [30] studied laminar freeconvection along a vertical isothermal plate with thermal radi-ation. The text books by Sparrow and Cess [31] and Howell

et al. [32] present the most essential features and state of theart applications of radiative heat transfer. Takhar et al. [33]analyzed the effect of radiation on MHD free convection flow

of a gas past a semi-infinite vertical plate. Raptis and Massalas[34] studied oscillatory magnetohydrodynamic flow of a gray,absorbing-emitting fluid with non-scattering medium past a

flat plate in the presence of radiation assuming the Rosselandflux model. Chamkha [35] discussed thermal radiation andbuoyancy effects on hydromagnetic flow over an accelerating

permeable surface with heat source or sink. Seddeek [28] stud-ied effects of thermal radiation and variable viscosity onunsteady hydromagnetic natural convection flow past a semi-infinite flat plate with an aligned magnetic field. Cookey

et al. [36] considered the influence of viscous dissipation andradiation on unsteady MHD free convection flow past an infi-nite heated vertical plate in a porous medium with time-

dependent suction. Suneetha et al. [37] studied effect of ther-mal radiation on unsteady hydromagnetic free convection flowpast an impulsively started vertical plate with variable surface

temperature and concentration. Ogulu and Makinde [38] con-sidered unsteady hydromagnetic free convection flow of a dis-sipative and radiative fluid past a vertical plate with constantheat flux. Mahmoud [39] investigated the effect of thermal

radiation on an unsteady MHD free convection flow past aninfinite vertical porous plate taking into account the effectsof viscous dissipation.

Most of the time Hall current was ignored while applyingOhm’s law because it has no significant effect for small andaverage values of the magnetic field. The effects of Hall current

are very important in the presence of a strong magnetic field[40], because for strong magnetic field electromagnetic forceis prominent. The recent research for the applications of

MHD is towards a strong magnetic field, due to which studyof Hall current is very important. Actually, in an ionized gasof low density subjected to a strong magnetic field, the conduc-tivity perpendicular to the magnetic field is decreased by free

n and Soret effects on MHD free convection heat and mass transfer flow past an//dx.doi.org/10.1016/j.asej.2016.03.005


u′

v′

g....................................................................................................................................................................................................................................................................

....................................................................................................

........................................

x′

y′

T∞′

C∞′

O

Porous Medium

0B

rqw′

z′

Ω

Figure 1 Geometry of the problem.

Effects of hall current, rotation and Soret effects 3

spiral movement of electrons and ions about the magnetic linesof force before suffering collisions. A current produced in adirection at right angle to the electric and magnetic fields is

called Hall current. The important engineering applicationsfor MHD boundary layer flows with heat transfer includingthe effects of Hall current are encountered in MHD power gen-

erators and pumps, Hall accelerators, refrigeration coils, elec-tric transformers, in flight MHD, solar physics involved in thesunspot development, the solar cycle, the structure of magnetic

stars, electronic system cooling, cool combustors, fiber andgranular insulation, oil extraction, thermal energy storageand flow through filtering devices and porous material regen-erative heat exchangers.

It is noticed that when the density of an electrically con-ducting fluid is low and/or applied magnetic field is strong,Hall current is produced in the flow-field which plays an

important role in determining flow features of the problemsbecause it induces secondary flow in the flow-field. Keepingin view this fact, significant investigations on hydromagnetic

free convection flow past a flat plate with Hall effects underdifferent thermal conditions are carried out by severalresearchers in the past. Mention may be made of the research

studies of Pop and Watanabe [41], Abo-Eldahab and Elbar-bary [42], Takhar et al. [43] and Saha et al. [44]. Satya Nar-ayana et al. [45] studied the effects of Hall current andradiation–absorption on MHD natural convection heat and

mass transfer flow of a micropolar fluid in a rotating frameof reference. Seth et al. [46] investigated effects of Hall cur-rent and rotation on unsteady hydromagnetic natural convec-

tion flow of a viscous, incompressible, electrically conductingand heat absorbing fluid past an impulsively moving verticalplate with ramped temperature in a porous medium taking

into account the effect of thermal diffusion. Seth et al. [47]investigated the effects of Hall current, thermal radiationand rotation on natural convection heat and mass transfer

flow past a moving vertical plate. Takhar et al. [48] investi-gated the effect of Hall current on MHD flow over a movingplate in a rotating fluid with magnetic field and free streamvelocity.

The present study deals with the study of the effects of Hallcurrent, rotation and Soret effect on an unsteady MHD freeconvection flow of a viscous, incompressible, electrically con-

ducting fluid past an impulsively moving vertical plate in aporous medium. The governing equations are first transformedinto a set of normalized equations and then solved analytically

by using Laplace transform technique and a general solution isobtained. The effects of different involved parameters such asmagnetic field parameter, Schmidt number, Prandtl number,Grashof number for heat transfer and mass transfer, chemical

reaction, thermal radiation, Hall current, rotation and Soreteffect on the fluid velocity, temperature and concentrationdistributions are plotted and discussed.

2. Formulation of the problem

Consider unsteady MHD natural convection flow with heat

and mass transfer of an electrically conducting, viscous,incompressible fluid past an infinite vertical plate embeddedin a uniform porous medium in a rotating system taking Hall

current into account. Assuming Hall currents, the generalizedOhm’s law [49] may be put in the following form:

Please cite this article in press as: Sarma D, Pandit KK, Effects of Hall current, rotatioaccelerated vertical plate through a porous medium, Ain Shams Eng J (2016), http:/

J!¼ r

1þm2E!þ V

!� B!� 1

rneJ!� B

!� �;

where V!

represents the velocity vector, E!

is the intensity vec-

tor of the electric field, B!

is the magnetic induction vector, J!

the electric current density vector, m is the Hall current param-eter, r the electrical conductivity and ne is the number density

of the electron. A very interesting fact that the effect of Hallcurrent gives rise to a force in the z0-direction which in turnproduces a cross flow velocity in this direction and thus the

flow becomes three-dimensional.Coordinate system is chosen in such a way that x0-axis is

considered along the plate in upward direction and y0-axis nor-mal to plane of the plate in the fluid. A uniform transversemagnetic field B0 is applied in a direction which is parallel toy0-axis. The fluid and plate rotate in unison with uniform angu-lar velocity X0 about y0-axis. Initially i.e., at time t0 6 0, both

the fluid and plate are at rest and are maintained at a uniform

temperature T01. Also species concentration at the surface of

the plate as well as at every point within the fluid is maintained

at uniform concentration C01. At time t0 > 0, plate starts mov-

ing in x0-direction with a velocity u0 ¼ Ut0 in its own plane. Thetemperature at the surface of the plate is raised to uniform

temperature T0w and species concentration at the surface of

the plate is raised to uniform species concentration C0w and is

maintained thereafter. Geometry of the problem is presentedin Fig. 1. Since plate is of infinite extent in x0 and z0 directionsand is electrically non-conducting, all physical quantities

except pressure depend on y0 and t0 only. Also no applied orpolarized voltages exist so the effect of polarization of fluidis negligible. This corresponds to the case where no energy is

added or extracted from the fluid by electrical means [50]. Itis assumed that the induced magnetic field generated by fluidmotion is negligible in comparison to the applied one. This

assumption is justified because magnetic Reynolds number isvery small for liquid metals and partially ionized fluids whichare commonly used in industrial applications [50].

Keeping in view the assumptions made above, governing

equations for natural convection flow with heat and masstransfer of an electrically conducting, viscous, incompressiblefluid past an infinite vertical plate embedded in a uniform

n and Soret effects on MHD free convection heat and mass transfer flow past an/dx.doi.org/10.1016/j.asej.2016.03.005



porous medium in a rotating system taking Hall current andSoret effect into account, are given by

2.1. Conservation of momentum

@u0

@t0þ 2Xw0 ¼ t

@2u0

@y02� rB2

0

qð1þm2Þ ðu0 þmw0Þ þ gbðT0 � T0

1Þ

þ gb0ðC0 � C01Þ � tu0

K01

ð1Þ

@w0

@t0� 2Xu0 ¼ t

@2w0

@y02þ rB2

0

qð1þm2Þ ðmu0 � w0Þ � tw0

K01

ð2Þ

2.2. Conservation of energy

@T0

@t0¼ k

qCp

@2T0

@y02� 1

qCp

@q0r@y0

ð3Þ

2.3. Conservation of species concentration (mass diffusion)

@C0

@t0¼ DM

@2C0

@y02þDMKT

TM

@2T0

@y02� Kr0ðC0 � C0

1Þ ð4Þ

where u0, w0, g, q, b, b0, k, Cp, r, t, m ¼ xese, xe, se, DM, KT,

TM, T0, C0, Kr0, q0r and K0

1 are, respectively, the fluid velocity in

the x0-direction, fluid velocity in z0-direction, acceleration dueto gravity, the fluid density, the volumetric coefficient of ther-mal expansion, the volumetric coefficient of expansion for con-

centration, thermal conductivity, specific heat at constantpressure, electrical conductivity, the kinematic viscosity, Hallcurrent parameter, cyclotron frequency, electron collisiontime, the coefficient of mass diffusivity, the thermal diffusion

ratio, the mean fluid temperature, the temperature of the fluid,species concentration, chemical reaction parameter, radiativeheat flux vector and permeability of the porous medium.

Initial and boundary conditions for the fluid flow problemare given below:

u0 ¼ w0 ¼ 0; T0 ¼ T01; C0 ¼ C0

1 for all y0 and t0 6 0

ð5aÞ

u0 ¼ Ut0; w0 ¼ 0; T0 ¼ T0w; C0 ¼ C0

w at y0 ¼ 0 for t0 > 0

ð5bÞ

u0 ! 0; w0 ! 0 T0 ! T01; C0 ! C0

1 as y0 ! 1 for t0 > 0

ð5cÞFor an optically thick fluid, in addition to emission there is

also self absorption and usually the absorption co-efficient iswavelength dependent and large so we can adopt the Rosse-land approximation for the radiative heat flux vector q0r. Thusq0r is given by

q0r ¼ � 4r1

3k1

@T0

@y0

4

ð6Þ

where k1 is Rosseland mean absorption co-efficient and r1 isStefan–Boltzmann constant.

We assume that the temperature differences within the flow

are sufficiently small so that T04 can be expressed as a linear

function. By using Taylor’s series, we expand T04 about the free


stream temperature T01 and neglecting higher order terms.

This results in the following approximation:

T04 � 4T031T0 � 3T04

1 ð7ÞEq. (3) with the help of (6) and (7) reduces to

@T0

@t0¼ k

qCp

@2T0

@y02þ 16r1T

031

3k1qCp

@2T0

@y02ð8Þ

Introducing the following non-dimensional quantities:

u ¼ u0

U0

; w ¼ w0

U0

; y ¼ y0U0

t; t ¼ t0U2

0

t; h ¼ T0 � T0

1T0

w � T01;

/ ¼ C0 � C01

C0w � C0

1; Gr ¼ gbtðT0

w � T01Þ

U30

;

Gm ¼ gb0tðC0w � C0

1ÞU3

0

; Pr ¼ lCp

k; K2 ¼ tX

U20

; Sc ¼ tDM

;

Kr ¼ tKr0

U20

; M2 ¼ rB20t

qU20

;

U ¼ U30

t; N ¼ kk1

4r1T031; k ¼ 3Nþ 4

3N;

Sr ¼ DMKTðT0w � T0

1ÞtTMðC0

w � C01Þ

; K1 ¼ K01U

20

t2:

where Gr, Gm, M2, K1, Pr, Sc, Sr, K2 and N are, respectively,

the thermal Grashof number, the solutal Grashof number, the

magnetic parameter, Permeability parameter, the Prandtl num-ber, the Schmidt number, the Soret number, the rotationparameter and radiation parameter.

Eqs. (1), (2), (4) and (8) in non-dimensional form are givenbelow:

@u

@tþ 2K2w ¼ @2u

@y2� M2

ð1þm2Þ ðuþmwÞ þ Grhþ Gm/� u

K1

ð9Þ

@w

@t� 2K2u ¼ @2w

@y2þ M2

ð1þm2Þ ðmu� wÞ � w

K1

ð10Þ

@h@t

¼ kPr

@2h@y2

ð11Þ

@/@t

¼ 1

Sc

@2/@y2

þ Sr@2h@y2

� Kr/ ð12Þ

The relevant initial and boundary conditions in non-dimensional form are given by:

u ¼ w ¼ 0; h ¼ 0; / ¼ 0; 8y and t 6 0 ð13aÞ

u ¼ t; w ¼ 0; h ¼ 1; / ¼ 1; at y ¼ 0 and t > 0 ð13bÞ

u ! 0; w ! 0; h ! 0; / ! 0; as y ! 1 and t

> 0 ð13cÞEqs. (9) and (10) are presented, in compact form, as

@F

@t¼ @2F

@y2� aFþ Grhþ Gm/ ð14Þ




where F= u + iw and a= M2(1 � im)/(1 + m2) +

1/K1 � 2iK2.The initial and boundary conditions (13a)–(13c), in com-

pact form, become

F ¼ 0; h ¼ 0; / ¼ 0; 8y and t 6 0 ð15aÞ

F ¼ t; h ¼ 1; / ¼ 1; at y ¼ 0 and t > 0 ð15bÞ

F ! 0; h ! 0; / ! 0; as y ! 1 and t > 0 ð15cÞThe system of differential Eqs. (11), (12) and (14) together

with the initial and boundary conditions (15a)–(15c) describesour model for the MHD free convective heat and mass transfer

flow of a viscous, incompressible, electrically conducting fluidpast an infinite vertical plate embedded in a porous mediumtaking Hall current, rotation and Soret effect intoconsideration.

3. Solution of the problem

The set of Eqs. (11), (12) and (14) subject to the initial and

boundary conditions (15a)–(15c) were solved analytically usingLaplace transforms technique. The exact solutions for the fluidtemperature h(y, t), species concentration /(y, t) and fluid

velocity Fðy; tÞ are obtained and expressed in the followingform:

hðy; tÞ ¼ erfcy

2

ffiffiffiffiffiffiPr

kt

r !ð16Þ

/ðy; tÞ ¼ A1

2A2

eA3A2

teyffiffiffiffiffiffiffiA7Sc

perfc

y

2

ffiffiffiffiffiSc

t

rþ

ffiffiffiffiffiffiffiA7t

p !þ e�y

ffiffiffiffiffiffiffiA7Sc

perfc

y

2

r 8<:

� e�y

ffiffiffiffiffiffiA3Pr

kA2

qerfc

y

2


kt

r�

ffiffiffiffiffiffiffiffiA3

A2

t

r !9=;� 1

2eyffiffiffiffiffiffiffiKrSc

perfc

y

2

r (

Fðy; tÞ ¼ 1

2A8e

A3A2

tey

ffiffiffiffiffiA16

perfc

y

2ffiffit

p þffiffiffiffiffiffiffiffiffiA16t

p� �þ e�y

ffiffiffiffiffiA16

perfc

y

2ffiffit

p ��

þ e�yffiffiffiffiffiA17

perfc

y

2ffiffit

p �ffiffiffiffiffiffiffiffiffiA17t

p� �� ey

ffiffiffiffiffiffiffiffiffiScA20

perfc

y

2

ffiffiffiffiffiSc

t

rþp

þ 1

2A10 ey

ffiffia

perfc

y

2ffiffit

p þ ffiffiffiffiat

p� �þ e�y

ffiffia

perfc

y

2ffiffit

p � ffiffiffiffiat

p� ��


perfc

y

2ffiffit


p� �� e

yffiffiffiffiPraA6

perfc

y

2


kt

rþ

ffiffiffikaA

s

þ t

2þ y

4ffiffiffia

p� �

eyffiffia

perfc

y

2ffiffit

p þ ffiffiffiffiat

p� �þ t

2� y

4ffiffiffia

p� �

e�yffiffia

pe

�


perfc

y

2


kt

r�


A2

t

r !)þ A13erfc

y

2


kt

r !� 1

2

þ e�yffiffiffiffiffiffiffiffiffiScA21

perfc

y

2

ffiffiffiffiffiSc

t

r�

ffiffiffiffiffiffiffiffiffiA21t

p !)� 1

2A15 ey

ffiffiffiffiffiffiffiScKr

perfc

y

2

(


3.1. Solution in case of unit Schmidt number

It is noticed from solution (18) that fluid velocity is not validfor the fluids with unit Schmidt number. Schmidt number isa measure of the relative strength of viscosity to molecular

(mass) diffusivity of fluid. Therefore, fluid flow problem withSc= 1 corresponds to those fluids for which both viscousand concentration boundary layer thicknesses are of sameorder of magnitude. There are some fluids of practical interest

which belong to this category [20]. Substituting Sc = 1 inEq. (12) and following the same procedure as before, exactsolution for species concentration /(y, t) and fluid velocity

F(y, t) is obtained and is presented in the following form:

/ðy; tÞ ¼ A22

2A6

eA24A6

tey

ffiffiffiffiffiA25

perfc

y

2ffiffit


p� ��


perfc

y

2ffiffit


p� �

� eyffiffiffiffiffiA26

perfc

y

2


kt

rþ

ffiffiffiffiffiffiffiffiffiffiA24

A6

t

r !

� e�yffiffiffiffiffiA26

perfc

y

2


kt

r�


A6

t

r !)

� 1

2eyffiffiffiffiKr

perfc

y

2ffiffit

p þffiffiffiffiffiffiffiKrt

p� ��

þ e�yffiffiffiffiKr

perfc

y

2ffiffit

p �ffiffiffiffiffiffiffiKrt

p� ��ð19Þ

ffiffiffiffiffiSc

t�


p !� e

y

ffiffiffiffiffiffiA3Pr

kA2

qerfc

y

2


kt

rþ


A2

t

r !

ffiffiffiffiffiSc

tþ

ffiffiffiffiffiffiffiKrt

p !þ e�y

ffiffiffiffiffiffiffiKrSc

perfc

y

2

ffiffiffiffiffiSc

t

r�


p !)ð17Þ


p �� 1

2A9e

�A5A4

tey

ffiffiffiffiffiA17

perfc

y

2ffiffit


p� ��ffiffiffiffiffiffiffiffiffiA20t

!� e�y

ffiffiffiffiffiffiffiffiffiScA20

perfc

y

2

ffiffiffiffiffiSc

t

r�


p !)

� 1

2A11e

kaA6

tey

ffiffiffiffiffiA18

perfc

y

2ffiffit


p� ��ffiffiffiffiffi6

t

!� e

�yffiffiffiffiPraA6

perfc

y

2


kt

r�

ffiffiffiffiffiffiffiffikaA6

t

s !)

rfcy

2ffiffit


p� ��þ 1

2A12e

A3A2

tey

ffiffiffiffiffiA19

perfc

y

2


kt

rþ


A2

t

r !(

A14eA3A2

teyffiffiffiffiffiffiffiffiffiScA21

perfc

y

2

ffiffiffiffiffiSc

t

rþ


p !(ffiffiffiffiffiSc

t

rþ


p !þ e�y

ffiffiffiffiffiffiffiScKr

perfc

y

2

ffiffiffiffiffiSc

t

r�


p !): ð18Þ



Fðy; tÞ ¼ A27

2eA24A6

tey

ffiffiffiffiffiffiffiffiffiaþA24

A6

qerfc

y

2ffiffit

p þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaþ A24

A6

� �t

s !þ e

�y

ffiffiffiffiffiffiffiffiffiaþA24

A6

qerfc

y

2ffiffit

p �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaþ A24

A6

� �t

s !8<:

9=;þ A28

2eyffiffia

perfc

y

2ffiffit

p þ ffiffiffiffiat

p� ��

þ e�yffiffia

perfc

y

2ffiffit


p� ��þ Gr

aerfc

y

2


kt

r !þ A29

2e

kaA6

teyffiffiffiffiffiffiffiffiaþ ka

A6

perfc

y

2ffiffit

p þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaþ ka

A6

� �t

s !(

þ e�y

ffiffiffiffiffiffiffiffiaþ ka

A6

perfc

y

2ffiffit

p �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaþ ka

A6

� �t

s !� e

yffiffiffiffiPraA6

perfc

y

2


kt

rþ


t

s !� e

�yffiffiffiffiPraA6

perfc

y

2


kt

r�


t

s !)

þ t

2þ y

4ffiffiffia

p� �

eyffiffia

perfc

y

2ffiffit

p þ ffiffiffiffiat

p� �þ t

2� y

4ffiffiffia

p� �

e�yffiffia

perfc

y

2ffiffit


p� �� þ A30

2eA24A6

tey

ffiffiffiffiffiffiffiPrA24kA6

qerfc

y

2


kt

rþ


A6

t

r !8<:

þ e�y

ffiffiffiffiffiffiffiPrA24kA6

qerfc

y

2


kt

r�


A6

t

r !9=;� A31

2eA24A6

tey

ffiffiffiffiffiffiffiffiffiffiffiKrþA24

A6

qerfc

y

2ffiffit

p þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKrþ A24

A6

� �t

s !8<:

þ e�y

ffiffiffiffiffiffiffiffiffiffiffiKrþA24

A6

qerfc

y

2ffiffit

p �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKrþ A24

A6

� �t

s !9=;� A32

2eyffiffiffiffiKr

perfc

y

2ffiffit

p þffiffiffiffiffiffiffiKrt

p� �þ e�y

ffiffiffiffiKr

perfc

y

2ffiffit

p �ffiffiffiffiffiffiffiKrt

p� �� : ð20Þ


3.2. Skin friction, Nusselt number and Sherwood number of theplate

The expressions for primary skin friction sx, secondary skinfriction sz, Nusselt number Nu, and Sherwood number Sh,which are measures of shear stress at the plate due to primary

flow, shear stress at the plate due to secondary flow, rate ofheat transfer at the plate and rate of mass transfer at the platerespectively, are presented in the following form:

sx þ isz ¼ @F

@y

��y¼0

¼ A8eA3A2

t �ffiffiffiffiffiffiffiA16

perf


p� � 1ffiffiffiffiffi

ptp e�A16t

� �� A10

þ 1ffiffiffiffiffipt

p e�A17t �ffiffiffiffiffiffiffiffiffiffiffiffiA20Sc

perf


p� �

ffiffiffiffiffiSc

pt

re�A20t

)þ A11e

� Pr

pkte� kaA6

t

r �� t

ffiffiffia

p þ 1

2ffiffiffia

p� �

erfffiffiffiffiat

p ��ffiffiffit

p

re�at � A12

� A13

ffiffiffiffiffiffiffiPr

pkt

rþ A14e

A3A2

tffiffiffiffiffiffiffiffiffiffiffiffiA21Sc

perf


p� þ

ffiffiffiffiffiSc

pt

re�A21t

( )

Sh ¼ @/@y

��y¼0

¼ A1

A2

eA3A2

tffiffiffiffiffiffiffiffiffiffiffiA7Sc

perfð


pÞ þ

ffiffiffiffiffiSc

pt

re�A7t �

ffiffiffiffiffiffiffiffiffiffiffiA3Pr

A2k

rerf


A2

t

r� ��

(


3.2.1. Nusselt number

The Nusselt number Nu, which measures the rate of heat trans-fer at the plate, is given by

Nu ¼ @h@y

��y¼0

¼ffiffiffiffiffiffiffiPr

pkt

rð22Þ

3.2.2. Sherwood number

The Sherwood number Sh, which measures the rate of masstransfer at the plate, is given by

ffiffiffia

perf

ffiffiffiffiat

p �þ 1ffiffiffiffiffipt

p e�at

� �þ A9e

�A5A4

tffiffiffiffiffiffiffiA17

perf


p� n

kaA6


perf


p� þ 1ffiffiffiffiffi

ptp e�A18t �

ffiffiffiffiffiffiffiffiPraA6

rerf


t

s !(

eA3A2


perf


A2

t

r� �þ


pkt

re�A3A2

t

( )

þ A15

ffiffiffiffiffiffiffiffiffiffiffiKrSc

perf


p� þ

ffiffiffiffiffiSc

pt

re�Krt

( )ð21Þ


pkt

re�A3A2

t

)þ

ffiffiffiffiffiffiffiffiffiffiffiKrSc

perfð


pÞ þ

ffiffiffiffiffiSc

pt

re�Krt

( )ð23Þ



Figure 3 Secondary velocity profiles when Gm = 5, Pr = 0.71,

Sc = 0.22, K1 = 0.5, K2 = 5, m = 0.5, t= 1.

Figure 4 Primary velocity profiles when Gr = 6, Pr = 0.71,

Sc = 0.22, K1 = 0.5, K2 = 5, m = 0.5, t= 1.


4. Results and discussion

In order to get the physical understanding of the problem andfor the purpose of analyzing the effect of Hall current, rota-

tion, thermal radiation, thermal buoyancy force, concentrationbuoyancy force, mass diffusion, thermal diffusion, chemicalreaction, Soret number and time on the flow field, numerical

values of the primary and secondary fluid velocities in theboundary layer region were computed from the analytical solu-tion of (18) and are displayed graphically versus boundarylayer co-ordinate y in Figs. 2–21 for various values of thermal

Grashof number Gr, solutal Grashof number Gm, Hall currentparameter m, rotation parameter K2, Schmidt number Sc, per-meability parameter K1, chemical reaction parameter Kr, ther-

mal radiation parameter N, Soret number Sr and time t takingmagnetic parameter M2 = 10.

Figs. 2–5 depict the influence of thermal and concentration

buoyancy forces on the primary and secondary fluid velocities.It is revealed from Fig. 2 that the primary fluid velocity uincreases on increasing Gr in a region near to the surface of

the plate and it decreases on increasing Gr in the region awayfrom the plate. It is revealed from Fig. 3 that, secondary fluidvelocity w decreases on increasing Gr throughout the boundarylayer region. From Figs. 4 and 5 it is revealed that u and w

increase on increasing Gm. Gr represents the relative strengthof thermal buoyancy force to viscous force and Gm representsthe relative strength of concentration buoyancy force to vis-

cous force. Therefore, Gr decreases on increasing the strengthsof thermal buoyancy force whereas Gm increases on increasingthe strength of concentration buoyancy force. In this problem

natural convection flow induced due to thermal and concentra-tion buoyancy forces; therefore, thermal buoyancy force tendsto retard the primary and secondary fluid velocities whereas

concentration buoyancy force tends to accelerate primaryand secondary fluid velocities throughout the boundary layerregion which is clearly evident from Figs. 2–5. Figs. 6 and 7demonstrate the effect of Hall current on the primary velocity

u and secondary velocity w respectively. It is perceived fromFigs. 6 and 7 that, the primary velocity u decreases on increas-ing m throughout the boundary layer region whereas sec-

ondary velocity w increases on increasing m throughout the

Figure 2 Primary velocity profiles when Gm = 5, Pr = 0.71,

Sc = 0.22, K1 = 0.5, K2 = 5, m= 0.5, t= 1.

Figure 5 Secondary velocity profiles when Gr = 6, Pr = 0.71,

Sc = 0.22, m = 0.5 K1 = 0.5, K2 = 5, t= 1.

Please cite this article in press as: Sarma D, Pandit KK, Effects of Hall current, rotation and Soret effects on MHD free convection heat and mass transfer flow past anaccelerated vertical plate through a porous medium, Ain Shams Eng J (2016), http://dx.doi.org/10.1016/j.asej.2016.03.005


Figure 6 Primary velocity profiles when Gr= 6, Gm = 5,

Pr= 0.71, Sc = 0.22, K2 = 5, K1 = 0.5, t= 1.

Figure 7 Secondary velocity profiles when Gr= 6, Gm = 5,

Pr= 0.71, K2 = 5, Sc = 0.22, K1 = 0.5, t= 1.


Pr= 0.71, m= 0.5, Sc = 0.22, K1 = 0.5, t= 1.


Pr= 0.71, Sc = 0.22, K1 = 0.5, m= 0.5, t= 1.


boundary layer region. This implies that, Hall current tends toaccelerate secondary fluid velocity throughout the boundarylayer region which is consistent with the fact that Hall currentinduces secondary flow in the flow-field whereas it has a

reverse effect on primary fluid velocity throughout the bound-ary layer region. Figs. 8 and 9 illustrate the effects of rotationon the primary and secondary fluid velocities respectively. It is

evident from Figs. 8 and 9 that, primary velocity u decreaseson increasing K2 whereas secondary velocity w increases onincreasing K2 in the region away from the plate. This implies

that rotation retards fluid flow in the primary flow directionand accelerates fluid flow in the secondary flow direction inthe boundary layer region. This may be attributed to the fact

that when the frictional layer at the moving plate is suddenlyset into the motion then the Coriolis force acts as a constraintin the main fluid flow i.e. in the fluid flow in the primaryflow direction to generate cross flow i.e. secondary flow. The


influences of the Schmidt number Sc on the primary velocity,

secondary velocity and concentration profiles are plotted inFigs. 10, 11 and 26 respectively. It is noticed from Figs. 10,11 and 26 that, u, w and / decrease on increasing Sc. The Sch-midt number embodies the ratio of the momentum to the mass

diffusivity. The Schmidt number therefore quantifies the rela-tive effectiveness of momentum and mass transport by diffu-sion in the hydrodynamic (velocity) and concentration

(species) boundary layers. As the Schmidt number increasesthe concentration decreases. This causes the concentrationbuoyancy effects to decrease yielding a reduction in the fluid

velocity. The reductions in the velocity and concentration pro-files are accompanied by simultaneous reductions in the veloc-ity and concentration boundary layers. These behaviors are

clear from Figs. 10, 11 and 26. Figs. 12 and 13 present the effectof permeability of porous medium on primary and secondaryfluid velocities. It is noticed from Figs. 12 and 13 that, primaryvelocity decreases on increasing permeability parameter




Pr = 0.71, K2 = 5, K1 = 0.5, m = 0.5, t = 1.


Pr = 0.71, K1 = 0.5, K2 = 5, m = 0.5, t = 1.


Pr = 0.71, m = 0.5, Sc = 0.22, K2 = 5, t= 1.

Figure 13 Secondary velocity profiles when Gr = 6, Gm= 5,

Pr= 0.71, m= 0.5, Sc = 0.22, K2 = 5, t= 1.

Figure 14 Primary velocity profiles when Gr= 6, Gm= 5,

Pr= 0.71, K2 = 5, Sc = 0.22, K1 = 0.5, m = 0.5, t= 1.



whereas it has reverse effect on secondary velocity. From the

flow configuration it is obvious that an increase in porosityof the medium assists the flow along the secondary directionthereby causing the secondary velocity to increase due to itsorientation through the porous medium. Figs. 14 and 15

demonstrate the influence of chemical reaction parameter Kron the primary fluid velocity (u) and secondary fluid velocity(w) respectively. As can be seen, an increase in the chemical

reaction parameter (Kr) leads to an increase in the thicknessof the velocity boundary layer; this shows that diffusion ratecan be tremendously altered by chemical reaction (Kr). A tem-

poral maximum of velocity profiles is clearly seen for increas-ing values of Kr. It should be mentioned here that physicallypositive values of Kr imply destructive reaction and negative

values of Kr imply generative reaction. We studied the caseof a destructive chemical reaction (Kr). Figs. 16, 17 and 23illustrate the influence of thermal radiation N on primary veloc-ity, secondary fluid velocity and fluid temperature respectively.



Figure 15 Secondary velocity profiles when Gr = 6, Gm = 5,

Pr= 0.71, m= 0.5, Sc = 0.22, K1 = 0.5, t= 1.


Pr= 0.71, m= 0.5, Sc = 0.22, K1 = 0.5, K2 = 5, t= 1.

Figure 17 Secondary velocity profiles when Gr = 6, Gm = 5,

Pr= 0.71, m= 0.5, Sc = 0.22, K2 = 5, K1 = 0.5, t = 1.

Figure 18 Primary velocity profiles when Gr = 6, Gm = 5,

Pr= 0.71, Sc = 0.22, K1 = 0.5, K2 = 5, m = 0.5, t = 1, N = 1.


It is evident from Figs. 16, 17 and 23 that, the thermal radia-tion leads to decrease in each of velocities and temperature.

Physically, thermal radiation causes a fall in temperature ofthe fluid medium and thereby causes a fall in kinetic energyof the fluid particles. This results in a corresponding decreasein fluid velocities. Thus, the Figs. 16, 17 and 23 are in excellent

agreement with the laws of Physics. Thus as N increases, h uand w also decreases. Now, from these figures, it may beinferred that radiation has a more significant effect on temper-

ature than on velocity. Thus, the thermal radiation does nothave a significant effect on the velocities but produces a com-paratively more pronounced effect on the temperature of the

mixture. Figs. 18 and 19 display the influence of Soret numberon primary and secondary fluid velocities. It is evident fromFigs. 18 and 19 that u and w increase on increasing Soret

number Sr. This implies that Soret number tends to accelerateprimary and secondary fluid velocities throughout the bound-ary layer region. Increasing Soret number indicates a fall in theviscosity of the mixture. This leads to increased inertia effects


and diminished viscous effects. Consequently the velocity com-ponents increase. Figs. 20 and 21 present the effect of time t on

the primary and secondary fluid velocities. It is evident fromFigs. 20 and 21 that, u and w increase on increasing t. Thisimplies that, primary and secondary fluid velocities are gettingaccelerated with the progress of time throughout the boundary

layer region.The numerical values of fluid temperature h, computed

from the analytic solution (16), are displayed graphically ver-

sus boundary layer co-ordinate y in Figs. 22–24 for variousvalues of Prandtl number Pr, thermal radiation and time t.It is evident from Fig. 22 that, fluid temperature h decreases

on increasing Pr. An increase in Prandtl number reduces thethermal boundary layer thickness. Prandtl number signifiesthe ratio of momentum diffusivity to thermal diffusivity. It

can be noticed that as Pr decreases, the thickness of the ther-mal boundary layer becomes greater than the thickness ofthe velocity boundary layer according to the well-known rela-tion dT=d ffi 1=Pr where dT the thickness of the thermal




Pr = 0.71, m = 0.5, Sc = 0.22, K2 = 5, K1 = 0.5, t= 1, N= 1.


Pr = 0.71, K1 = 0.5, Sc = 0.22, K2 = 5, m= 0.5.

Figure 21 Secondary velocity profiles when Gr= 6, Gm= 5,

Pr= 0.71, m= 0.5, Sc = 0.22, K2 = 5, K1 = 0.5.

Figure 22 Temperature profiles when N= 1 and t= 0.7.

Figure 23 Temperature Profiles when Pr= 0.71 and t= 0.7.


boundary layer and d the thickness of the velocity boundary

layer, so the thickness of the thermal boundary layer increasesas Prandtl number decreases and hence temperature profiledecreases with increase in Prandtl number. In heat transferproblems, the Prandtl number controls the relative thickening

of momentum and thermal boundary layers. When Prandtlnumber is small, it means that heat diffuses quickly comparedto the velocity (momentum), which means that for liquid met-

als, the thickness of the thermal boundary layer is much biggerthan the momentum boundary layer. Hence Prandtl numbercan be used to increase the rate of cooling in conducting flows.

Fig. 23 is plotted to depict the influence of the thermal radia-tion parameter N on the temperature profile. We observe inthis figure that increasing thermal radiation parameter pro-

duces a significant decrease in the thermal condition of thefluid flow. This can be explained by the fact that a decreasein the values of N means a decrease in the Rosseland radiationabsorptivity k1. Thus the divergence of the radiative heat flux



Figure 24 Temperature profiles when Pr= 0.71 and N= 1.

Figure 25 Concentration profiles when Sc = 0.22 and t= 0.7.

Figure 26 Concentration profiles when Pr= 0.71 and t= 0.7.

Figure 27 Concentration profiles when Pr = 0.71, Sc = 0.22,

N= 1 and t= 0.7.

Figure 28 Concentration profiles when Pr = 0.71, Sc = 0.22,

Kr= 1 and t= 0.7.

Figure 29 Concentration profiles when Pr = 0.71, Sc = 0.22

and t= 0.7.




Figure 30 Concentration profiles when Pr= 0.71 and

Sc = 0.22.


decreases as k1 increases the rate of radiative heat transferred

from the fluid and consequently the fluid temperaturedecreases. Fig. 24 shows that fluid temperature h increaseson increasing time t. This implies that, there is an enhancementin fluid temperature with the progress of time throughout the

thermal boundary layer region.The numerical values of species concentration /, computed

from the analytical solution (17), are depicted graphically ver-

sus boundary layer co-ordinate y in Figs. 25–30 for various

Table 1 Skin friction.

Gr Gm Sc K2 K1 Kr

5 5 0.22 5 0.5 1

10 5 0.22 5 0.5 1

15 5 0.22 5 0.5 1

6 5 0.22 5 0.5 1

6 10 0.22 5 0.5 1

6 15 0.22 5 0.5 1

6 5 0.22 5 0.5 1

6 5 0.6 5 0.5 1

6 5 0.78 5 0.5 1

6 5 0.22 3 0.5 1

6 5 0.22 5 0.5 1

6 5 0.22 7 0.5 1

6 5 0.22 5 0.2 1

6 5 0.22 5 0.5 1

6 5 0.22 5 0.8 1

6 5 0.22 5 0.5 1

6 5 0.22 5 0.5 2

6 5 0.22 5 0.5 3

6 5 0.22 5 0.5 1

6 5 0.22 5 0.5 1

6 5 0.22 5 0.5 1

6 5 0.22 5 0.5 1

6 5 0.22 5 0.5 1

6 5 0.22 5 0.5 1

6 5 0.22 5 0.5 1

6 5 0.22 5 0.5 1

6 5 0.22 5 0.5 1


values of the Prandtl number Pr, the Schmidt number Sc,chemical reaction parameter Kr, thermal radiation parameterN, Soret number Sr and time t. It is evident from Fig. 25 that

species concentration / decreases on increasing Prandtl num-ber Pr. Prandtl number Pr, which is a measure of relativeimportance of viscosity and thermal conductivity of the fluid,

has tendency to reduce the species concentration of the fluid.As Pr increases, the viscous effects increase thereby reducingspecies molecular activity. This leads to reduction in mass

average velocity that in turn leads to a fall in concentrationlevels. Fig. 27 shows the influence of a chemical reaction onconcentration profiles. In this study, we are analyzing theeffects of a destructive chemical reaction (Kr > 0). It is noticed

that concentration distributions decrease when the chemicalreaction increases. Physically, for a destructive case, chemicalreaction takes place with many disturbances. This, in turn,

causes high molecular motion, which results in an increase inthe transport phenomenon, thereby reducing the concentrationdistributions in the fluid flow. It is evident from Fig. 28 that,

the species concentration / decreases on increasing the thermalradiation parameter N. As thermal radiation increases the fluidtemperature decreases thereby causing reduced molecular

activity. Decrease in fluid temperature lessens the impetus ofthermal flux on concentration flux. This reduces the molecularconcentration. This implies that the thermal radiation tends toretard the species concentration of the fluid. It is noticed from

Fig. 29 that, the species concentration / increases on increas-ing Soret number Sr. An increase in Soret effect indicatesincreasing molar mass diffusivity, as seen from definition of

Sr. The surge in molecular mass diffusivity causes the concen-tration to rise. This implies that, Soret number tends to

Sr m t sx sz

1 0.5 0.5 �0.8224 0.2258

1 0.5 0.5 0.033 1.0918

1 0.5 0.5 0.8884 1.9579

1 0.5 0.5 �0.6513 0.399

1 0.5 0.5 �1.4044 �0.2413

1 0.5 0.5 �2.1576 �0.8815

1 0.5 0.5 �0.6513 0.399

1 0.5 0.5 0.6172 1.6239

1 0.5 0.5 0.5329 1.5262

1 0.5 0.5 �0.9897 0.5756

1 0.5 0.5 �0.6513 0.399

1 0.5 0.5 �0.4952 0.1079

1 0.5 0.5 �0.4086 0.6463

1 0.5 0.5 �0.6513 0.399

1 0.5 0.5 �0.6922 0.3504

1 0.5 0.5 �0.6513 0.399

1 0.5 0.5 �2.401 �1.1741

1 0.5 0.5 �9.2697 �7.0361

1 0.5 0.5 �0.6513 0.399

3 0.5 0.5 �2.4361 �1.2741

5 0.5 0.5 �4.2209 �2.9471

1 0.5 0.5 �0.6513 0.399

1 1 0.5 �0.7266 0.1856

1 1.5 0.5 �0.7697 0.0808

1 0.5 0.3 �0.5894 0.5277

1 0.5 0.5 �0.6513 0.399

1 0.5 0.7 �0.6625 �0.3947



Table 2 Nusselt number.

Pr N t Nu

0.3 1 0.5 0.2861

0.5 1 0.5 0.3693

0.71 1 0.5 0.4401

0.71 2 0.5 0.5208

0.71 4 0.5 0.5822

0.71 6 0.5 0.6081

0.71 1 0.5 0.4401

0.71 1 1 0.3112

0.71 1 1.5 0.2541

Table 3 Sherwood number.

Pr Sc Kr N Sr t Sh

0.3 0.22 1 1 1 0.5 1.5939

0.71 0.22 1 1 1 0.5 1.3975

7 0.22 1 1 1 0.5 0.8293

0.71 0.1 1 1 1 0.5 1.3975

0.71 0.22 1 1 1 0.5 1.4206

0.71 0.78 1 1 1 0.5 1.5453

0.71 0.22 1 1 1 0.5 1.3975

0.71 0.22 2 1 1 0.5 2.1567

0.71 0.22 3 1 1 0.5 2.7345

0.71 0.22 1 2 1 0.5 1.1464

0.71 0.22 1 4 1 0.5 1.0004

0.71 0.22 1 6 1 0.5 0.9475

0.71 0.22 1 1 1 0.5 1.3975

0.71 0.22 1 1 3 0.5 1.1238

0.71 0.22 1 1 5 0.5 0.8501

0.71 0.22 1 1 1 0.5 1.3975

0.71 0.22 1 1 1 1 1.525

0.71 0.22 1 1 1 1.5 1.5788


enhance the species concentration of the fluid. From Fig. 30, itis clear that with unabated mass diffusion into the fluid stream,

the molar concentration of the mixture rises with increasingtime and so there is an enhancement in species concentrationwith the progress of time throughout the boundary layer

region.The numerical values of primary skin friction sx and sec-

ondary skin friction sz, computed from the analytical expres-

sion (21), are presented as tabular form in Table 1 forvarious values of Gr, Gm, Sc, m, K2, K1, Kr, Sr and t takingM2 = 10. It is evident from Table 1 that, primary skin frictionsx decreases on increasing Gm, m, K1, Kr, Sr and t whereas it

increases on increasing Gr, Sc and K2. Secondary skin frictionsz decreases on increasing Gm, m, K2, K1, Kr, Sr and t whereasit increases on increasing Gr and Sc. This implies that, Hall

current, concentration buoyancy force, permeability of theporous medium, chemical reaction, Soret number and timehave the tendency to reduce primary and secondary skin fric-

tions. Rotation tends to reduce primary kin friction whereasit has reverse effect on secondary skin friction. Thermal buoy-ancy force and mass diffusion have the tendency to enhancethe primary and secondary skin friction at the plate.

The numerical values of Nusselt number Nu, computedfrom the analytic expression (22), are presented in Table 2


for different values Pr, N and t. It is noticed from Table 2 that,Nusselt number Nu increases on increasing Pr and N whereasit decreases on increasing time t. This implies that, thermal dif-

fusion and thermal radiation tend to enhance rate of heattransfer at the plate. As time progresses, the rate of heat trans-fer is getting reduced at the plate.

The numerical values of Sherwood number Sh, computedfrom the analytic expression (23), are presented in Table 3for various values of Pr, Sc, Kr, N, Sr and t. It is revealed from

Table 3 that, the rate of mass transfer increases on increasingSc, Kr and t whereas it decreases on increasing Pr, N and Sr.This implies that mass diffusion, chemical reaction parameterand time tend to enhance the rate of mass transfer at the plate

whereas thermal diffusion, thermal radiation and Soret num-ber have reverse effect on it.

5. Conclusions

An investigation of the effects of Hall current, rotation andSoret number on an unsteady MHD natural convection flow

with heat and mass transfer of a viscous, incompressible, elec-trically conducting fluid past an infinite vertical plate embed-ded in a porous medium, is carried out. Exact solutions of

the governing equations were obtained using Laplace trans-form technique. A comprehensive set of graphical for the fluidvelocity, fluid temperature and species concentration is pre-

sented and their dependence on some physical parameters isdiscussed. Significant finding are as follows:

� Hall current tends to accelerate secondary fluid velocity

throughout the boundary layer region whereas it has areverse effect on the primary fluid velocity throughout theboundary layer region.

� Rotation tends to accelerate secondary fluid velocitythroughout the boundary layer region whereas it has areverse effect on the primary fluid velocity throughout the

boundary layer region.� Thermal buoyancy force tends to retard the secondary fluidvelocity throughout the boundary layer region. Thermal

buoyancy force tends to accelerate primary fluid velocityin the region near to the plate whereas it has a reverse effecton primary fluid velocity in the region away from the plate.

� Concentration buoyancy force tends to accelerate both the

primary and secondary fluid velocities throughout theboundary layer region whereas mass diffusion has reverseeffect on it throughout the boundary layer region.

� Permeability of the porous medium tends to accelerate thesecondary fluid velocity throughout the boundary layerregion whereas it has reverse effect on primary fluid velocity

throughout the boundary layer region.� Soret number tends to accelerate both the primary and sec-ondary fluid velocities throughout the boundary layerregion.

� Primary and secondary fluid velocities are getting acceler-ated with the progress of time throughout the boundarylayer region.

� Thermal diffusion and thermal radiation tend to retard thefluid temperature and there is an enhancement in fluid tem-perature with the progress of time throughout the boundary

layer region.




� Thermal and mass diffusion tends to retard species concen-

tration and there is an enhancement in species concentra-tion due to increasing Soret number and time throughoutthe boundary layer region.

� Concentration buoyancy force, hall current, permeability ofthe porous medium, chemical reaction, Soret number andtime have tendency to reduce both primary and secondaryskin friction whereas thermal buoyancy force and mass dif-

fusion have reverse effect on it.� Rotation tends to enhance the primary skin friction whereasit has reverse effect on secondary skin friction.

� Thermal diffusion and thermal radiation tend to enhancerate of heat transfer whereas as time progress the rate ofheat transfer is getting reduced.

� Mass diffusion, chemical reaction and time tend to enhancethe rate of mass transfer whereas thermal diffusion, thermalradiation and Soret number have reverse effect on it.

References

[1] Bejan A. Convection heat transfer. 2nd ed. New York: Wiley;

1993.

[2] Gebhart B, Pera L. The nature of vertical natural convection flows

resulting from the combined buoyancy effects of thermal and mass

diffusion. Int J Heat Mass Transfer 1971;14(12):2025–50.

[3] Raptis AA. Free convection and mass transfer effects on the

oscillatory flow past an infinite moving vertical isothermal plate

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D. Sarma is an Associate Professor in the

Department of Mathematics, Cotton College,

Guwahati-781001, Assam, India. He received

his Ph.D. in Mathematics from Gauhati

University, Guwahati-781014, Assam, India.

He has more than twenty years of teaching

experience and around thirteen years of

research studies. His current area of research

studies includes Fluid Dynamics, Magneto-

hydrodynamics and Heat and Mass transfer.

He has published more than thirty research

papers in national/International journals of repute.

K.K. Pandit has received M.Sc. Degree in

Mathematics from Gauhati University,

Guwahati-781014, Assam, India in 2007. He

has received M.Phil. degree in applied Math-

ematics from Gauhati University, Guwahati

in 2009. In the year 2012, he has joined

Department of Mathematics, Gauhati

University, Guwahati as a full time research

fellow to do research work leading to Ph.D.

Degree. He has three years of teaching and

research experience. He has published four

research papers in the peer reviewed International Journals of repute.















































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Effects of Hall current, rotation and Soret effects on MHD ... · heat and mass transfer in MHD free convection ﬂow from a vertical surface with Ohmic heating and viscous dissipation