Heat and Mass Transfer on MFD Flow through Porous … · · 2017-08-25Heat and Mass Transfer on...

ISSN 1810-2328, Journal of Engineering Thermophysics, 2012, Vol. 21, No. 2, pp. 119–130. c© Pleiades Publishing, Ltd., 2012.

Heat and Mass Transfer on MFD Flow through PorousMedia over an Accelerating Surface

in the Presence of Suction and Blowing

J. P. Panda1*, N. Dash2, and G. C. Dash1

1Department of Mathematics, Institute of Technical Education and Research,Faculty of Engineering, Siksha ‘O’ Anusandhan University,

Khandagiri, Bhubaneswar-751030, Orissa, INDIA2Department of Mathematics, Kaniha College, Kaniha, Talcher, Orissa, INDIA

Received January 17, 2011

Abstract—The heat and mass transfer of electrically conducting fluid through porous media overan accelerating surface subject both to power law surface temperature and power law heat fluxvariations with a temperature-dependent heat source in the presence of a transverse uniformmagnetic field is studied. A series solution to the energy and species concentration equation in termsof Kummer’s function is studied. The effect of Prandtl number and Schmidt number is studied withthe help of graphs.

DOI: 10.1134/S1810232812020038

1. INTRODUCTION

The flow through porous medium has become of principal interest in many scientific and engineeringapplications, i.e., in the fields of agricultural engineering to study the underground water resources,seepage of water in river beds, in chemical engineering for filtration and purification process in petroleumtechnology to study the movement of natural gas, oil, and water through the oil reservoirs. Porous mediameans a mixture of fluid and solid phases that have interconnected pores or intergranular spaces throughwhich the fluid can move. The problem of a stretching constant surface temperature was studied byCrane [1]. Chakrabarti and Gupta [2] have studied hydromagnetic flow and heat transfer over a stretchedsheet. Soundalgekar and Ramanmurty [3] have investigated the problem with constant surface velocityand power law temperature variations. The power law temperature variations in the case of a stretchingcontinuous surface was studied by Grubka and Bobba [4]. Anderson [5] has studied an exact solution ofthe Navier–Stokes equation for magnetohydrodynamic flow. Acharya et al. [6] have discussed the heatand mass transfer over an accelerating surface with a heat source in the presence of suction and blowing.

Dandapat and Singh [7] have investigated heat transfer due to a permeable stretching wall in thepresence of transverse magnetic field. Panda et al. [8] have discussed free convection of conductingviscous fluid between two vertical walls filled with a porous material. In another paper, Panda et al.[9] have investigated hydromagnetic unsteady free convective flow and mass transfer of an elastico-viscous fluid past an infinite vertical porous plate in rotating porous medium. Sharma and Yadav [10]have reported unsteady MHD forced and free convection three-dimensional flow heat and mass transferalong an infinite vertical porous plate with periodic suction. Chaudhary and Chand [11] have studiedhydromagnetic flow past a long vertical channel embedded in porous medium with transpiration cooling.Panda and Das [12] have discussed mass transfer effects of unsteady free convective MHD flow of arotating elastico-viscous fluid past an infinite vertical porous plate with constant suction and fluctuatingtemperature. In another paper, Panda and Das [13] have investigated a magnetohydrodynamic steadyfree convective flow and mass transfer in a rotating elastico-viscous fluid past an infinite vertical porousflat plate with constant suction.

*E-mail: [email protected]

119

120 PANDA et al.

In this analysis, heat and mass transfer of electrically conducting fluid through porous media over anaccelerating surface subject to both power law surface temperature and power law heat flux variationswith a temperature-dependent heat source in the presence of a transverse uniform magnetic field isstudied. A series solution to the energy and species concentration equation in terms of Kummer’sfunction [15] is studied. The solutions that govern the flow are obtained analytically for different values ofPrandtl number (especially for air Pr = 0.71 at 20◦C) and Schmidt number Sc (in air due to H2O, NH3

and CO2) [16].

2. MATHEMATICAL FORMULATIONA steady flow of viscous incompressible and electrically conducting fluid through porous media over

an accelerating flat plate in the presence of a temperature-dependent heat source is considered. Thefollowing assumptions are made to formulate the problem:

(i) The fluid properties are assumed to be constant.(ii) No chemical reactions are taking place in the flow.(iii) The concentration of diffusing species is very small in comparison to other chemical species and,

hence, the species thermal diffusion and diffusion thermal energy effects are neglected.(iv) A uniform magnetic field is applied normally to the direction of the flow.(v) The induced magnetic field is neglected in comparison to the applied magnetic field.(vi) The external electric field is zero and due to the polarization of charges, the electric field is

negligible.Under the above assumptions and taking the usual Boussinesq approximation, the governing

equations for the continuity, motion, energy, and species diffusion can be written as:Continuity Equation:

∂u

∂x+

∂v

∂y= 0, (1)

Momentum Equation:

u∂u

∂x+ v

∂u

∂y= ν

∂2u

∂y2− σ.B2

ou

ρ− υu

K∗ , (2)

Energy Equation:

u∂T

∂x+ v

∂T

∂y= α

∂2T

∂y2+

βu

ρcp(T∞ − T ), (3)

Concentration Equation:

u∂C

∂x+ v

∂C

∂y= D

∂2C

∂y2. (4)

Here, the x-axis is on the accelerating surface and the y-axis is perpendicular to it, u and v are thecomponents of the velocity along the directions of the x-axis and the y-axis, respectively, υ is thekinematic coefficient of viscosity, σ is the electrical conductivity of the medium, ρ is the density of thefluid, Bo is the applied uniform magnetic field strength imposed along the y-axis, T is the temperature, αis the thermal diffusivity, β is a constant either positive or negative, cp is the specific heat at constantpressure, T∞ is the free stream temperature, C is the concentration of the fluid, D is the chemicalmolecular diffusivity, and K∗ is the permeability of the medium.

The relevant boundary conditions for Eqs. (1) and (2) are

u = ax, v = vw at y = 0;

u → 0 as y → ∞ . . . , (5)

where a is a constant stretching rate. We have considered two cases, namely, prescribed surfacetemperature (PST) and prescribed wall heat flux (WHF) to study the heat and mass transfer.

JOURNAL OF ENGINEERING THERMOPHYSICS Vol. 21 No. 2 2012

HEAT AND MASS TRANSFER ON MFD FLOW THROUGH POROUS MEDIA 121

3. METHOD OF SOLUTION

Introducing the stream function ψ, the solution of Eq. (1) can be written as:

u =∂ψ

∂y, v = −∂ψ

∂x. (6)

Equations (2) to (4) are decoupled and can be solved consecutively as the fluid is incompressible.Considering f as the dimensionless stream function and η as the similarity variable depending on yonly, we introduce the similarity transformation

ψ = (υa)12 xf(η) and η =

(a

υ

)y

12 . (7)

Substituting Eqs. (6) and (7) into (2), we get

f ′′′ + ff ′′ − f ′2 −(

M2 +1

Kp

)f ′ = 0 ⇒ f ′′′ + ff ′ − f ′2 − Q2f ′ = 0, (8)

where M = B0

(σρa

)(Magnetic parameter), Kp = K∗a

ν (Permeability parameter), and Q2 = M2 +1

Kp.

The corresponding boundary conditions are

f ′(0) = 1, f(0) =−vw

(aν)1/2, f ′(∞) = 0, (9)

where primes denote the differentiation with respect to η.The solution to Eq. (8) is given by:

f =λ2 − Q2 − e−λη

λ, f ′ = e−λη, (10)

where λ (a positive number) is the suction or blowing parameter.

When λ = 1 and M = 0, f = 1 − e−η and f ′ = e−η (the case of crane [5]). (11)

Now,

vw = −(aν)1/2

(λ − Q2 + 1

λ

). (12)

Obviously,

(i) λ >√

Q2 + 1 (suction),

(ii) λ <√

Q2 + 1 (blowing),

(iii) λ =√

Q2 + 1 (impermeable surface).Two heating processes are discussed to solve Eqs. (3) and (4).

Case (A): Prescribed surface temperature (PST).For this the boundary conditions are given by

Tw = T∞ + A0xr, Cw = C∞ + A2x

r at y = 0,

T → T∞ , C → C∞ as y → ∞, (13)

where A1, A2 are proportionality constants and r is the temperature parameter, r = 0 corresponds to theisothermal state. The nondimensional temperature and concentration are defined as:

θ(η) =T − T∞Tw − T∞

, C(η) =C − C∞Cw − C∞

. (14)


122 PANDA et al.

Substituting Eqs. (6), (7), (10), and (14) in (3) and (4), we get

θII + Pr(

λ − Q2

λ− e−λη

λ

)θI − (r + δ) Pr e−ληθ = 0, (15)

C ′′ + Sc(

λ − Q2

λ− e−λη

λ

)C ′ − rSce−ληC = 0, (16)

where δ =βx

ρCpis the nondimensional longitudinal coordinate.

The corresponding boundary conditions are obtained as:

θ(0) = 1, θ(∞) = 0 and C(0) = 1, C(∞) = 0. (17)

Let us introduce some new variables ξ1 and ξ2 such that

ξ1 =−Pr e−λη

λ2, ξ2 =

−Sce−λη

λ2.

Now Eqs. (15) and (16) reduce to

ξ1d2θ

dξ21

+(

1 − Pr +Q2 Pr

λ2− ξ1

)dθ

dξ1+ (r + δ)θ = 0, (18)

ξ2 =d2C

dξ22

+(

1 − Sc +Q2Scλ2

− ξ2

)dC

dξ22

+ rC = 0. (19)

The corresponding boundary conditions are:

θ

(−Pr

λ2

)= 1, θ(0) = 0 and C

(−Scλ2

)= 1, C(0) = 0. (20)

Equations (18) and (19) are similar to Kummer’s equation

zd2w

dz2+ (b∗ − z)

dw

dz− a∗z = 0,

which has a regular singularity at z = 0 and irregular singularity at z = ∞.The solution of the above equation is given by

w(z) = AK(a∗, b∗, z) + Bz1−b∗K(a∗ + 1 − b∗, 2 − b∗, z),

where A, B are arbitrary constants and b∗ �= −n.

K(a∗, b∗, z) = 1 +∞∑

n=1

(a∗)nzn

(b∗)nn!.

Here, (a∗)n = a∗(a∗ + 1)(a∗ + 2). . . (a∗ + n − 1), (b∗)n = b∗(b∗ + 1)(b∗ + 2) . . . (b∗ + n − 1), (a∗)0 =

1, and K1(a∗, b∗, z) =a∗

b∗K(a∗ + 1, b∗ + 1, z), where the prime denotes the differentiation with respect

to z.Hence, the solutions of Eqs. (18) and (19) satisfying the boundary condition (20) in terms of

Kummer’s function [12] are:

θ (ξ1) =(−λ2ξ1

Pr

)Pr−Q2 Pr

λ2 K(Pr−r − δ − Q2 Pr

λ2 , 1 + Pr−Q2 Prλ2 , ξ1

)

K(Pr−r − δ − Q2 Pr

λ2 , 1 + Pr−Q2 Prλ2 , −pr

λ2

) , (21)



C (ξ2) =(−λ2ξ2

Sc

)Sc−Q2sc

λ2 K(Sc − r − Q2Sc

λ2 , 1 + Sc − Q2Scλ2 , ξ2

)

K(Sc−r − Q2Sc

λ2 , 1 + Sc − Q2Scλ2 , −Sc

λ2

) . (22)

In terms of η, Eqs. (21) and (22) can be written as:

θ(η) = e−λ Pr η(1−Q2/λ2)K

(Pr−r − δ − Q2 Pr

λ2 , 1 + Pr−Q2 Prλ2 , −Pr e−λη

λ2

)


λ2 , 1 + Pr−Q2 Prλ2 , −Pr

λ2

) , (23)

C(η) = e−λScη(1−Q2/λ2) −K

(Sc − r − Q2Sc

λ2 , 1 + Sc − Q2Scλ2 , −Sce−λη

λ2

)

K(Sc − r − Q2Sc

λ2 , 1 + Sc − Q2Scλ2 , −Sc

λ2

) . (24)

The dimensionless wall temperature gradient and dimensionless concentration gradient derived fromEqs. (23) and (24):

θ′(0) = −λPr(

1 − Q2

λ2

)+


λ2

1 + Pr−Q2 Prλ2

)(Prλ

)

×K


λ2 + 1, Pr−Q2 Prλ2 + 2, −Pr

λ2

)


λ2 , Pr−Q2 Prλ2 + 1, −Pr

λ2

) , (25)

C ′(0) = −λSc(

1 − Q2

λ2

)+

(Sc − r − Q2Sc

λ2

1 + Sc − Q2Scλ2

)(Scλ

)

×K

(Sc − r − Q2Sc

λ2 + 1, Sc − Q2Scλ2 + 2, −Sc

λ2

)

K(Sc − r − Q2Sc

λ2 , Sc − Q2Scλ2 + 1, −Sc

λ2

) . (26)

Now, the local heat flux qw and the local mass flux mw in terms of dimensionless variables used inEq. (14) are given as:

qw = −k

(∂T

∂y

)

w

= k(a

υ

) 12 (Tw − T∞)[−θ′(0)], (27)

mw = −D

(∂C

∂y

)

w

= D(a

υ

) 12 (Cw − C∞)[−C ′(0)]. (28)

Considering Q∗ = qw/k(a

υ

) 12

and M∗ = mw/D(a

υ

) 12

, we have

Q∗

Tw − T∞= −θ′(0),

M∗

Cw − C∞= −C ′(0),

where Q∗ and M∗ are heat flux and mass flux, respectively.

Case B: Prescribed wall heat flux (WHF). The boundary conditions are:

−k∂T

∂y= qw = E1x

s, −D∂C

∂y= mw = E2x

s for y = 0 and T → T∞, C → C∞ as y → ∞, (29)


124 PANDA et al.

where E1, E2 are proportionality constants and s is the heat flux parameter. For s = 0, the acceleratingsheet is subject to uniform heat flux. In order to obtain the temperature and concentration distributions,we introduce the similarity variables

T − T∞ =E1x

s

k

(υ

a

) 12g(η),

C − C∞ =E2x

s

D

(υ

a

) 12h(η). (30)

Substituting Eqs. (6), (10), and (30) into (3) and (4), we obtain

g′′ + Pr(

λ − Q2

λ− e−λη

λ

)g′ − (s + δ) Pr e−ληg = 0, (31)

h′′ + Sc(

λ − Q2

λ− e−λη

λ

)h′ − sSc.e−ληh = 0 (32)

with boundary conditions

g′(0) = −1, g(∞) → 0, h′(0) = −1, h(∞) → 0. (33)

Using the values of ξ1 and ξ2 in Eqs. (31) and (32), we get

ξ1d2g

dξ21

+(

1 − Pr+Q2 Pr

λ2− ξ1

)dg

dξ1+ (s + δ)g = 0, (34)

ξ2d2h

dξ2+

(1 − Sc +

Q2Scλ2

− ξ2

)dh

dξ2+ sh = 0. (35)

The corresponding boundary conditions are:

d

dξ1g

(−Prλ2

)= − λ

Pr, g(0) = 0,

d

dξ2h

(−Scλ2

)= − λ

Sc, h(0) = 0. (36)

The solutions for temperature and concentration distributions in terms of η are:

g(η) =e−λprη

(1−Q2

λ2

)

λPr(1 − Q2

λ2

)K

(Pr−s − δ − Q2 Pr

λ2 , 1 + Pr−Q2 Prλ2 , −Pr e−λη

λ2

)

K(Pr−s − δ − Q2 Pr

λ2 , Pr−Q2 Prλ2 , −Pr

λ2

) , (37)

h (η) =e−λScη

(1−Q2

λ2

)

λSc(1 − Q2

λ2

)K

(Sc − s − Q2Sc

λ2 , 1 + Sc − Q2Scλ2 , −Sce−λη

λ2

)

K(Sc − s − Q2Sc

λ2 , Sc − Q2Scλ2 , −Sc

λ2

) . (38)

Again, the wall temperature Tw and the wall concentration Cw are

Tw = T∞ +E1x

s

k

(υ

a

) 12g(0),

Cw = C∞ +E2x

s

D

(υ

a

) 12h(0).



4. RESULTS AND DISCUSSION

In this chapter, the heat and mass transfer from an accelerating surface of a conducting viscous fluidthrough a uniform porous medium in the presence of temperature-dependant heat source/sink subjectto suction or blowing, and transverse magnetic field is studied. Two different heating processes areconsidered: (i) an accelerating plate with a prescribed surface temperature (PST) and (ii) wall heat flux(WHF). The resulting temperature and concentration equations have been solved in terms of Kummer’sfunction. The heat and mass transfer phenomena for this problem are characterized by:

(i) Prandtl number, Pr;(ii) Suction or blowing parameters, λ;(iii) The temperature parameter, r (PST) and the heat flux parameter, s (WHF);(iv) The nondimensional longitudinal coordinate, δ;(v) The magnetic parameter, M;(vi) The permeability of the medium, Kp;(vii) The mass transfer parameter, Sc.In order to interpret the result in a meaningful way, we have prescribed the values of the parameters

representing the substances commonly used in the industry and laboratories. The value of Pr is chosento be 0.71, 1.0, and 7.0, which represent air at 20◦C, electrolytic solution, and water, respectively. Thevalues of Schmidt number, Sc, are chosen to represent the diffusing chemical species of most commoninterest in air due to the presence of H2, H2O, NH3, and CO2, the values of Sc are 0.22, 0.60, 0.78, and0.94, respectively. The main objective of the present discussion is to bring out the effect of magnetic fieldand the permeability of the medium. For nonisothermal heat transfer, generation or absorption is mostsuitable. In all heat transfer processes, there is a lot of heat generation, which may be expressed in theform of Eq. (3). The form of heat source βu(T∞ − T ) varying with temperature is chosen such that theambient temperature of the fluid is not constant, but varies along with the plate.

Figure 1 illustrates the effect of Pr in the case of suction and blowing under prescribed surfacetemperature (PST) and prescribed wall heat flux (WHF) on the temperature distribution.

Case of PST: It is observed that for high Prandtl number fluid (Pr = 7.0), abrupt fall of temperatureis marked in the layers close to the surface under suction, but in the case of blowing, the process slowsdown. As the Prandtl number decreases, the temperature increases. The effect is enhanced in the caseof blowing.

Case of WHF: For suction, the effect remains the same as that of PST. But in the case of blowing,the temperature remains positive for high Prandtl number fluid, whereas, for low Pr, (Pr = 0.71 andPr = 1.0), the temperature becomes negative. This indicates that temperature adsorption occurs in thecase of blowing.

Figure 2 shows the effect of the temperature parameter r for PST and the heat flux parameter s forWHF both for suction (λ = 2.0) and blowing (λ = 1.5). The temperature distribution attains unity atthe wall for PST and has different values at the wall for WHF. In the case of PST r > 0, heat flowsfrom the accelerating surface to surroundings. The temperature decreases as the temperature parameterincreases under suction and blowing (PST). In the case of WHF, the reverse effect is observed.

It is interesting to note that under the suction effect, profiles almost overlap, which indicates thatsuction has a thinning effect on the thermal boundary layer (PST). For r < 0 (i.e., r = −2), thewall temperature gradient becomes positive and heat flows into the accelerating surface from thesurroundings. The temperature of the fluid adjacent to the wall may exceed the wall temperature forr = −2 under PST. But in the case of WHF, the reverse effect is observed. For both PST and WHFconditions, suction reduces the thermal boundary layer thickness more, as compared to blowing. It isinteresting to observe that the phenomenon of reverse temperature distribution is observed in the caseof blowing for value of s (i.e., s = −2). However, this effect disappears in the suction condition for WHF.Comparing with Acharya et al. [6], i.e., the case of M = 0 and Kp → ∞, it may be concluded that theresistance due to Lorentz force and permeability reduces the high fluctuation of temperature particularlyin the PST condition.

Figure 3 shows that temperature increases as the magnetic parameter increases in the presence ofsuction and blowing under both PST and WHF. The only exception is the case of suction for M = 1.0,where temperature decreases in comparison with the case of M = 0.0.

Figure 4 shows the effect of porosity parameter on the temperature distribution.


126 PANDA et al.

Fig. 1. Dimensionless temperature profiles for various Pr at r = 1 (PST), s = 1 (WHF), δ = −0.5, M = 1, Kp = 1;— (suction, λ = 2.0) and − · − · − (blowing, λ = 1.5).

Fig. 2. Dimensionless temperature profiles for various r (PST) and various s (WHF) at Pr = 0.71, δ = −0.5, M = 1,Kp = 1; — (suction, λ = 2.0) and − · − · − (blowing, λ = 1.5).



Fig. 3. Dimensionless temperature profiles for various M at r = 1 (PST), s = 1 (WHF), δ = −1.0, Pr = 0.71, Kp = 1;— (suction, λ = 1.5, 2.0, 2.6) and − · − · − (blowing, λ = 1.2, 1.5, 2.3).

Fig. 4. Dimensionless temperature profiles for various Kp at r = 1 (PST), s = 1 (WHF), δ = −1.0, Pr = 0.71, M = 1;— (suction, λ = 1.5, 2.0, 1.7) and − · − · − (blowing, λ = 1.2, 1.5, 1.2).


128 PANDA et al.

Fig. 5. Dimensionless wall temperature gradient for Pr = 0.71, M = 1 at r = 1 (PST) and dimensionless walltemperature for Pr = 0.71, M = 1 at s = 1 (WHF) for various values of Kp; — (suction, λ = 2.0, 1.7, 1.5) and − ·− ·−(blowing, λ = 1.5, 1.2, 1.2).

Fig. 6. Dimensionless concentration profiles for various Sc at r = −1 (PST), s = −1 (WHF), Kp = 1, M = 1; —(suction, λ = 2) and − · − · − (blowing, λ = 1.5).



Fig. 7. Dimensionless concentration profiles for various Kp at r = −1 (PST), s = −1 (WHF), Sc = 0.6, M = 1; —(suction, λ = 2, 1.7, 1.5) and − · − · − (blowing, λ = 1.5, 1.2.)

Case of PST: An increase in the porosity parameter leads to an increase in temperature under suction.The case of without porous medium is represented by the curve (Kp = 100). In this case, the temperatureincreases slightly under suction, whereas in the case of blowing, the temperature falls significantly.

Case of WHF: The same effect as in PST is observed under WHF.Thus, it may be concluded that the heat adsorption takes place under blowing in the absence of porous

medium, whereas the opposite effect is observed in the case of suction.Figure 5 shows the dimensionless wall temperature gradient θ′(0), for air Pr = 0.71, for constant

temperature parameter r = 1 and wall temperature g(0), for air with constant heat flux parameter ats = 1.0, for various values of porosity parameter under suction and blowing. High fluctuation rangingfrom negative values to positive values is marked in the case of wall temperature gradient under suctionas well as blowing with or without porous medium. Moreover, wall temperature g(0) under WHF exhibitshigh fluctuation ranging from negative to positive values at the wall and then maintains positive valuesafterwards. Thus, it may be concluded that temperature adsorption with high fluctuation occurs in afew layers close to the wall due to suction and blowing at the wall, then the thermoequilibrium sets in,whether the flow takes place in the presence of porous medium or not.

Concentration profiles are shown in Fig. 6 for various values of Schimdt number, Sc. It is seen that forheavier species, i.e., higher value of Sc, concentration decreases under the effect of suction and blowingin both cases, i.e., PST and WHF. Further, it is observed that concentration is always higher in the caseof blowing than in the case of suction. Now, we are to study the concentration profiles for PST and WHFprocess in the presence of foreign species such as H2, H2O, NH3, and CO2 with Sc = 0.23, 0.60, 0.78,and 0.94, respectively for r = −1 and S = −1 both for suction and blowing.

A comparative study of the curves reveals that the species diffusion layer thickness is thinner forheavier particles (high Sc) than for lighter species. Further, for a given value of Sc, the concentration


130 PANDA et al.

is less for suction than for blowing; physically this implies that even in the presence of foreign speciessuction reduces the species diffusion layer thickness, whereas blowing increases both for PST and WHCconditions.

Figure 7 shows the effect of porosity parameter on the concentration profile. It is interesting to recordthat the concentration variation is almost the same as the temperature variation shown in Fig. 4.

5. CONCLUSIONS

1. With the present choice of heat source βu(T∞ − T ) varying with temperature attributes abrupt fallof temperature near the surface to high Prandtl number fluid.

2. The effects of suction under PST and WHF are the same. But under blowing, low Prandtl numberfluid suffers from temperature adsorption.

3. The Lorentz force enhances the temperature in the presence of suction and blowing under bothPST and WHF.

4. In the presence of porous medium, adsorption takes place under suction, whereas under blowing itis duly compensated.

5. High fluctuation in temperature gradient with positive and negative values accounts for the riseand fall of temperature on the plate.

6. Under the influence of mass diffusion for heavier species, fall of concentration takes place undersuction, blowing, PST, and WHF leading to thinning of species diffusion layer.

REFERENCES1. Crane, L.E., Flow past a Stretching Plane, ZAMP, 1970, vol. 21, pp. 645–647.2. Chakrabarti, A. and Gupta, A.S., Hydromagnetic Flow and Heat Transfer over a Stretched Sheet, Quart.

Appl. Math., 1979, vol. 37, pp. 73–78.3. Soundalgekar, V.M. and Ramanmurty, T.V., Heat Transfer past a Continuous Moving Plate with Variable

Temperature, Warmeundstol–Fubertragung, 1980, pp. 14–19.4. Gruba, L.J. and Bobba, K.M., Heat Transfer Characteristics of a Continuous Stretching Surface with

Variable Temperature, J. Heat Transfer, 1985, pp. 248–250.5. Anderson, H.I., An Exact Solution of the Navier–Stokes Equation for Magnetohydrodynamic Flow, Acta

Mech., 1995, vol. 113, pp. 241–244.6. Acharya, M., Singh, L.P., and Dash, G.C., Heat and Mass Transfer over an Accelerating Surface with Heat

Source in Presence of Suction and Blowing, Int. J. Eng. Sci., 1999, vol. 37, pp. 189–211.7. Dandapat, S., Singh, S.N., and Singh, R.P., Heat Transfer due to Permeable Stretching Wall in Presence of

Transverse Magnetic Field, Arch. Mech., 2004, vol. 56, no. 2, pp. 87–101.8. Panda, J.P., Patnaik, A.B., and Acharya, A., Free Convection of Conducting Viscous Fluid between Two

Vertical Walls Filled with a Porous Material, AMSE, Mod., Meas. Cont., B, 2006, vol. 75, no. 3, pp. 31–44.9. Panda, J.P., Das, S.S., Pattnaik, A.B., and Satpathy, P.R., Hydromagnetic Unsteady Free Convective Flow

and Mass Transfer of an Elastico-Viscous Fluid past an Infinite Vertical Porous Plate in Rotating PorousMedium, AMSE, Mod., Meas. Cont., B, 2006, vol. 75, no. 4, pp. 57–74.

10. Sharma, P.R. and Yadav, G.R., Unsteady MHD Forced and Free Convection Three-Dimensional Flow Heatand Mass Transfer along an Infinite Vertical Porous Plate with Periodic Suction, AMSE, Mod., Meas. Cont.,B, 2007, vol. 76, no. 2, pp. 11–33.

11. Chaudhary, R.C. and Chand, T., Hydromagnetic Flow past a Long Vertical Channel Embedded in PorousMedium with Transpiration Cooling, AMSE, Mod., Meas. Cont., B, 2005, vol. 74, no. 3, pp. 43–54.

12. Panda, J.P. and Das, S.S., Mass Transfer Effects of Unsteady Free Convective MHD Flow of a RotatingElastico-Viscous Fluid past an Infinite Vertical Porous Plate with Constant Suction and FluctuatingTemperature, AMSE. Mod., Meas. Cont., B, 2008, vol. 77, no. 3, pp. 1–18.

13. Panda, J.P. and Das, S.S., Magnetohydrodynamic Steady Free Convective Flow and Mass Transfer in aRotating Elastico-Viscous Fluid past an Infinite Vertical Porous Flat Plate with Constant Suction, AMSE.Mod., Meas. Cont., B, 2009, vol. 78, no. 2, pp. 1–19.


Heat and Mass Transfer on MFD Flow through Porous … · · 2017-08-25Heat and Mass Transfer on...

Documents

Transcript of Heat and Mass Transfer on MFD Flow through Porous … · · 2017-08-25Heat and Mass Transfer on...