effects of hall current, rotation and soret effects on mhd ... · heat and mass transfer in mhd...
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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
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
4 D. Sarma, K.K. Pandit
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
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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
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
Effects of hall current, rotation and Soret effects 5
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
ffiffiffiffiffiffiPr
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� �� �
þ e�yffiffiffiffiffiA18
perfc
y
2ffiffit
p �ffiffiffiffiffiffiffiffiffiA18t
p� �� e
yffiffiffiffiPraA6
perfc
y
2
ffiffiffiffiffiffiPr
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
�
þ e�yffiffiffiffiffiA19
perfc
y
2
ffiffiffiffiffiffiPr
kt
r�
ffiffiffiffiffiffiffiffiA3
A2
t
r !)þ A13erfc
y
2
ffiffiffiffiffiffiPr
kt
r !� 1
2
þ e�yffiffiffiffiffiffiffiffiffiScA21
perfc
y
2
ffiffiffiffiffiSc
t
r�
ffiffiffiffiffiffiffiffiffiA21t
p !)� 1
2A15 ey
ffiffiffiffiffiffiffiScKr
perfc
y
2
(
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:/
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 þffiffiffiffiffiffiffiffiffiA25t
p� ��
þ e�yffiffiffiffiffiA25
perfc
y
2ffiffit
p �ffiffiffiffiffiffiffiffiffiA25t
p� �
� eyffiffiffiffiffiA26
perfc
y
2
ffiffiffiffiffiffiPr
kt
rþ
ffiffiffiffiffiffiffiffiffiffiA24
A6
t
r !
� e�yffiffiffiffiffiA26
perfc
y
2
ffiffiffiffiffiffiPr
kt
r�
ffiffiffiffiffiffiffiffiffiffiA24
A6
t
r !)
� 1
2eyffiffiffiffiKr
perfc
y
2ffiffit
p þffiffiffiffiffiffiffiKrt
p� ��
þ e�yffiffiffiffiKr
perfc
y
2ffiffit
p �ffiffiffiffiffiffiffiKrt
p� ��ð19Þ
ffiffiffiffiffiSc
t�
ffiffiffiffiffiffiffiA7t
p !� e
y
ffiffiffiffiffiffiA3Pr
kA2
qerfc
y
2
ffiffiffiffiffiffiPr
kt
rþ
ffiffiffiffiffiffiffiffiA3
A2
t
r !
ffiffiffiffiffiSc
tþ
ffiffiffiffiffiffiffiKrt
p !þ e�y
ffiffiffiffiffiffiffiKrSc
perfc
y
2
ffiffiffiffiffiSc
t
r�
ffiffiffiffiffiffiffiKrt
p !)ð17Þ
ffiffiffiffiffiffiffiffiffiA16t
p ��� 1
2A9e
�A5A4
tey
ffiffiffiffiffiA17
perfc
y
2ffiffit
p þffiffiffiffiffiffiffiffiffiA17t
p� ��ffiffiffiffiffiffiffiffiffiA20t
!� e�y
ffiffiffiffiffiffiffiffiffiScA20
perfc
y
2
ffiffiffiffiffiSc
t
r�
ffiffiffiffiffiffiffiffiffiA20t
p !)
� 1
2A11e
kaA6
tey
ffiffiffiffiffiA18
perfc
y
2ffiffit
p þffiffiffiffiffiffiffiffiffiA18t
p� ��ffiffiffiffiffi6
t
!� e
�yffiffiffiffiPraA6
perfc
y
2
ffiffiffiffiffiffiPr
kt
r�
ffiffiffiffiffiffiffiffikaA6
t
s !)
rfcy
2ffiffit
p � ffiffiffiffiat
p� ��þ 1
2A12e
A3A2
tey
ffiffiffiffiffiA19
perfc
y
2
ffiffiffiffiffiffiPr
kt
rþ
ffiffiffiffiffiffiffiffiA3
A2
t
r !(
A14eA3A2
teyffiffiffiffiffiffiffiffiffiScA21
perfc
y
2
ffiffiffiffiffiSc
t
rþ
ffiffiffiffiffiffiffiffiffiA21t
p !(ffiffiffiffiffiSc
t
rþ
ffiffiffiffiffiffiffiKrt
p !þ e�y
ffiffiffiffiffiffiffiScKr
perfc
y
2
ffiffiffiffiffiSc
t
r�
ffiffiffiffiffiffiffiKrt
p !): ð18Þ
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
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 � ffiffiffiffiat
p� ��þ Gr
aerfc
y
2
ffiffiffiffiffiffiPr
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
ffiffiffiffiffiffiPr
kt
rþ
ffiffiffiffiffiffiffiffikaA6
t
s !� e
�yffiffiffiffiPraA6
perfc
y
2
ffiffiffiffiffiffiPr
kt
r�
ffiffiffiffiffiffiffiffikaA6
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 � ffiffiffiffiat
p� �� �þ A30
2eA24A6
tey
ffiffiffiffiffiffiffiPrA24kA6
qerfc
y
2
ffiffiffiffiffiffiPr
kt
rþ
ffiffiffiffiffiffiffiffiffiffiA24
A6
t
r !8<:
þ e�y
ffiffiffiffiffiffiffiPrA24kA6
qerfc
y
2
ffiffiffiffiffiffiPr
kt
r�
ffiffiffiffiffiffiffiffiffiffiA24
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
6 D. Sarma, K.K. Pandit
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
ffiffiffiffiffiffiffiffiffiA16t
p� � 1ffiffiffiffiffi
ptp e�A16t
� �� A10
þ 1ffiffiffiffiffipt
p e�A17t �ffiffiffiffiffiffiffiffiffiffiffiffiA20Sc
perf
ffiffiffiffiffiffiffiffiffiA20t
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
ffiffiffiffiffiffiffiffiffiA21t
p� þ
ffiffiffiffiffiSc
pt
re�A21t
( )
Sh ¼ @/@y
����y¼0
¼ A1
A2
eA3A2
tffiffiffiffiffiffiffiffiffiffiffiA7Sc
perfð
ffiffiffiffiffiffiffiA7t
pÞ þ
ffiffiffiffiffiSc
pt
re�A7t �
ffiffiffiffiffiffiffiffiffiffiffiA3Pr
A2k
rerf
ffiffiffiffiffiffiffiffiA3
A2
t
r� ��
(
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:
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
ffiffiffiffiffiffiffiffiffiA17t
p� n
kaA6
tffiffiffiffiffiffiffiA18
perf
ffiffiffiffiffiffiffiffiffiA18t
p� þ 1ffiffiffiffiffi
ptp e�A18t �
ffiffiffiffiffiffiffiffiPraA6
rerf
ffiffiffiffiffiffiffiffikaA6
t
s !(
eA3A2
tffiffiffiffiffiffiffiA19
perf
ffiffiffiffiffiffiffiffiA3
A2
t
r� �þ
ffiffiffiffiffiffiffiPr
pkt
re�A3A2
t
( )
þ A15
ffiffiffiffiffiffiffiffiffiffiffiKrSc
perf
ffiffiffiffiffiffiffiKrt
p� þ
ffiffiffiffiffiSc
pt
re�Krt
( )ð21Þ
ffiffiffiffiffiffiffiPr
pkt
re�A3A2
t
)þ
ffiffiffiffiffiffiffiffiffiffiffiKrSc
perfð
ffiffiffiffiffiffiffiKrt
pÞ þ
ffiffiffiffiffiSc
pt
re�Krt
( )ð23Þ
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
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.
Effects of hall current, rotation and Soret effects 7
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.
Figure 8 Primary velocity profiles when Gr= 6, Gm = 5,
Pr= 0.71, m= 0.5, Sc = 0.22, K1 = 0.5, t= 1.
Figure 9 Secondary velocity profiles when Gr= 6, Gm = 5,
Pr= 0.71, Sc = 0.22, K1 = 0.5, m= 0.5, t= 1.
8 D. Sarma, K.K. Pandit
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
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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
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
Figure 10 Primary velocity profiles when Gr= 6, Gm = 5,
Pr = 0.71, K2 = 5, K1 = 0.5, m = 0.5, t = 1.
Figure 11 Secondary velocity profiles when Gr= 6, Gm = 5,
Pr = 0.71, K1 = 0.5, K2 = 5, m = 0.5, t = 1.
Figure 12 Primary velocity profiles when Gr= 6, Gm = 5,
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.
Effects of hall current, rotation and Soret effects 9
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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.
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
Figure 15 Secondary velocity profiles when Gr = 6, Gm = 5,
Pr= 0.71, m= 0.5, Sc = 0.22, K1 = 0.5, t= 1.
Figure 16 Primary velocity profiles when Gr= 6, Gm = 5,
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.
10 D. Sarma, K.K. Pandit
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
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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
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
Figure 19 Secondary velocity profiles when Gr= 6, Gm = 5,
Pr = 0.71, m = 0.5, Sc = 0.22, K2 = 5, K1 = 0.5, t= 1, N= 1.
Figure 20 Primary velocity profiles when Gr= 6, Gm = 5,
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.
Effects of hall current, rotation and Soret effects 11
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
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 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.
12 D. Sarma, K.K. Pandit
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 30 Concentration profiles when Pr= 0.71 and
Sc = 0.22.
Effects of hall current, rotation and Soret effects 13
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
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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
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
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
14 D. Sarma, K.K. Pandit
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
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:
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.
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
Effects of hall current, rotation and Soret effects 15
� 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
with constant suction and heat sources. Astro Phys Space Sci
1982;86(1):43–53.
[4] Bejan A, Khair KR. Heat and mass transfer by natural convection
in a porous medium. Int J Heat Mass Transfer 1985;28(5):909–18.
[5] Jang JY, Chang WJ. Buoyancy-induced inclined boundary layer
flow in a porous medium resulting from combined heat and mass
buoyancy effects. IntCommunHeatMassTransfer 1988;15(1):17–30.
[6] Lai FC, Kulacki FA. Non-Darcy mixed convection along a
vertical wall in a saturated porous medium. J Heat Transfer
1991;113:252–5.
[7] Nakayama A, Hossain MA. An integral treatment for combined
heat and mass transfer by natural convection in a porous medium.
Int J Heat Mass Transfer 1995;38(4):761–5.
[8] Yih KA. The effect of transpiration on coupled heat and mass
transfer in mixed convection over a vertical plate embedded in a
saturated porous medium. Int Commun Heat Mass Transfer
1997;24(2):265–75.
[9] Chamkha AJ, Takhar HS, Soundalgekar VM. Radiation effects
on free convection flow past a semi-infinite vertical plate with
mass transfer. Chem Eng J 2001;84(3):335–42.
[10] Ganesan P, Palani G. Natural convection effects on impulsively
started inclined plate with heat and mass transfer. Heat Mass
Transfer 2003;39(4):277–83.
[11] Oreper GM, Szekely J. The effect of an externally imposed
magnetic field on buoyancy driven flow in a rectangular cavity. J
Cryst Growth 1983;64(3):505–15.
[12] Hossain MA, Mandal AC. Mass transfer effects on the unsteady
hydromagnetic free convection flow past an accelerated vertical
porous plate. J Phys D Appl Phys 1985;18(7):163–9.
[13] Jha BK. MHD free convection and mass transform flow through
a porous medium. Astrophys Space Sci 1991;175(2):283–9.
[14] Elbashbeshy EMA. Heat and mass transfer along a vertical plate
with variable surface tension and concentration in the presence of
the magnetic field. Int J Eng Sci 1997;35(5):515–22.
[15] Chen CH. Combined heat and mass transfer in MHD free
convection from a vertical surface with Ohmic heating and viscous
dissipation. Int J Eng Sci 2004;42(7):699–713.
[16] Ibrahim FS, Hassanien IA, Bakr AA. Unsteady magnetohydro-
dynamic micropolar fluid flow and heat transfer over a vertical
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:/
porous plate through a porous medium in the presence of thermal
and mass diffusion with a constant heat source. Can J Phys
2004;82(10):775–90.
[17] Chamkha AJ. Unsteady MHD convective heat and mass transfer
past a semi-infinite vertical permeable moving plate with heat
absorption. Int J Eng Sci 2004;42(2):217–30.
[18] Makinde OD, Sibanda P. Magnetohydrodynamic mixed convec-
tive flow and heat and mass transfer past a vertical plate in a
porous medium with constant wall suction. J Heat Transfer
2008;130(11):8 112602.
[19] Makinde OD. On MHD boundary layer flow and mass transfer
past a vertical plate in a porous medium with constant heat flux.
Int J Numer Meth Heat Fluid Flow 2009;19(3–4):546–54.
[20] Eldabe NTM, Elbashbeshy EMA, Hasanin WSA, Elsaid EM.
Unsteady motion of MHD viscous incompressible fluid with heat
and mass transfer through porous medium near a moving vertical
plate. Int J Energy Tech 2011;35(3):1–11.
[21] Singh AK. MHD free-convection flow in the Stokes problem for a
vertical porous plate in a rotating system. Astrophys Space Sci
1983;95(2):283–9.
[22] Singh AK. MHD free convection flow past an accelerated vertical
porous plate in a rotating fluid. Astrophys Space Sci 1984;103
(1):155–63.
[23] Raptis AA, Singh AK. Rotation effects on MHD free-convection
flow past an accelerated vertical plate. Mech Res Commun
1985;12(1):31–40.
[24] Kythe PK, Puri P. Unsteady MHD free convection flows on a
porous plate with time-dependent heating in a rotating medium.
Astrophys Space Sci 1988;143(1):51–62.
[25] Tokis JN. Free convection and mass transfer effects on the
magnetohydrodynamic flows near a moving plate in a rotating
medium. Astrophys Space Sci 1988;144(1–2):291–301.
[26] Nanousis N. Thermal diffusion effects on MHD free convective
and mass transfer flow past a moving infinite vertical plate in a
rotating fluid. Astrophys Space Sci 1992;191(2):313–22.
[27] Singh AK, Singh NP, Singh U, Singh H. Convective flow past an
accelerated porous plate in rotating system in presence of
magnetic field. Int J Heat Mass Transfer 2009;52(13–14):3390–5.
[28] Seddeek MA. Effects of radiation and variable viscosity on a
MHD free convection flow past a semi-infinite flat plate with an
aligned magnetic field in the case of unsteady flow. Int J Heat
Mass Transfer 2002;45(4):931–5.
[29] Viskanta R, Grosh RJ. Boundary layer in thermal radiation
absorbing and emitting media. Int J Heat Mass Transfer 1962;5
(9):795–806.
[30] Cess RD. The interaction of thermal radiation with free convec-
tion heat transfer. Int J Heat Mass Transfer 1966;9(11):1269–77.
[31] E.M. Sparrow, R.D. Cess, Radiation heat transfer. Belmont,
Calif.: Brooks/Cole; 1966.
[32] Howell JR, Siegel R, Menguc MP. Thermal radiation heat
transfer. 5th ed. FL: CRC Press; 2010.
[33] Takhar HS, Gorla RSR, Soundalgekar VM. Short communica-
tion radiation effects on MHD free convection flow of a gas past a
semi-infinite vertical plate. Int J Numer Meth Heat Fluid Flow
1996;6(2):77–83.
[34] Raptis A, Massalas CV. Magnetohydrodynamic flow past a plate
by the presence of radiation. Heat Mass Transfer 1998;34(2–
3):107–9.
[35] Chamkha AJ. Thermal radiation and buoyancy effects on
hydromagnetic flow over an accelerating permeable surface with
heat source or sink. Int J Eng Sci 2000;38(15):1699–712.
[36] Cookey CI, Ogulu A, Omubo-Pepple VB. Influence of viscous
dissipation and radiation on unsteady MHD free convection flow
past an infinite heated vertical plate in a porous medium with
time-dependent suction. Int J Heat Mass Transfer 2003;46
(13):2305–11.
[37] Suneetha S, Bhaskar Reddy N, Ramachandra Prasad V. Thermal
radiation effects on MHD free convection flow past an impulsively
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
16 D. Sarma, K.K. Pandit
started vertical plate with variable surface temperature and
concentration. J Naval Arch Marine Eng 2008;5(2):57–70.
[38] Ogulu A, Makinde OD. Unsteady hydromagnetic free convection
flow of a dissipative and radiating fluid past a vertical plate with
constant heat flux. Chem Eng Commun 2008;196(4):454–62.
[39] Mahmoud MAA. Thermal radiation effect on unsteady MHD
free convection flow past a vertical plate with temperature
dependent viscosity. Can J Chem Eng 2009;87(1):47–52.
[40] Cramer K, Pai S. Magnetofluid dynamics for engineers and
applied physicists. New York: McGraw-Hill; 1973.
[41] Pop I, Watanabe T. Hall Effect on magnetohydrodynamic free
convection about a semi-infinite vertical flat plate. Int J Eng Sci
1994;32(12):1903–11.
[42] Abo-Eldahab EM, Elbarbary EME. Hall current effect on
magnetohydrodynamic free-convection flow past a semi-infinite
vertical plate with mass transfer. Int J Eng Sci 2001;39
(14):1641–52.
[43] Takhar HS, Roy S, Nath G. Unsteady free convection flow over
an infinite vertical porous plate due to the combined effects of
thermal and mass diffusion, magnetic field and Hall currents.
Heat Mass Transfer 2003;39(10):825–34.
[44] Saha LK, Siddiqa S, Hossain MA. Effect of Hall current on
MHD natural convection flow from vertical permeable flat plate
with uniform surface heat flux. Appl Math Mech Engl Ed 2011;32
(9):1127–46.
[45] Satya Narayana PV, Venkateswarlu B, Venkataramana S. Effects
of Hall current and radiation absorption on MHD micropolar
fluid in a rotating system. Ain Shams Eng J 2013;4(4):843–54.
[46] Seth GS, Mahato GK, Sarkar S. Effects of Hall current and
rotation on MHD natural convection flow past an impulsively
moving vertical plate with ramped temperature in the presence of
thermal diffusion with heat absorption. Int J Energy Tech 2013;5
(16):1–12.
[47] Seth GS, Sarkar S, Hussain SM. Effects of Hall current, radiation
and rotation on natural convection heat and mass transfer flow
past a moving vertical plate. Ain Shams Eng J 2014;5(2):489–503.
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:
[48] Takhar HS, Chamkha AJ, Nath G. MHD flow over a moving
plate in a rotating fluid with magnetic field, Hall currents and free
stream velocity. Int J Eng Sci 2002;40(13):1511–27.
[49] Cowling TG. Magnetohydrodynamics. New York: Interscience
Publishers; 1957.
[50] Cramer KR, Pai SI. Magnetofluid dynamics for engineers and
applied physicists. New York: McGraw Hill Book Company;
1973.
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.
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