Journal of the Egyptian Mathematical Society (2015) 23, 197–207
Egyptian Mathematical Society
Journal of the Egyptian Mathematical Society
www.etms-eg.orgwww.elsevier.com/locate/joems
ORIGINAL ARTICLE
Hydromagnetic natural convection flow with heat
and mass transfer of a chemically reacting and heat
absorbing fluid past an accelerated moving vertical
plate with ramped temperature and ramped surface
concentration through a porous medium
* Corresponding author.E-mail addresses: [email protected] (G.S. Seth), hussain.modassir@
yahoo.com (S.M. Hussain), [email protected] (S. Sarkar).
Peer review under responsibility of Egyptian Mathematical Society.
Production and hosting by Elsevier
1110-256X ª 2014 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society.
http://dx.doi.org/10.1016/j.joems.2014.03.006
G.S. Seth a,*, S.M. Hussain b, S. Sarkar a
a Department of Applied Mathematics, Indian School of Mines, Dhanbad 826004, Indiab Department of Mathematics, O. P. Jindal Institute of Technology, Raigarh 496109, India
Received 11 January 2014; revised 21 March 2014; accepted 26 March 2014Available online 29 April 2014
KEYWORDS
Ramped temperature;
Ramped surface concentra-
tion;
Chemical reaction;
Heat absorption;
Nusselt number;
Sherwood number
Abstract Unsteady hydromagnetic natural convection flow with heat and mass transfer of an elec-
trically conducting, viscous, incompressible, chemically reacting and heat absorbing fluid past an
accelerated moving vertical plate with ramped temperature and ramped surface concentration
through a porous medium in the presence of thermal and mass diffusions is studied. The exact solu-
tions of momentum, energy and concentration equations, under the Boussinesq approximation, are
obtained in closed form by Laplace Transform technique. The expressions for skin friction, Nusselt
number and Sherwood number are also derived. The variations in fluid velocity, fluid temperature
and species concentration are displayed graphically whereas numerical values of skin friction, Nus-
selt number and Sherwood number are presented in tabular form for various values of pertinent
flow parameters. Natural convection flow near a ramped temperature plate with ramped surface
concentration is also compared with the flow near an isothermal plate with uniform surface concen-
tration.
2010 MATHEMATICS SUBJECT CLASSIFICATION: 76W05; 76R10; 80A20; 80A32
ª 2014 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society.
1. Introduction
The problems of hydromagnetic convective flow in a porousmedium have drawn considerable attentions of severalresearchers owing to its importance in various scientific andtechnological applications viz. problems of boundary layer
Nomenclature
B0 uniform magnetic field
C0 species concentrationD chemical molecular diffusivityGc solutal Grashof numberK1 permeability parameter
K01 permeability of porous mediumPr Prandtl numberR dimensionless constant
T dimensionless fluid temperatureu dimensionless fluid velocity in g directionC dimensionless species concentration
Cp specific heat at constant pressureg acceleration due to gravityGr thermal Grashof numberK2 chemical reaction parameter
M magnetic parameter
Q0 heat absorption coefficient
Sc Schmidt numberT0 fluid temperatureu0 fluid velocity in g direction
Greek symbols
a0 thermal diffusivityb0 volumetric coefficient of thermal expansion for
species concentration
t kinematic coefficient of viscosity/ dimensionless heat absorption coefficientq fluid densityb* volumetric coefficient of expansion
r electrical conductivity
198 G.S. Seth et al.
flow control, plasma studies, geothermal energy extraction,metallurgy, chemical, mineral and petroleum engineering,
etc. and on the performance of so many engineering devicesusing electrically conducting fluids, namely, MHD generators,MHD pumps, MHD accelerators, MHD flow-meters, nuclear
reactors, plasma jet engines, etc. Raptis and Kafousias [1]investigated steady hydromagnetic free convection flowthrough a porous medium bounded by an infinite vertical platewith constant suction velocity. Raptis [2] discussed unsteady
two-dimensional natural convection flow of an electrically con-ducting, viscous and incompressible fluid along an infinite ver-tical plate embedded in a porous medium. Chamkha [3]
studied unsteady MHD free convection flow through a porousmedium supported by a surface. Chamkha [4] also studiedMHD natural convection flow near an isothermal inclined sur-
face adjacent to a thermally stratified porous medium. Aldosset al. [5] investigated combined free and forced convection flowfrom a vertical plate embedded in a porous medium in the
presence of a magnetic field. Kim [6] analyzed unsteadyMHD free convection flow past a moving semi-infinite verticalporous plate embedded in a porous medium with variable suc-tion. Theoretical/experimental investigations of convective
boundary layer flow with heat and mass transfer induced dueto a moving surface with a uniform or non-uniform velocityplay an important role in several manufacturing processes in
industry which include the boundary layer flow along materialhandling conveyers, extrusion of plastic sheets, cooling of aninfinite metallic plate in cooling bath, glass blowing, continu-
ous casting and levitation, design of chemical processingequipment, formation and dispersion of fog, distribution oftemperature and moisture over agricultural fields and grovesof trees, damage of crops due to freezing, common industrial
sight especially in power plants, etc. Keeping in view theimportance of such study, Jha [7] considered hydromagneticfree convection and mass transfer flow past a uniformly accel-
erated moving vertical plate through a porous medium. Ibra-him et al. [8] investigated unsteady hydromagnetic freeconvection flow of micro-polar fluid and heat transfer past a
vertical porous plate through a porous medium in the presence
of thermal and mass diffusions with a constant heat source.Makinde and Sibanda [9] studied MHD mixed convective flow
with heat and mass transfer past a vertical plate embedded in aporous medium with constant wall suction. Makinde [10] ana-lyzed hydromagnetic mixed convection flow and mass transfer
over a vertical porous plate with constant heat flux embeddedin a porous medium. Makinde [11] also investigated MHDboundary layer flow with heat and mass transfer over a mov-ing vertical plate with a convective surface boundary
condition.It is noticed that there may be an appreciable temperature
difference between the surface of the solid body and ambient
fluid in so many fluid flow problems of practical interests. Thisprompted many researchers to consider temperature depen-dent heat sources and/or sinks, which may have strong influ-
ence on heat transfer characteristics [12]. The researchstudies related to heat generating and/or heat absorbing fluidflow are of considerable importance in several physical prob-
lems viz. fluids undergoing exothermic and/or endothermicchemical reaction [12], its applications in the field of nuclearenergy [13], convection in Earth’s mantle [14], post accidentheat removal [15], fire and combustion modeling [16], develop-
ment of metal waste from spent nuclear fuel [17], etc. Exactmathematical modeling of internal heat generation/absorptionis very much complicated. It is found that some simple
mathematical models yet idealized may present their averagebehavior for most of the physical situations. Taking into con-sideration of this fact Sparrow and Cess [18] discussed the
effects of temperature dependent heat absorption in theirresearch study on steady stagnation point flow and heat trans-fer. Moalem [19] investigated steady heat transfer in a porousmedium with temperature dependent heat source. Chamkha
and Khaled [20] considered hydromagnetic combined heatand mass transfer by natural convection from a permeable ver-tical plate embedded in a fluid saturated porous medium in the
presence of heat generation or absorption. Kamel [21] investi-gated unsteady hydromagnetic convection flow due to heatand mass transfer through a porous medium bounded by an
infinite vertical porous plate with temperature dependent heat
Hydromagnetic natural convection flow with heat and mass transfer of a chemically reacting and heat absorbing fluid 199
sources and sinks. Chamkha [22] studied unsteady hydromag-netic two dimensional convective laminar boundary layer flowwith heat and mass transfer of a viscous, incompressible, elec-
trically conducting and temperature dependent heat absorbingfluid along a semi infinite vertical permeable moving plate inthe presence of a uniform transverse magnetic field. Makinde
[23] investigated heat and mass transfer by MHD mixed con-vection stagnation point flow toward a vertical plate embeddedin a highly porous medium with radiation and internal heat
generation. In all these investigations, numerical/analyticalsolution is obtained by assuming conditions for the velocityand temperature at the plate as continuous and well defined.However, there are several problems of practical interests
which may require non-uniform or arbitrary conditions atthe plates. Keeping in view this fact, several researchers,namely, Hayday et al. [24], Kelleher [25], Kao [26], Lee and
Yovanovich [27] and Chandran et al. [28] studied natural con-vection flow from a vertical plate with step discontinuities inthe surface temperature considering different aspects of the
problem. Patra et al. [29] investigated the effects of radiationon natural convection flow of a viscous and incompressiblefluid near a vertical flat plate with ramped temperature. They
compared the effects of radiative heat transfer on natural con-vection flow near a ramped temperature plate with the flownear an isothermal plate. Seth and Ansari [30] investigatedunsteady hydromagnetic natural convection flow of a viscous,
incompressible, electrically conducting and temperaturedependent heat absorbing fluid past an impulsively movingvertical plate with ramped temperature in a porous medium
taking into account the effects of thermal diffusion. Subse-quently, Seth et al. [31] extended the problem studied by Sethand Ansari [30] to consider the effects of rotation on flow-field.
In many chemical engineering processes, there does occurthe chemical reaction between a foreign mass and the fluid.Chemical reactions can be classified as either heterogeneous
or homogeneous processes. This depends on whether theyoccur at an interface or as a single phase volume reaction.These processes take place in numerous industrial applicationsviz. polymer production, manufacturing of ceramics or glass-
ware, food processing, etc. Afify [32] studied the effect of radi-ation on free convective flow and mass transfer past a verticalisothermal cone surface with chemical reaction in the presence
of a transverse magnetic field. Muthucumaraswamy and Chan-drakala [33] investigated radiative heat and mass transfereffects on moving isothermal vertical plate in the presence of
chemical reaction. Ibrahim et al. [34] analyzed the effect ofchemical reaction and radiation absorption on the unsteadyMHD free convection flow past a semi-infinite vertical perme-able moving plate with heat source. Bakr [35] discussed the
effects of chemical reaction on MHD free convection and masstransfer flow of a micro-polar fluid with oscillatory plate veloc-ity and constant heat source in a rotating frame of reference.
Chamkha [36] studied MHD flow of a uniformly stretched ver-tical permeable surface in the presence of heat generation/absorption and chemical reaction. Chamkha et al. [37] dis-
cussed the effects of Joule heating, chemical reaction and ther-mal radiation on unsteady hydromagnetic natural convectionboundary layer flow with heat and mass transfer of a micro-
polar fluid from a semi-infinite heated vertical porous platein the presence of a uniform transverse magnetic field. Bhat-tacharya and Layek [38] obtained similarity solution ofMHD boundary layer flow with mass diffusion and chemical
reaction over a porous flat plate with suction/blowing.Recently, Mohamed et al. [39] investigated unsteady MHDfree convection heat and mass transfer boundary layer flow
of viscous, incompressible, optically thick and electrically con-ducting fluid through a porous medium along an impulsivelymoving hot vertical plate in the presence of homogeneous
chemical reaction of first order and temperature dependentheat sink. They obtained analytical solution of the governingequations in closed form by Laplace transform technique.
Objective of present investigation is to study unsteadyhydromagnetic natural convection flow with heat and masstransfer of a chemically reacting and heat absorbing fluid pastan accelerated moving vertical plate with ramped temperature
and ramped surface concentration through a porous medium.Such study may find application in solar collection systems,fire dynamics in insulations, geothermal energy systems, cata-
lytic reactors, nuclear waste repositories, recovery of petro-leum products and gases (e.g. CBM: Coal Bed Methane andUCG: Underground Coal Gasification), etc.
2. Formulation of the problem and its solution
Consider unsteady hydromagnetic natural convection flow
with heat and mass transfer of an electrically conducting, vis-cous, incompressible, chemically reacting and temperaturedependent heat absorbing fluid past an accelerated moving infi-
nite vertical plate embedded in a porous medium in the pres-ence of thermal and mass diffusions. Choose the co-ordinatesystem in such a way that x0-axis is along the plate in upwarddirection, y0-axis normal to the plane of the plate and z0-axis
perpendicular to x0y0-plane. The fluid is permeated by uniformtransverse magnetic field B0 applied parallel to y0-axis. Initially,i.e. at time t0 6 0, both the fluid and plate are at rest and main-
tained at uniform temperature T01 and uniform surface concen-tration C01. At time t0 > 0, plate starts moving in x0-directionagainst the gravitational field with time dependent velocity
U(t0). Temperature of the plate is raised or lowered toT01 þ ðT0w � T01Þt0=t0 and the level of concentration at the sur-face of the plate is raised or lowered to C01 þ ðC0w � C01Þt0=t0when 0 < t0 6 t0. Thereafter, i.e. at t
0 > t0, plate is maintainedat the uniform temperature T0w and the level of concentration atthe surface of the plate is preserved at uniform concentrationC0w. Here t0 is characteristic time. It is assumed that there exists
a homogeneous chemical reaction of first order with constantrate K02 between the diffusing species and the fluid. The sche-matic diagram of the physical problem is shown in Fig. 1. Since
the plate is of infinite extent along x0 and z0-directions and iselectrically non-conducting, all physical quantities except pres-sure depend on y0 and t0 only. The induced magnetic field pro-
duced by fluid motion is neglected in comparison with appliedone. This statement is justified because magnetic Reynoldsnumber is very small for liquid metals and partially ionizedfluid [40]. Also no external electric field is applied so the effect
of polarization of fluid is negligible. This corresponds to thecase where no energy is added or extracted from the fluid byelectrical means [40].
Taking into consideration the assumptions made above, thegoverning equations for natural convection flow with heat andmass transfer of an electrically conducting, viscous, incom-
pressible, chemically reacting and temperature dependent heatabsorbing fluid through a porous medium in the presence of
Figure 1 Schematic diagram of the physical problem.
200 G.S. Seth et al.
thermal and mass diffusions, under Boussinesq approxima-tion, are given by
@u0
@t0¼ t
@2u0
@y02� rB2
0
qu0 � t
K01u0 þ gb0ðT0 �T01Þ þ gb�ðC0 �C01Þ; ð2:1Þ
@T0
@t0¼ a0
@2T0
@y02� Q0
qCp
ðT0 � T01Þ; ð2:2Þ
@C0
@t0¼ D
@2C0
@y02� K02ðC0 � C01Þ; ð2:3Þ
Appropriate initial and boundary conditions for fluid velocity,fluid temperature and species concentration are given by
u0 ¼ 0;T0 ¼ T01;C0 ¼ C01 for y0 P 0 and t0 6 0; ð2:4aÞ
u0 ¼ Uðt0Þ at y0 ¼ 0 for t0 > 0; ð2:4bÞ
T0 ¼ T01 þ ðT0w � T01Þt0=t0;C0 ¼ C01 þ ðC0w � C01Þt0=t0;
�at y0 ¼ 0 for 0 < t0 6 t0;
ð2:4cÞ
T0 ¼ T0w;C0 ¼ C0w at y0 ¼ 0 for t0 > t0; ð2:4dÞ
u0 ! 0; T0 ! T01; C0 ! C01 as y0 ! 1 for t0 > 0: ð2:4eÞ
In order to represent Eqs. (2.1)–(2.3) along with initial and
boundary conditions Eq. (2.4) in dimensionless form, follow-ing dimensionless quantities and parameters are introduced.
y¼y0=U0t0; u¼u0=U0; t¼ t0=t0; T¼ðT0 �T01Þ=ðT0w�T01Þ;C¼ðC0 �C01Þ=ðC0w�C01Þ; Gr¼ tgb0ðT0w�T01Þ=U3
0;
Gc¼ tgb�ðC0w�C01Þ=U30; M¼rB2
0t=qU20; K1¼K01U
20=t
2;
Pr¼ t=a0; Sc¼ t=D; K2¼ tK02=U20 and/¼ tQ0=qCpU
20;
9>>>>=>>>>;ð2:5Þ
Using dimensionless quantities and parameters defined in(2.5), Eqs. (2.1)–(2.3), in dimensionless form, become
@u
@t¼ @
2u
@y2�Mu� u
K1
þ GrTþ GcC; ð2:6Þ
@T
@t¼ 1
Pr
@2T
@y2� /T; ð2:7Þ
@C
@t¼ 1
Sc
@2C
@y2� K2C: ð2:8Þ
According to above non-dimensionalization process, charac-teristic time t0 may be defined as
t0 ¼ t=U20: ð2:9Þ
Appropriate initial and boundary conditions (2.4), in dimen-sionless form, assume the following form
u ¼ 0; T ¼ 0; C ¼ 0 for y P 0 and t 6 0; ð2:10aÞ
u ¼ FðtÞ at y ¼ 0 for t > 0; ð2:10bÞ
T ¼ t; C ¼ t at y ¼ 0 for 0 < t 6 1; ð2:10cÞ
T ¼ 1; C ¼ 1 at y ¼ 0 for t > 1; ð2:10dÞ
u! 0; T! 0; C! 0 as y!1 for t > 0; ð2:10eÞ
where F(t) = U(t0)/U0.The fluid flow described by the Eqs. (2.6)–(2.8) subject to
initial and boundary conditions (2.10) is quite general. In orderto analyze the flow features of the fluid flow we now consider a
particular case of interest, namely, uniformly acceleratedmovement of the plate, i.e. F(t) = Rt where R is dimensionlessconstant.
Eqs. (2.6)–(2.8) are solved analytically with the help ofLaplace transform technique subject to the initial and bound-ary conditions (2.10) and the exact solutions for fluid velocity
F(y, t), fluid temperature T(y, t) and species concentrationC(y, t) are obtained and are presented after simplification inthe following form
uðy; tÞ ¼ R
2tþ y
2ffiffiffikp
� �d1 þ t� y
2ffiffiffikp
� �d2
� �
� d1
d23
fF1ðy; tÞ �Hðt� 1ÞF1ðy; t� 1Þg
� d2
d24
fF2ðy; tÞ �Hðt� 1ÞF2ðy; t� 1Þg; ð2:11Þ
Hydromagnetic natural convection flow with heat and mass transfer of a chemically reacting and heat absorbing fluid 201
Tðy; tÞ ¼ Pðy; tÞ �Hðt� 1ÞPðy; t� 1Þ; ð2:12Þ
Cðy; tÞ ¼ Qðy; tÞ �Hðt� 1ÞQðy; t� 1Þ; ð2:13Þ
where
F1ðy; tÞ ¼ed3t
2ey
ffiffiffiffiffiffiffiffiffiffiðkþd3Þp
erfcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkþ d3Þt
pþ y
2ffiffitp
� ��
þe�yffiffiffiffiffiffiffiffiffiffiðkþd3Þp
erfc �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkþ d3Þt
pþ y
2ffiffitp
� �
�eyffiffiffiffiffiffiffiffiffiffiffiffiffiffiPrð/þd3Þp
erfcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið/þ d3Þt
pþ y
2
ffiffiffiffiffiPr
t
r !
�e�yffiffiffiffiffiffiffiffiffiffiffiffiffiffiPrð/þd3Þp
erfc �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið/þ d3Þt
pþ y
2
ffiffiffiffiffiPr
t
r !)
� d3
2tþ 1
d3
þ y
2ffiffiffikp
� �d1
�þ tþ 1
d3
� y
2ffiffiffikp
� �d2
� d3
2tþ 1
d3
þ y
2ffiffiffikp
� �d1
�þ tþ 1
d3
� y
2ffiffiffikp
� �
d2� tþ 1
d3
þ y
2
ffiffiffiffiffiPr
/
s !d3 � tþ 1
d3
� y
2
ffiffiffiffiffiPr
/
s !d4
);
F2ðy; tÞ ¼ed4t
2ey
ffiffiffiffiffiffiffiffiffiffiðkþd4Þp
erfcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkþ d4Þt
pþ y
2ffiffitp
� ��
þe�yffiffiffiffiffiffiffiffiffiffiðkþd4Þp
erfc �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkþ d4Þt
pþ y
2ffiffitp
� �
�eyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiScðK2þd4Þp
erfcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðK2 þ d4Þt
pþ y
2
ffiffiffiffiffiSc
t
r !
�e�yffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiScðK2þd4Þp
erfc �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðK2 þ d4Þt
pþ y
2
ffiffiffiffiffiSc
t
r !)
� d4
2tþ 1
d4
þ y
2ffiffiffikp
� �d1
�þ tþ 1
d4
� y
2ffiffiffikp
� �
d2� tþ 1
d4
þ y
2
ffiffiffiffiffiffiSc
K2
r� �d5 � tþ 1
d4
� y
2
ffiffiffiffiffiffiSc
K2
r� �d6
�;
Pðy; tÞ ¼ 1
2tþ y
2
ffiffiffiffiffiPr
/
s !d3 þ t� y
2
ffiffiffiffiffiPr
/
s !d4
( );
and
Cðy; tÞ ¼ 1
2tþ y
2
ffiffiffiffiffiffiSc
K2
r� �d5 þ t� y
2
ffiffiffiffiffiffiSc
K2
r� �d6
��;
where
k ¼ Mþ 1
K1
� �; d1 ¼ Gr=ð1� PrÞ; d2 ¼ Gc=ð1� ScÞ;
d3 ¼ ðPr/� kÞ=ð1� PrÞ; d4 ¼ ðScK2 � kÞ=ð1� ScÞ;
d1 ¼ eyffiffikperfc
ffiffiffiffiktpþ y
2ffiffitp
� �;
d2 ¼ e�yffiffikperfc �
ffiffiffiffiktpþ y
2ffiffitp
� �;
d3 ¼ eyffiffiffiffiffiffiPr/p
erfcffiffiffiffiffi/t
pþ y
2
ffiffiffiffiffiPr
t
r !;
d4 ¼ e�yffiffiffiffiffiffiPr/p
erfc �ffiffiffiffiffi/t
pþ y
2
ffiffiffiffiffiPr
t
r !;
d5 ¼ eyffiffiffiffiffiffiffiScK2
perfc
ffiffiffiffiffiffiffiK2t
pþ y
2
ffiffiffiffiffiSc
t
r !;
d6 ¼ e�yffiffiffiffiffiffiffiScK2
perfc �
ffiffiffiffiffiffiffiK2t
pþ y
2
ffiffiffiffiffiSc
t
r !:
3. Solution in the case of isothermal plate with uniform surface
concentration
In order to highlight the effects of ramped temperature andramped surface concentration on fluid flow, it may be worth-while to compare such flow with the one near an accelerated
moving vertical plate with uniform temperature and uniformsurface concentration. Keeping in view the assumptions madein this paper, the solution for fluid velocity, fluid temperature
and species concentration for natural convection flow past anaccelerated moving vertical isothermal plate with uniform sur-face concentration is obtained and is expressed in the followingform
uðy; tÞ ¼ 1
2R tþ y
2ffiffiffikp
� �þ d1
d3
þ d2
d4
� �d1
�þ R t� y
2ffiffiffikp
� �þ d1
d3
þ d2
d4
� �d2
�
� d1
2d3
ed3t eyffiffiffiffiffiffiffiffiffiffiðkþd3Þp
erfcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkþ d3Þt
pþ y
2ffiffitp
� ��
þe�yffiffiffiffiffiffiffiffiffiffiðkþd3Þp
erfc �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkþ d3Þt
pþ y
2ffiffitp
� �
� eyffiffiffiffiffiffiffiffiffiffiffiffiffiffiPrð/þd3Þp
erfcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið/þ d3Þt
pþ y
2
ffiffiffiffiffiPr
t
r !
� e�yffiffiffiffiffiffiffiffiffiffiffiffiffiffiPrð/þd3Þp
erfcð�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið/þ d3Þt
pþ y
2
ffiffiffiffiffiPr
t
rÞ#
� d1
2d3
ðd3 þ d4Þ �d2
2d4
ed4t eyffiffiffiffiffiffiffiffiffiffiðkþd4Þp
erfcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkþ d4Þt
pþ y
2ffiffitp
� ��
þe�yffiffiffiffiffiffiffiffiffiffiðkþd4Þp
� erfc �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkþ d4Þt
pþ y
2ffiffitp
� �
� eyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiScðK2þd4Þp
erfcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðK2 þ d4Þt
pþ y
2
ffiffiffiffiffiSc
t
r !
� e�yffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiScðK2þd4Þp
�erfc �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðK2 þ d4Þt
pþ y
2
ffiffiffiffiffiSc
t
r !#� d2
2d4
ðd5 þ d6Þ;
ð3:1Þ
Tðy; tÞ ¼ 1
2ðd3 þ d4Þ; ð3:2Þ
Cðy; tÞ ¼ 1
2ðd5 þ d6Þ: ð3:3Þ
4. Skin friction, Nusselt number and Sherwood number
The expressions for the skin friction s, Nusselt number Nu andSherwood number Sh, which are measures of the shear stress,rate of heat transfer and rate of mass transfer at the plate
respectively, are presented in the following form for the platewith ramped temperature and ramped surface concentrationand isothermal plate with uniform surface concentration:
Figure 2 Velocity profiles when K2 = 0.2, Gr = 4, Gc = 5, /= 1 and t= 0.5.
202 G.S. Seth et al.
For the plate with ramped temperature and ramped surfaceconcentration
s ¼ R1
2ffiffiffikp þ t
ffiffiffikp� �
erfcffiffiffiffiktp
� 1n o
�ffiffiffit
p
re�kt
" #
� d1
d23
fF5ð0; tÞ �Hðt� 1ÞF5ð0; t� 1Þg
� d1
d23
fF6ð0; tÞ �Hðt� 1ÞF6ð0; t� 1Þg; ð4:1Þ
Nu ¼ P2ð0; tÞ �Hðt� 1ÞP2ð0; t� 1Þ; ð4:2Þ
Sh ¼ Q2ð0; tÞ �Hðt� 1ÞQ2ð0; t� 1Þ; ð4:3Þ
where
F5ð0; tÞ ¼ ed3tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkþ d3Þ
perfc
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkþ d3Þt
p � 1
n o� 1ffiffiffiffiffi
ptp e�ðkþd3Þt
�
þffiffiffiffiffiPr
pt
re�ð/þd3Þt�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPrð/þ d3Þ
perfc
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið/þ d3Þt
p � 1
n o#
� d3
1
2ffiffiffikp þ
ffiffiffikp
tþ 1
d3
� �� ��� erfc
ffiffiffiffiktp
� 1n o
� 1
2
ffiffiffiffiffiPr
/
sþ
ffiffiffiffiffiffiffiffiPr/
ptþ 1
d3
� �( )erfc
ffiffiffiffiffi/t
p � 1
n o
� tþ 1
d3
� �� 1ffiffiffiffiffi
ptp e�kt þ tþ 1
d3
� � ffiffiffiffiffiPr
pt
re�/t
#;
F6ð0; tÞ ¼ ed4tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkþ d4Þ
perfc
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkþ d4Þt
p � 1
n o� 1ffiffiffiffiffi
ptp e�ðkþd4Þt
�
þffiffiffiffiffiSc
pt
re�ðK2þd4Þt�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiScðK2þ d4Þ
perfc
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðK2þ d4Þt
p � 1
n o#
� d4
1
2ffiffiffikp þ
ffiffiffikp
tþ 1
d4
� �� ��� erfc
ffiffiffiffiktp
� 1n o
� 1
2
ffiffiffiffiffiffiSc
K2
rþ
ffiffiffiffiffiffiffiffiffiffiScK2
ptþ 1
d4
� �� �erfc
ffiffiffiffiffiffiffiK2t
p � 1
n o
� tþ 1
d4
� �� 1ffiffiffiffiffi
ptp e�ktþ tþ 1
d4
� � ffiffiffiffiffiSc
pt
re�K2t
#;
P2ð0;tÞ¼1
2
ffiffiffiffiffiPr
/
sþ2t
ffiffiffiffiffiffiffiffiPr/
p !erfc
ffiffiffiffiffi/t
p �1
n o�2t
ffiffiffiffiffiPr
pt
re�/t
" #;
Q2ð0; tÞ¼1
2
ffiffiffiffiffiffiSc
K2
rþ2t
ffiffiffiffiffiffiffiffiffiffiScK2
p� �erfcð
ffiffiffiffiffiffik2t
p�1
n o�2t
ffiffiffiffiffiSc
pt
re�K2t
" #;
and for isothermal plate with uniform surface concentration
s¼ R
2ffiffiffikp þ
ffiffiffikp
Rtþ d1
d3
þ d2
d4
� �� �erfc
ffiffiffiffiffiktp
� 1n o
� Rtþ d1
d3
þ d2
d4
� �1ffiffiffiffiffiptp e�kt � d1
d3
ed3tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkþ d3Þ
perfc
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkþ d3Þt
p � 1
n oh� 1ffiffiffiffiffi
ptp e�ðkþd3Þt �
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPrð/þ d3Þ
perfc
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið/þ d3Þt
p � 1
n o
þffiffiffiffiffiPr
pt
re�ð/þd3Þt
#� d1
d3
ffiffiffiffiffiffiffiffiPr/
perfcð
ffiffiffiffiffi/t
pÞ � 1
n o�
ffiffiffiffiffiPr
pt
re�/t
" #
� d2
d4
ed4t
ffiffiffiffiffiSc
pt
re�ðK2þd4Þt
"þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkþ d4Þ
p� erfc
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkþ d4Þt
p � 1
n o
� 1ffiffiffiffiffiptp e�ðkþd4Þt �
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiScðK2 þ d4Þ
perfc
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðK2 þ d4Þt
p � 1
n o�
� d2
d4
ffiffiffiffiffiffiffiffiffiffiScK2
perfc
ffiffiffiffiffiffiffiK2t
p � 1
n o�
ffiffiffiffiffiSc
pt
re�K2t
" #; ð4:4Þ
Nu ¼ffiffiffiffiffiffiffiffiPr/
perfc
ffiffiffiffiffi/t
p � 1
n o�
ffiffiffiffiffiPr
pt
re�/t; ð4:5Þ
Sh ¼ffiffiffiffiffiffiffiffiffiffiScK2
perfc
ffiffiffiffiffiffiffiK2t
p � 1
n o�
ffiffiffiffiffiSc
pt
re�K2t: ð4:6Þ
5. Results and discussion
In order to study the influence of magnetic field, thermal buoy-
ancy force, species buoyancy force, heat absorption, chemicalreaction and time on flow-field in the boundary layer region,the numerical values of fluid velocity, computed from the ana-lytical solutions reported in Sections 2 and 3, are displayed
graphically versus boundary layer co-ordinate y in Figs. 2–7for various values of magnetic parameter M, thermal Grashofnumber Gr, solutal Grashof number Gc, heat absorption co-
efficient /, chemical reaction parameter K2 and time t takingK1 = 0.5, Pr = 0.71 (ionized air), Sc = 0.22 and R = 1. It isevident from Figs. 2–7 that, for both ramped temperature plate
with ramped surface concentration and isothermal plate withuniform surface concentration, fluid velocity u increasesrapidly in the region near the surface of the plate, attains a
maximum distinctive value and then decreases properly onincreasing boundary layer coordinate y to approach freestream value. Fig. 2 illustrates the influence of magnetic fieldon the fluid velocity. For both ramped temperature plate with
ramped surface concentration and isothermal plate with uni-form surface concentration, fluid velocity decreases on increas-ing magnetic parameterM. This implies that magnetic field has
retarding influence on fluid flow for both ramped temperatureplate with ramped surface concentration and isothermal platewith uniform surface concentration. This is due to the fact that
the application of a magnetic field to an electrically conductingfluid gives rise to a resistive force called Lorentz force whichhas a tendency to slow down the motion of fluid in the bound-
ary layer region. Figs. 3 and 4 depict the influence of thermalbuoyancy force and species buoyancy force on fluid velocity.The thermal Grashof number Gr signifies the relative effectof the thermal buoyancy force to the viscous hydrodynamic
force. The solutal Grashof number Gc characterizes the ratioof the species buoyancy force and viscous hydrodynamic force.As expected, it is observed that, for both ramped temperature
Figure 3 Velocity profiles when M = 4, K2 = 0.2, Gc = 5, /= 1 and t= 0.5.
Figure 4 Velocity profiles when M= 4, K2 = 0.2, Gr = 4, /= 1 and t= 0.5.
Figure 5 Velocity profiles when M= 4, K2 = 0.2, Gr = 4,
Gc = 5 and t = 0.5.
Figure 6 Velocity profiles when M= 4, Gr = 4, Gc = 5, / = 1
and t= 0.5.
Figure 7 Velocity profiles when M= 4, K2 = 0.2, Gr = 4,
Gc = 5 and / = 1.
Hydromagnetic natural convection flow with heat and mass transfer of a chemically reacting and heat absorbing fluid 203
plate with ramped surface concentration and isothermal plate
with uniform surface concentration, fluid velocity increases onincreasing either Gr or Gc. This implies that fluid velocity is get-ting accelerated due to enrichment in either thermal buoyancy
force or species buoyancy force.Fig. 5 demonstrates the effects of heat absorption on fluid
velocity. It is noticed that, for both ramped temperature platewith ramped surface concentration and isothermal plate with
uniform surface concentration, fluid velocity decreases onincreasing /. This implies that, both for ramped temperatureplate with ramped surface concentration and isothermal plate
with uniform surface concentration, heat absorption tends toretard fluid velocity throughout the boundary layer region.This may be attributed to the fact that the tendency of heat
absorption (thermal sink) is to reduce the fluid temperaturewhich causes the strength of thermal buoyancy force todecrease resulting in a net reduction in the fluid velocity.
Fig. 6 displays the effect of chemical reaction on fluid veloc-ity. It is observed that fluid velocity decreases on increasing K2
for both ramped temperature plate with ramped surface con-centration and isothermal plate with uniform surface concen-
tration. This implies that chemical reaction has a retardinginfluence on fluid flow for both ramped temperature plate withramped surface concentration and isothermal plate with uni-
form surface concentration. Fig. 7 depicts the effects of timeon fluid velocity. It is observed that fluid velocity increaseson increasing time t. This implies that there is an enhancement
in fluid velocity for both ramped temperature plate withramped surface concentration and isothermal plate with uni-form surface concentration with the progress of time. It is
noted from Figs. 2–7 that fluid velocity is faster in the caseof isothermal plate with uniform surface concentration thanthat in the case of ramped temperature plate with ramped sur-face concentration.
Figure 8 Temperature profiles when Pr = 0.71 and t = 0.5.
Figure 9 Temperature profiles when / = 1 and Pr = 0.71.
Figure 10 Concentration profiles when Sc = 0.22 and t= 0.5.
Figure 11 Concentration profiles when K2 = 0.2 and Sc = 0.22.
204 G.S. Seth et al.
The numerical solutions for fluid temperature and speciesconcentration, computed from analytical solutions reportedin Sections 2 and 3, are depicted graphically in Figs. 8–11
for different values of heat absorption coefficient /, chemicalreaction parameter K2 and time t. It is observed fromFigs. 8–11 that, fluid temperature and species concentration
are maximum at the surface of the plate and decrease properlyon increasing boundary layer coordinate y to approach freestream value. Figs. 8 and 9 illustrate the effects of heat absorp-tion coefficient / and time t on fluid temperature. It is evident
that, for both ramped temperature plate with ramped surface
concentration and isothermal plate with uniform surface con-centration, fluid temperature decreases on increasing /whereas it increases on increasing time t. This implies that,
for both ramped temperature plate with ramped surface con-centration and isothermal plate with uniform surface concen-tration, heat absorption has a tendency to reduce the fluidtemperature and there is a rise in fluid temperature with the
progress of time in the boundary layer region. Figs. 10 and11 demonstrate the effects of chemical reaction parameter K2
and time t on species concentration. It is noticed that, for both
ramped temperature plate with ramped surface concentrationand isothermal plate with uniform surface concentration, spe-cies concentration decreases on increasing K2 whereas it
increases on increasing time t. This implies that, for bothramped temperature plate with ramped surface concentrationand isothermal plate with uniform surface concentration,
chemical reaction tends to reduce species concentration andthere is enrichment in species concentration with the progressof time. It may be noted from Figs. 8–11 that the fluid temper-ature and species concentration are lower in the case of
ramped temperature plate with ramped surface concentrationthan that in the case of isothermal plate with uniform surfaceconcentration.
The numerical values of skin friction s, computed from theanalytical expressions reported in Section 4, are presented intabular form in Tables 1–3 for various values of M, Gr, Gc,
K2, / and t taking R= 1, K1 = 0.5, Sc = 0.22 andPr = 0.71 whereas those of Nusselt number Nu and Sherwoodnumber Sh, calculated from the analytical expressions reportedin Section 4, are exhibited in tabular form in Tables 4 and 5 for
different values of /, K2 and t.It is found from Table 1 that, for ramped temperature plate
with ramped surface concentration, skin friction �s increases
on increasing time t whereas, for isothermal plate with uniformsurface concentration, it decreases in magnitude on increasingt whenM P 4. This implies that, for ramped temperature plate
with ramped surface concentration, skin friction is gettingenhanced with the progress of time whereas, for isothermalplate with uniform surface concentration, skin friction is
getting reduced with the progress of time when M P 4. It isworthy to note from Table 1 that there exists flow separationat the isothermal plate on increasing time t when M = 6 andon increasing M when t= 0.7. It is observed from Tables 2
Table 1 Skin friction �s when K1 = 0.5, Gr = 4, Gc = 5, K2 = 0.2 and / = 1.
Ramped Isothermal
Mfltfi 0.3 0.5 0.7 0.3 0.5 0.7
2 0.361179 0.298291 0.186799 �0.52415 �0.94968 �0.880144 0.484248 0.554626 0.594741 �0.79709 �0.70894 �0.359606 0.595863 0.777144 0.937804 �0.71756 �0.38275 0.10460
Table 2 Skin friction �s when M = 4, Gc = 5, K1 = 0.5, K2 = 0.2 and t= 0.5.
Ramped Isothermal
Grfl/fi 1 3 5 1 3 5
2 0.554626 0.568566 0.579712 �0.708944 �0.644484 �0.5830824 0.339164 0.367043 0.389335 �1.22794 �1.09902 �0.9762146 0.123702 0.165521 0.198959 �1.74693 �1.55355 �1.36935
Table 3 Skin friction �s when M = 4, Gr = 4, K1 = 0.5, / = 1 and t= 0.5.
Ramped Isothermal
GcflK2fi 0.2 2 5 0.2 2 5
3 0.817907 0.834995 0.857408 �0.066691 0.003292 0.087601
5 0.554626 0.583106 0.620462 �0.708944 �0.592306 �0.4517907 0.291346 0.331217 0.383515 �1.35120 �1.18790 �0.991182
Table 4 Nusselt number �Nu when Pr = 0.71.
Ramped Isothermal
/fltfi 0.3 0.5 0.7 0.3 0.5 0.7
1 0.571348 0.779133 0.969291 1.11605 0.983021 0.925311
3 0.664544 0.966961 1.26243 1.55005 1.48794 1.47004
5 0.749004 1.12943 1.50704 1.92093 1.89157 1.88595
Table 5 Sherwood number �Sh when Sc = 0.22.
Ramped Isothermal
K2fltfi 0.3 0.5 0.7 0.3 0.5 0.7
0.2 0.295649 0.386593 0.463189 0.525702 0.428415 0.379505
2 0.344659 0.488076 0.625355 0.839945 0.785973 0.757863
5 0.416933 0.628694 0.838894 1.1897 1.12945 1.09522
Hydromagnetic natural convection flow with heat and mass transfer of a chemically reacting and heat absorbing fluid 205
and 3 that, for ramped temperature plate with ramped surfaceconcentration, skin friction �s increases on increasing either /or K2 and it decreases on increasing either Gr or Gc. For
isothermal plate with uniform surface concentration, �sdecreases in magnitude, attains a minimum and then increaseson increasing K2 when Gc = 3 and it decreases in magnitude
on increasing K2 when Gc = 5and7. �s increases in magnitudeon increasing either Gr or Gc for isothermal plate with uniformsurface concentration. This implies that, for ramped tempera-
ture plate with ramped surface concentration, heat absorptionand chemical reaction have tendency to enhance skin frictionwhereas thermal and species buoyancy forces have reverseeffect on it. For isothermal plate with uniform surface
concentration, heat absorption has a tendency to reduce skinfriction whereas chemical reaction tends to reduce skin frictionwhen Gc P 5 and thermal and species buoyancy forces have
tendency to enhance skin friction. It is evident from Table 3that there exists flow separation at the isothermal plate onincreasing K2 when Gc = 3 and on increasing Gc when
K2 = 5. It is perceived from Table 4 that, for ramped temper-ature plate with ramped surface concentration, Nusselt num-ber �Nu increases on increasing / or t. For isothermal plate
with uniform surface concentration, �Nu increases on increas-ing / and decreases on increasing t. This implies that, for bothramped temperature plate with ramped surface concentrationand isothermal plate with uniform surface concentration, heat
206 G.S. Seth et al.
absorption tends to enhance rate of heat transfer at the plate.Rate of heat transfer at the plate is getting enhanced forramped temperature plate with ramped surface concentration
whereas rate of heat transfer at the plate is getting reducedfor isothermal plate with uniform surface concentration withthe progress of time.
It is observed from Table 5 that, for both ramped temper-ature plate with ramped surface concentration and isothermalplate with uniform surface concentration, Sherwood number
�Sh increases on increasing K2. On increasing time t, �Sh
increases for ramped temperature plate with ramped surfaceconcentration whereas it decreases for isothermal plate withuniform surface concentration. This implies that, for both
ramped temperature plate with ramped surface concentrationand isothermal plate with uniform surface concentration,chemical reaction tends to enhance rate of mass transfer at
the plate. For ramped temperature plate with ramped surfaceconcentration, rate of mass transfer at the plate is gettingenhanced whereas for isothermal plate with uniform surface
concentration rate of mass transfer at the plate is gettingreduced with the progress of time.
6. Conclusions
The present study brings out the following significant findings:
(i) For both ramped temperature plate with ramped surfaceconcentration and isothermal plate with uniform surface
concentration:Magnetic field, heat absorption and chemical reactionhave retarding influence on fluid flow. Fluid velocity isgetting accelerated with the progress of time. Heat
absorption has tendency to reduce fluid temperatureand there is a rise in fluid temperature with the progressof time. Chemical reaction tends to reduce species con-
centration and there is enrichment in species concentra-tion with the progress of time.
(ii) For ramped temperature plate with ramped surface con-
centration: skin friction is getting enhanced with theprogress of time when M P 4 whereas heat absorptionand chemical reaction have an accelerating influence
on fluid flow whereas thermal and species buoyancyforces have reverse effect on it.
(iii) For isothermal plate with uniform surface concentra-tion: Skin friction is getting reduced with the progress
of time when M P 4. Heat absorption has a tendencyto reduce skin friction whereas chemical reaction tendsto reduce skin friction when Gc P 5 and thermal and
species buoyancy forces have tendency to enhance skinfriction. There exists flow separation at the plate onincreasing time t when M = 6 and on increasing M
when t = 0.7. Also, there exists flow separation at theplate on increasing K2 when Gc = 3 and on increasingGc when K2 = 5.
(iv) For both ramped temperature plate with ramped surface
concentration and isothermal plate with uniform surfaceconcentration, heat absorption tends to enhance rate ofheat transfer at the plate. Rate of heat transfer at the
plate is getting enhanced for ramped temperature platewith ramped surface concentration whereas rate of heattransfer at the plate is getting reduced for isothermal
plate with uniform surface concentration with the pro-
gress of time.(v) For both ramped temperature plate with ramped surface
concentration and isothermal plate with uniform surface
concentration, chemical reaction tends to enhance rateof mass transfer at the plate. For ramped temperatureplate with ramped surface concentration, rate of masstransfer at the plate is getting enhanced whereas for iso-
thermal plate with uniform surface concentration rate ofmass transfer at the plate is getting reduced with the pro-gress of time.
Acknowledgement
Authors are grateful to the reviewers for their comments andsuggestions which helped them to improve the quality of the
research paper.
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