Radiation and mass transfer eﬁects on an unsteady MHD · PDF fileRadiation and mass transfer...

Radiation and mass transfer effectson an unsteady MHD free

convection flow past a heatedvertical plate in a porous medium

with viscous dissipation

V. Ramachandra Prasad ∗ N. Bhaskar Reddy †

Theoret. Appl. Mech., Vol.34, No.2, pp. 135–160, Belgrade 2007

Abstract

An unsteady, two-dimensional, hydromagnetic, laminar free con-vective boundary-layer flow of an incompressible, Newtonian, electrically-conducting and radiating fluid past an infinite heated verticalporous plate with heat and mass transfer is analyzed, by takinginto account the effect of viscous dissipation. The dimensionlessgoverning equations for this investigation are solved analyticallyusing two-term harmonic and non-harmonic functions. Numeri-cal evaluation of the analytical results is performed and graphicalresults for velocity, temperature and concentration profiles withinthe boundary layer and tabulated results for the skin-friction coef-ficient, Nusselt number and Sherwood number are presented anddiscussed. It is observed that, when the radiation parameter in-creases, the velocity and temperature decrease in the boundarylayer, whereas when thermal and solutal Grashof increases the ve-locity increases.Keywords: Radiation, viscous dissipation, heat and mass trans-fer

∗Department of Mathematics, ICFAI Institute of Science and Technology, 4- 102,Jeedimetla, Hyderabad-500055, India, e-mail: [email protected]

†Department of Mathematics, Sri Venkateswara University, Tirupathi-517502, In-dia

135

136 V. Ramachandra Prasad, N. Bhaskar Reddy

1 Introduction

Coupled heat and mass transfer (or double-diffusion) driven by buoy-ancy, due to temperature and concentration variations in a saturatedporous medium, has several important applications in geothermal andgeophysical engineering such as the migration of moisture through theair contained in fibrous insulation, the extraction of geothermal energy,underground disposal of nuclear wastes, and the spreading of chemicalcontaminants through water-saturated soil. Bejan and Khair [1] investi-gated the vertical free convection boundary layer flow in porous mediaowing to combined heat and mass transfer. The suction and blowing ef-fects on free convection coupled heat and mass transfer over a verticalplate in a saturated porous medium was studied by Raptis et al. [2] andLai and Kulacki [3] respectively.

Hydromagnetic flows and heat transfer have become more importantin recent years because of its varied applications in agriculture, engineer-ing and petroleum industries. Raptis [4] studied mathematically the caseof time varying two-dimensional natural convective flow of an incompress-ible, electrically conducting fluid along an infinite vertical porous plateembedded in a porous medium. Soundalgekar [5] obtained approximatesolutions for two-dimensional flow of an incompressible viscous flow pastan infinite porous plate with constant suction velocity, the difference be-tween the temperature of the plate and the free stream is moderately largecausing free convection currents. Takhar and Ram [6] studied the MHDfree convection heat transfer of water at 4˚C through a porous medium.Soundalgekar et al. [7] analyzed the problem of free convection effectson Stokes problem for a vertical plate under the action of transverselyapplied magnetic field with mass transfer. Elbashbeshy [8] studied heatand mass transfer along a vertical plate under the combined buoyancyeffects of thermal and species diffusion, in the presence of magnetic filed.

In all these investigations, the radiation effects are neglected. Forsome industrial applications such as glass production and furnace designand in space technology applications, such as cosmical flight aerodynam-ics rocket, propulsion systems, plasma physics and spacecraft re-entryaerothermodynamics which operate at higher temperatures, radiation ef-fects can be significant. Alagoa et al. [9] studied radiative and freeconvection effects on MHD flow through porous medium between infinite

Radiation and mass transfer... 137

parallel plates with time-dependent suction. Bestman and Adjepong [10]analyzed unsteady hydromagnetic free convection flow with radiative heattransfer in a rotating fluid.

In all these investigations, the viscous dissipation is neglected. Theviscous dissipation heat in the natural convective flow is important, whenthe flow field is of extreme size or at low temperature or in high gravita-tional field. Gebhart [11] shown the importance of viscous dissipative heatin free convection flow in the case of isothermal and constant heat fluxat the plate. Soundalgekar [12] analyzed the effect of viscous dissipativeheat on the two-dimensional unsteady, free convective flow past an infinitevertical porous plate when the temperature oscillates in time and thereis constant suction at the plate. Israel Cookey et al. [13] investigatedthe influence of viscous dissipation and radiation on unsteady MHD freeconvection flow past an infinite heated vertical plate in a porous mediumwith time dependent suction.

The objective of the present chapter is to study the radiation and masstransfer effects on an unsteady two-dimensional laminar free convectiveflow of a viscous, incompressible, electrically conducting fluid past verticala heated porous plate with suction, embedded in a porous medium, underthe influence of a uniform transverse magnetic field, by taking into ac-count the effect of viscous dissipation. The equations of continuity, linearmomentum, energy and diffusion, which govern the flow field are solvedby using a regular perturbation method. The behaviour of the velocity,temperature, concentration, skin-friction, Nusselt number and Sherwoodnumber has been discussed for variations in the physical parameters.

2 Mathematical analysis

An unsteady two-dimensional hydromagnetic laminar free convection bound-ary layer flow of a viscous, incompressible, electrically conducting andradiating fluid in an optically thin environment past an infinite heatedvertical porous plate, embedded in a uniform porous medium, in the pres-ence of thermal and concentration buoyancy effects, is considered. Thex′- axis is taken along the vertical plate and the y′- axis normal to theplate. A uniform magnetic field is applied in the direction perpendicularto the plate. The fluid is assumed to be slightly conducting, and hence


the magnetic Reynolds number is much less than unity and the inducedmagnetic field is negligible in comparison with the transverse magneticfield. It is further assumed that there is no applied voltage, so that elec-tric field is absent. The level of concentration of foreign mass is assumedto be low, so that the Soret and Dufour effects are negligible. Since theplate is of infinite length, all the physical variables are functions of y′ andt′ only. Now, under the usual Boussinesq’s approximation, the flow fieldis governed by the following equations.

∂ v′

∂ y′= 0 (2.1)

∂ u′

∂ t′+ v′

∂ u′

∂ y′=

∂U ′

∂ t′+ ν

∂2 u′

∂ y′2

+ g β(T ′ − T ′∞) + gβ∗(C ′ − C ′

∞) −(

σµ2eH

20

ρ+

ν

K ′

)(u′ − U ′) (2.2)

∂ T ′

∂ t′+ v′

∂ T ′

∂ y′=

k

ρ cp

[∂2T ′

∂ y′2− 1

k

∂ q′

∂ y′

]+

ν

cp

(∂ u′

∂ y′

)2

(2.3)

∂2q′

∂ y′2− 3α2q′ − 16σ∗αT ′3

∞∂ T ′

∂ y′= 0 (2.4)

∂ C ′

∂ t′+ v′

∂ C ′

∂ y′= D

∂2 C ′

∂ y′2(2.5)

where u′, v′ are the velocity components in x′,y′ directions respectively, t′-the time, p′- the pressure, ρ - the fluid density, σ - the electrical conductiv-ity of the fluid, ν - the kinematic viscosity, µe- the magnetic permeability,K ′ - the permeability of the porous medium, g - the acceleration due togravity, β and β∗ - the thermal and concentration expansion coefficients,T ′ - the temperature of the fluid in the boundary layer, T ′

∞ - the temper-ature of the fluid far away from the plate,C ′ - the species concentration inthe boundary layer, C ′

∞ - the species concentration in the fluid far awayfrom the plate, H2

0 - the constant transverse magnetic field, k - the ther-mal conductivity, q′- the radiative heat flux, σ∗- the Stefan- Boltzmannconstant and D - the mass diffusivity. The second and third terms on theright hand side of the momentum equation (2) denote the thermal and


concentration buoyancy effects respectively. Also, the second term onright hand side of the energy equation (2.3) represents the radiative heatflux and the third term represents viscous dissipation. Equation (2.4) isthe differential approximation for radiation under fairly broad realisticassumptions. In one space coordinate y′, the radiative flux q′ satisfiesthis nonlinear differential Equation [10].

The boundary conditions for the velocity, temperature, and concen-tration fields are

u′ = 0, T ′ = Tw C ′ = Cw at y′ = 0

u′ = U ′(t′) = V0(1 + ε en′t′) , T ′ → T ′∞, C ′ → C ′

∞ as y′ →∞ (2.6)

where T ′w and C ′

w are the temperature and concentration of the platerespectively. Equation (2.1) asserts that, the suction velocity at the plateis either a constant or a function of time. Hence the suction velocitynormal to the plate is assumed to be of the form

v′ = −V0(1 + εA en′t′) (2.7)

where A is a real positive constant, ε is small such that ε << 1, andεA << 1 and V0 is scale of suction velocity which is non-zero positiveconstant. The negative sign indicates that the suction is towards theplate.

Since the medium is optically thin with relatively low density and α(absorption coefficient) << 1, the radiative heat flux given by Equation(2.4), in the spirit of Cogley et al. [14] becomes

∂ q′

∂ y′= 4α2(T ′ − T ′

∞) (2.8)

where

α2 =

∞∫

0

δλ∂ B

∂ T ′ (2.9)

where B is Planck’s function.In order to write the governing equations and the boundary condi-

tions in dimensionless form, the following non-dimensional quantities areintroduced.

y =V0

νy′ , n =

4ν

V 20

n′, u =u′

U0

, t =V 2

0

4νt′ , v =

4ν

V 20

V ′, θ =T ′ − T ′

∞T ′

w − T ′∞,


C =C ′ − C ′

∞C ′

w − C ′∞, χ2 =

v2

K ′V 20

, Pr =µ cp

k=

ν

α, Sc =

ν

D,

M2 =µ2

eσH ′20

ρV 20

, Gr =ν β g(T ′

w − T ′∞)

U0 V 20

, Gm =νβ∗g(C ′

w − C ′∞)

U0V 20

,

Ec =U2

0

cp(T ′w − T ′∞)

, R2 =4α2

ρ kcpV 20

(T ′w − T ′∞) (2.10)

In view of Equations (2.4), (2.7), (2.8), (2.9) and (10), Equations (2),(2.3) and (2.5) reduce to the following dimensionless form.

1

4

∂ u

∂ t−(1+εA ent)

∂ u

∂ y=

1

4

∂ U

∂ t+

∂2u

∂ y2+ Grθ + Gm C+(M2+χ2)(u−U)

(2.11)

1

4

∂ θ

∂ t− (1 + εA ent)

∂ θ

∂ y=

1

Pr

[∂2θ

∂ y2−R2

]+ Ec

(∂ u

∂ y

)2

(2.12)

1

4

∂ C

∂ t− (1 + ε A ent)

∂ C

∂y=

1

Sc

∂2C

∂ y2(2.13)

where Gr,Gm, Pr,R,Ec and Sc are the thermal Grashof number, solutalGrashof Number, Prandtl Number, radiation parameter, Eckert numberand the Schmidt number respectively.

The corresponding boundary conditions are

u = 0, θ = 1, C = 1 at y = 0

u → 1 + εent , θ → 0 , C → 0 as y →∞ (2.14)

3 Solution of the problem

Equations (2.11) - (2.13) represent coupled, non-linear partial differen-tial equations, which cannot be solved in closed-form. So, in order tosolve these equations, we may represent the velocity, temperature andconcentration of the fluid in the neighbourhood of the plate as

u(y, t) = u0(y) + ε ent u1(y) + o(ε2) + ...


θ(y, t) = θ0(y) + ε ent θ1(y) + o(ε2) + ... (3.1)

C(y, t) = C0(y) + ε ent C1(y) + o(ε2) + ...

Substituting (3.1) in Equations (2.11) - (2.13), and equating the har-monic and non - harmonic terms, and neglecting the higher-order termsof O (ε2), we obtain

u′′0 + u′0 − (χ2 + M2) u0 = −N − Gr θ0 − GmC0 (3.2)

u′′1 +u′1 − (χ2 +M2 +n

4) u1 = −(N +n) −Au′0 − Gr θ1− GmC1 (3.3)

θ′′0 + Pr θ′0 −R2θ0 = −Pr Ec(u′0)2 (3.4)

θ′′1 + Pr θ′1 − (R +n Pr

4)θ1 = −Pr Aθ′0 − 2 Pr Ecu′0u

′1 (3.5)

C ′′0 + Sc C ′

0 = 0 (3.6)

C ′′1 + Sc C ′

1 − nScC ′1 = −ASc C ′

0 (3.7)

where prime denotes ordinary differentiation with respect to y. The cor-responding boundary conditions can be written as

u0 = 0, u1 = 0 , θ0 = 1 , θ1 = 0 , C0 = 1 , C1 = 0 at y = 0

u0 = 1 , u1 = 1 , θ0 → 0 , θ1 → 0 , C0 → 0 , C1 → 0 as y →∞ (3.8)

The Equations (3.2) - (3.7) are still coupled non-linear equations,whose exact solutions are not possible. So we expand u0, u1, θ0, θ1, C0, C1

in terms of Ec in the following way, as the Eckert number for incompress-ible fluid is always very small.

u0(y) = u01(y) + Ec u02(y)

u1(y) = u11(y) + Ec u12(y)

θ0(y) = θ01(y) + Ec θ02(y)

θ1(y) = θ11(y) + Ec θ12(y)

C0(y) = C01(y) + Ec C02(y)


C1(y) = C11(y) + Ec C12(y) (3.9)

Substituting (3.9) in Equations (3.2) - (3.7), equating the coefficientsof Ec to zero and neglecting the terms in Ec2 and higher order, we getthe following equations.

The zeroth order equations are

u′′01 + u′01 − (χ2 + M2) u01 = −(χ2 + M2) − Gr θ01 − Gm C01 (3.10)

u′′02 + u′02 − (χ2 + M2) u02 = −Gr θ02 − Gm C02 (3.11)

θ′′01 + Pr θ′01 −R2θ01 = 0 (3.12)

θ′′02 + Pr θ′02 −R2θ02 = −Pr u′201 (3.13)

C ′′01 + Sc C ′

01 = 0 (3.14)

C ′′02 + Sc C ′

02 = 0 (3.15)

and the respective boundary conditions are

u01 = 0, u02 = 0, θ01 = 1, θ02 = 0, C01 = 1, C02 = 0 at y = 0 (3.16)

u01 → 1, u02 → 0, θ01 → 0, θ02 → 0, C01 → 0, C02 → 0 as y →∞

The first order equations are

u′′11 + u′11 − N1 u11 = −N1 − Gr θ11 − GmC11 − Au′01 (3.17)

u′′12 + u′12 −N1 u12 = −Gr θ12 −GmC12 − Au′02 (3.18)

θ′′11 + Pr θ′11 −N2 θ11 = −A Pr θ′01 (3.19)

θ′′12 + Pr θ′12 −N2θ12 = −Pr Aθ′02 − 2 Pr u′01u′11 (3.20)

C ′′11 + ScC ′

11 − nSc C11 = −AScC ′01 (3.21)

C ′′12 + Sc C ′

12 − nScC12 = −AScC ′02 (3.22)

and the respective boundary conditions are

u11 = 0, u12 = 0, θ11 = 0, θ12 = 0, C11 = 0, C12 = 0 at y = 0


u11 → 1, u12 → 0, θ11 → 0, θ12 → 0, C11 → 0, C12 → 0 as y →∞(3.23)

where N1 = χ2 + M2 + n4, N2 = R2 + n Pr

4

Solving Equations (3.10) - (3.15) under the boundary conditions (3.16),and Equations (3.17) - (3.22) under the boundary conditions (3.23), andusing Equations (3.9) and (3.1), we obtain the velocity, temperature andconcentration distributions in the boundary layer as

u(y, t) = P3e−(m3y) + P1e

−m2y + P2e−Scy + 1 +

Ec{J10e−(m3y) + J1e

−m2y + J2e−2m3y + J3e

−(2m2)y

+J4e−2Scy + J5e

−(m2+m3)y + J6e−(m2+Sc)y+

J7e−(m3+Sc)y}+ εent[{D10e

−(m5y) + D6e−m3y

+D7e−m2y + D8e

−Scy + D9e−m4y + D5e

−m1y + 1}+Ec{Z20e

−m5y + Z1e−m4y + Z2e

−m2y

+Z3e−2m3y + Z4e

−m3y + Z5e−2m2y+

Z6e−2Scy + Z7e

−(m2+m3)y + Z8e−(m2+Sc)y

+Z9e−(m3+Sc)y + Z10e

−(m5+m3)y + Z11e−(m4+m3)y

+Z12e−(m2+m5)y + Z13e

−(m2+m4)y

+Z14e−(m3+m4)y + Z15e

−(m5+Sc)y + Z16e−(m1+m3)y+

Z17e−(m1+m2)y + Z18e

−(m1+Sc)y}]θ(y, t) = e−m2y + Ec{S7e

−m2y + S1e−2m3y + S2e

−2m2y

+S3e−2Scy + S4e

−(m3+m2)y + S5e−(m2+Sc)y

+S6e−(m3+Sc)y}+ εent[{D1 e−m2y −D2e

−m4y}+Ec{G1e

−m2y + G2e−2m3y

+G3e−2m2y + G4e

−2Scy + G5e−(m3+m2)y

+G6e−(m2+Sc)y + G7e

−(m3+Sc)y+

+G8e−(m3+m5)y + G9e

−(m3+m4)y + G10e−(m5+m2)y

+G11e−(m4+m2)y + G12e

−(m4+Sc)y

+G13e−(m5+Sc)y + G14e

−(m1+m3)y + G15e−(m2+m1)y


+G16e−(m1+Sc)y + G20e

−m4y}]

C(y, t) = e−Scy + εent

{4ASc

n(e−Scy − e−m1y)

}

where the expressions for the constants are given in Appendix.The skin-friction, the Nusselt number and the Sherwood number are

important physical parameters for this type of boundary layer flow.Knowing the velocity field, the skin-friction at the plate can be ob-

tained, which in non-dimensional form is given by

Cf =

(µ

∂ u′

∂ y′

)

y′=0

=

(∂ u

∂ y

)

y=0

=

(∂ u0

∂ y+ ε ent ∂ u1

∂ y

)

y=0

= −P3m3−P1m2−P2Sc+Ec {−J10m3−J1m2−2J2m3−2J3m2−2J4Sc

−J5(m2+m3)−J6(m2+Sc)−J7(m3+Sc)+ε ent[{−D10m5−D6m3−D7m2

−D8Sc−D9m4−D5m1}+Ec{−Z20m5−Z1m4−Z2m2−2Z3m3−Z4m3−2Z5m2

−2Z6Sc− Z7(m2 + m3)− Z8(m2 + Sc)− Z9(m3 + Sc)− Z10(m5 + m3)

−Z11(m4 + m3)− Z12(m2 + m5)− Z13(m2 + m4)− Z14(m3 + m4)

−Z15(m5 + Sc)− Z16(m1 + m3)− Z17(m1 + m2)− Z18(m1 + Sc)}]Knowing the temperature field, the rate of heat transfer coefficient can

be obtained, which in the non-dimensional form, in terms of the Nusseltnumber, is given by

Nu =q′ν

kV0(T ′w − T ′∞)

, where q′ = −k

(∂ T ′

∂ y′

)

y′=0

= −(

∂ θ

∂ y

)

y=0

= −(

∂θ0

∂ y+ ε ent ∂θ1

∂ y

)

y=0

= − (−m2+Ec{−S7m2−2S1m3−2S2m2−2S3Sc−S4(m3+m2)−S5(m2+Sc)

+ε ent[{−D1m2 + D1m4}+ Ec{−G1m2 − 2G2m3 − 2G3m2 − 2G4Sc

−G5(m2 + m3)−G6(m2 + Sc)−G7(m3 + Sc)−G8(m3 + m5)

−G9(m3 + m4)−G10(m2 + m5)−G11(m2 + m4)−G12(m4 + Sc)


−G13(m5 +Sc)−G14(m1 +m3)−G15(m1 +m2)−G16(m1 +Sc)−G20Sc]

Knowing the concentration field, the rate of mass transfer coefficientcan be obtained, which in the non-dimensional form, in terms of theSherwood number, is given by

Sh =C∗ν

DV0(T ′w − T ′∞)

, where C∗ = −D

(∂ C ′

∂ y′

)

y′=0

= −(

dC

d y

)

y=0

= −(

∂ C0

∂ y+ ε ent ∂ C1

∂ y

)

y=0

= −[−Sc + εent

{4ASc

n(−Sc + m1)

}]

4 Results and discussion

In the preceding sections, the problem of MHD unsteady free convec-tive flow of a viscous, incompressible, radiating and dissipating fluid pastan infinite porous plate embedded in a porous medium was formulatedand solved by means of a perturbation method, by applying Cogley etal. [14] approximation for the radiative heat flux. The expressions forthe velocity, temperature and concentration were obtained. To illustratethe behaviour of these physical quantities, numerical values were com-puted with respect to the variations in the governing parameters viz., thethermal Grashof number Gr, the solutal Grashof number Gm, Prandtlnumber Pr, Schmidt number Sc, the radiation parameter R and the Eck-ert number Ec.

Fig.1 presents the typical velocity profiles in the boundary layer forvarious values of the thermal Grashof number. It is observed that an in-crease in Gr, leads to a rise in the values of velocity due to enhancementin buoyancy force. Here the positive values of Gr correspond to coolingof the surface. In addition, the curve show that the peak value of thevelocity increases rapidly near the wall of the porous plate as Grashofnumber increases and then decays the free stream velocity. For the caseof different values of the solutal Grashof number, the velocity profiles inthe boundary layer are shown in Fig.2.The velocity distribution attains


a distinctive maximum value in the vicinity of the plate and then de-creases properly to approach a free stream value. As expected, the fluidvelocity increases and the peak value becomes more distinctive due toincrease in the buoyancy force represented by Gm. Figs.3(a) and 3(b)display the effects of Schmidt number on the velocity and concentrationprofiles respectively. As the Schmidt number increases, the concentrationdecreases. This causes the concentration buoyancy effects to decreaseyielding a reduction in the fluid velocity. Reductions in the velocity andconcentration profiles are accompanied by simultaneous reductions in thevelocity and concentration boundary layers. These behaviors are evidentfrom Figs.3 (a) and 3(b).

For different values of the radiation parameter R, the velocity andtemperature profiles are plotted in Figs.4(a) and 4(b). It is obvious thatan increase in the radiation parameter results in decreasing velocity andtemperature within the boundary layer, as well as decrease the thicknessof the velocity and temperature boundary layers. For various values of themagnetic parameter, the velocity profiles are plotted in Fig.5. It is obviousthat existence of the magnetic field decreases the velocity. Fig.6 showsthe velocity profiles for different values of the permeability parameter χ.Clearly as χ increases, the velocity tends to decrease.

Figs.7(a) and 7(b) illustrate the velocity and temperature profiles fordifferent values of Prandtl number. The numerical results show thatthe effect of increasing values of Prandtl number results in a decreasingvelocity. From Fig.7(b), the numerical results show that an increase inthe Prandtl number results a decrease of the thermal boundary layer andin general lower average temperature with in the boundary layer. Thereason is that smaller values of Pr are equivalent to increase in the thermalconductivity of the fluid and therefore heat is able to diffuse away fromthe heated surface more rapidly for higher values of Pr. Hence in the caseof smaller Prandtl numbers the thermal boundary layer is thicker and therate of heat transfer is reduced.

The effects of the viscous dissipation parameter i.e., the Eckert numberon the velocity and temperature are shown in Figs.8(a) and 8(b). Greaterviscous dissipative heat causes a rise in the temperature as well as thevelocity profiles.

Tables 1-5 show the effects of the thermal Grashof Number, solutalGrashof number, radiation parameter, the Schmidt number and Eckert


number on the skin-friction Cf , the Nusselt number Nu, and the Sher-wood numberSh. From Tables 1 and 2, it is observed that as Gr orGm increases, the skin-friction coefficient increases. However, from Ta-ble 3, it can be easily seen that as radiation parameter increases, theskin-friction decreases and the Nusselt number increases. From Table 4,it is noticed that an increase in the Schmidt number reduces the skin-friction and increases the Sherwood number. Finally, Table 5 shows thatas Eckert number increases, the skin-friction increases, the Nusselt num-ber decreases.

0 1 2 3 4 50

0.25

0.50

0.75

1.00

1.25

1.50

y

u

Gr=0,1.0,2.0,3.0

Gm=1.0 Pr=0.71 Sc=0.6M=1.0 R=0.5 t=1.0A=0.01 Ec=0.001 n=0.1

ε=0.001 χ=0.1

Figure 1: Effect of Gr on velocity.

References

[1] Bejan. A and Khair. K.R.(1985), Heat and mass transfer by naturalconvection in a porous medium, Int. J. Heat Mass Transfer, Vol.28,pp.909-918.


0 1 2 3 4 5 6 70

0.25

0.50

0.75

1.00

1.25

1.50

1.75

y

u

Gm=0.0,1.0,2.0,3.0

Gr=2.0 Pr=0.71 Sc=0.6M=1.0 R=0.5 t=1.0A=0.01 Ec=0.001 n=0.1

ε=0.001 χ=0.1

Figure 2: Effect of Gm on velocity

Gr Cf

0 2.52781 3.01972 3.47753 3.8868

Table 1: Effects of Gr on skin-friction for reference values in Fig.1

Gm Cf

0 2.80121 3.47752 4.14843 4.8138

Table 2: Effects of Gm on skin-friction for reference values in Fig.2.


0 1 2 3 4 5 6 70

0.25

0.50

0.75

1.00

1.25

1.50

y

u

Gr=2.0 Gm=1.0 Pr=0.71 n=0.1 t=1.0 R=0.5M=1.0 A=0.01 Ec=0.001

ε=0.001 χ=0.5

Sc=0.22,0.6,0.78,0.94

(a) Effect of Sc on velocity

0 1 2 3 4 5 6 7 80

0.25

0.50

0.75

1.00

y

u

Gr=2.0 Gm=1.0 Pr=0.71 t=1.0 R=0.5M=1.0 A=0.01 Ec=0.001

ε=0.001 χ=0.5

Sc=0.22

Sc=0.6,0.78,0.94

n=0.1

(b) Effect of Sc on concentration

Figure 3:


0 1 2 3 4 5 60

0.25

0.50

0.75

1.00

1.25

1.50

y

u

Gr=2.0 Gm=1.0 Pr=0.71 Sc=0.6 n=0.1 t=1.0M=1.0 A=0.01 Ec=0.001

ε=0.001 χ=0.5

R=0.0,0.5,1.0,1.5

(a) Effect of radiation on velocity profiles

0 1 2 3 4 5 6 7 80

0.25

0.50

0.75

1.00

y

Gr=2.0 Gm=1.0 Pr=0.71 Sc=0.6 n =0.1 t=1.0M=1.0 A=0.01 Ec=0.001

ε=0.001 χ=0.5

R=0.0,0.5,1.0,1.5 θ

(b) Effect of radiation on temperature

Figure 4:


0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

y

u

M=0.0,1.0,2.0,3.0,4.0

Gm=1.0 Pr=0.71 Gr=2.0 Sc=0.6 n=0.1 t=1.0A=0.01 R=0.5 Ec=0.001

ε=0.001 χ=0.5

Figure 5: Effect of M on velocity

R Cf Nu0.0 3.7670 0.67210.5 3.4775 1.10641.0 3.3291 1.35991.5 3.0956 1.4836

Table 3: Effects of radiation on skin-friction and Nusselt number forreference values in Fig.4(a) and 4(b).

Sc Cf Sh0.22 3.7316 0.22010.60 3.4775 0.60180.78 3.3980 0.78040.94 3.3400 0.9403

Table 4: Effects of Sc on skin-friction and Sherwood number for referencevalues in Fig.3 (a) and 3(b).


0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1.0

1.2

1.4

y

u

χ=0.1,0.5,1.0,2.0

Gr=2.0 Gm=1.0 Pr=0.71Sc=0.6 n=0.1 t=1.0R=0.5 M=1.0 A=0.01Ec=0.001

ε=0.001

Figure 6: Effect of χ on velocity.

Ec Cf Nu0 1.6302 3.33960.001 1.4836 3.09560.01 0.1644 0.8996

Table 5: Effects of Ec on skin-friction and Nusselt number for referencevalues in Fig.8(a) and 8(b).


0 1 2 3 4 50

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Gr=2.0 Gm=1.0 Sc=0.6 n=0.1 t=1.0 M=1.0R=0.5 A=0.01 Ec=0.001

ε=0.001 χ=0.5

y

u

Pr=0.71,1.0,2.0,3.0

(a) Effect of Pr on velocity

0 1 2 3 4 50

0.25

0.50

0.75

1.00

y

θ

Pr=0.71,1.0,2.0,3.0

Gr=2.0 Gm=1.0 Sc=0.6 n=0.1 t=1.0 M=1.0R=0.5 A=0.01 Ec=0.001

ε=0.001 χ=0.5

(b) Effect of Pr on temperture

Figure 7:


0 1 2 3 4 50

0.25

0.50

0.75

1.00

1.25

1.50

y

u

Ec=0.0,0.001,0.01

Gr=2.0 Gm=1.0 Pr=0.71 Sc=0.6 n=0.1 t=1.0A=0.01 R=0.5 M=1.0

ε=0.001 χ=0.5

(a) Effect of Ec on vleocity

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Ec=0.0,0.001,0.01

Gr=2.0 Gm=1.0 Pr=0.71 Sc=0.6 n=0.1 t=1.0A=0.01 R=0.5 M=1.0

ε=0.001 χ=0.5

y

θ

(b) Effect of Ec on temperature

Figure 8:


[2] Raptis A., Tzivanidis. G.and Kafousias N.(1981), Free convectionand mass transfer flow through a porous medium bounded by aninfinite vertical limiting surface with constant suction, Lett. HeatMass Transfer, Vol.8, pp.417-424.

[3] Lai F.C. and Kulacki F.A.(1991), Coupled heat and mass transfer bynatural convection from vertical surfaces in a porous medium, Int.J. Heat Mass Transfer,Vol.34, pp.1189-1194.

[4] Raptis A.(1986), Flow through a porous medium in the presence ofmagnetic field, Int. J. Energy Res., Vol.10, pp.97-101.

[5] Soundalgekar V.M., Free-convection effects on the oscillatory flowpast an infinite vertical porous plate with constant suction,J.Roy.Soc.London ,Vol.A333, pp.25-36.

[6] Takhar H.S. and Ram P.C.(1994), Magneto hydrodynamic freeconvection flow of water at 4˚C through a porous medium,Int.Comm.Heat Mass Transfer,Vol.21, pp. 371-376.

[7] Soundalgekar V.M., Gupta S.K, Birajdar N.S.(1979), Effects of masstransfer and free convection effects on MHD Stokes problem for avertical plate, Nucl.Eng.Design,Vol.53, pp. 309-346.

[8] Eilbashbeshy E.M.A.(1997), Heat and Mass transfer along a verticalplate with variable surface tension and concentration in the presenceof magnetic field,Int.J.Eng Science,Vol. 35, pp. 515-522.

[9] Alagoa K.D., Tay G. and Abbey T.M.(1999), Radiative and freeconvection effects of a MHD flow through porous medium betweeninfinite parallel plates with time dependent suction, Astrophysics.Space. Sci.,Vol.260, pp.455-468.

[10] Bestman A.R. and Adjepong S.K.(1998), Unsteady hydromagneticfree convection flow with radiative transfer in a rotating fluid, As-trophysics. Space Sci.,Vol.143, pp.73-80.

[11] Gebhar B.(1962), Effects of viscous dissipation in natural convection,J. Fluid Mech.,Vol.14, pp.225-232.


[12] Soundalgekar V.M.(1972), Viscous dissipation effects on unsteadyfree Convective flow past an infinite vertical porous plate with con-stant suction, Int. J. Heat Mass Transfer,Vol.15, pp.1253-1261.

[13] Israel-Cookey C., Ogulu A. and Omubo-Pepple V.B.(2003), Influenceof viscous dissipation on unsteady MHD free-convection floe past aninfinite heated vertical plate in porous medium with time-dependentsuction, Int. J. Heat Mass Transfer,Vol.46, pp. 2305-2311.

[14] Cogley A.C.,Vincenti W.G. and Gill S.E.(1968), Differential approx-imation for radiative transfer in a non-gray-gas near equilibrium,AIAA. J.,Vol.6, pp.551-553.

Appendix

m1 =Sc +

√Sc2 + 4nSc

2, m2 =

Pr +√

Pr2 +4R2

2,

m3 =1 +

√1 + 4(χ2 + M2)

2, m4 =

Pr +√

Pr2 +4N2

2,

m5 =1 +

√1 + 4N1

2.

P2 =−Gm

Sc2 − Sc− (χ + M), P3 = P1 + P2 − 1.

S1 =−Pr m2

3P23

4m23 − 2 Pr m3 −R

, S2 =−Pr m2

2P21

4m22 − 2 Pr m2 −R

,

S3 =−Pr Sc2P 2

2

4Sc2 − 2 Pr Sc−R, S4 =

2 Pr m3P3P1m2

(m3 + m2)2 − Pr(m3 + m2)−R,

S5 = − −2 Pr m3P2P1Sc

(m2 + Sc)2 − Pr(m2 + Sc)−R,


S6 =2 Pr m3P3P2Sc

(m3 + Sc)2 − Pr(m3 + Sc)−R,

S7 = −(S1 + S2 + S3 + S4 + S5 + S6).

D1 =m2A Pr

m22 − Pr m2 −N2

, D5 =GmASc

n

m21 −m1 −N1

,

D6 =−Am3P3

m23 −m3 −N1

, D8 =−AP2Sc− GmASc

n

Sc2 − Sc−N1

,

D9 =AP1m2 + GrD1

m22 −m2 −N1

, D9 = −(D5 + D6 + D7 + D8 + D9).

J1 =−GrS7

m22 −m2 − (χ + M)

, J2 =GrS1

4m23 − 2m3 − (χ + M)

,

J3 =GrS2

4m22 − 2m2 − (χ + M)

, J4 =GrS3

4Sc2 − 2Sc− (χ + M),

J5 =−GrS4

(m2 + m3)2 − (m2 + m3)− (χ + M),

J6 =GrS5

(m2 + Sc)2 − (m2 + Sc)− (χ + M),

J7 =−GrS6

(m3 + Sc)2 − (m3 + Sc)− (χ + M),

J10 = −(J1 + J2 + J3 + J4 + J5 + J6 + J7).

G1 =RAS7m2

m22 − Pr m2 −N2

,

G3 = −2 Pr AS2m2 + 2 Pr m22P1D7

4m22 − 2 Pr m2 −N2

,

G4 = −2 Pr AS3Sc + 2 Pr Sc2P2D8

4Sc2 − 2 Pr Sc−N2

,

G5 =2 Pr m3P3D7m2 + 2 Pr P1D6m2m3 + Pr AS4(m2 + m3)

(m2 + m3)2 − Pr(m2 + m3)−N2

,


G6 = −2 Pr m2P2D7Sc + 2 Pr P1D7m2Sc + Pr AS5(m2 + Sc)

(m2 + Sc)2 − Pr(m2 + Sc)−N2

,

G7 =2 Pr m3P2D6Sc + 2 Pr P3D8m3Sc + Pr AS6(m3 + Sc)

(m3 + Sc)2 − Pr(m3 + Sc)−N2

,

G8 =−2 Pr m3P3D10m5

(m3 + m5)2 − Pr(m3 + m5)−N2

,

G9 =−2 Pr m3P3D9m4

(m3 + m4)2 − Pr(m3 + m4)−N2

,

G10 =2 Pr m2P1D10m5

(m2 + m5)2 − Pr(m2 + m5)−N2

,

G11 =2 Pr m2P1D9m4

(m2 + m4)2 − Pr(m2 + m4)−N2

,

G12 =2 Pr ScP2D9m4

(m4 + Sc)2 − Pr(m4 + Sc)−N2

,

G14 =−2 Pr m1P3D5m3

(m1 + m3)2 − Pr(m1 + m3)−N2

,

G15 =2 Pr m2P1D5m1

(m1 + m2)2 − Pr(m1 + m2)−N2

,

G16 =2 Pr ScP2D5m1

(m1 + Sc)2 − Pr(m1 + Sc)−N2

,

G20 = −(G1 + G2 + G3 + G4 + G5 + G6 + G7 + G8 + G9

+G10 + G11 + G12 + G13 + G14 + G15 + G16).

Z1 =−GrG20

m24 −m4 −N1

, Z2 =GrG1 − AJ1m2

m22 −m2 −N1

,

Z3 =2AJ2m3 −GrG2

m23 −m3 −N1

, Z4 =m3AJ10

m23 −m3 −N1

,

Z5 =2AJ3m2 −GrG3

4m22 −m2 −N1

, Z6 =2AJ4Sc−GrG4

4Sc2 − 2Sc−N1

,

Z7 = − AJ5(m2 + m3) + GrG5

(m2 + m3)2 − (m2 + m3)−N1

,


Z8 =AJ6(m2 + Sc) + GrG6

(m2 + Sc)2 − (m2 + Sc)−N1

,

Z9 = − AJ7(m3 + Sc) + GrG7

(m3 + Sc)2 − (m3 + Sc)−N1

,

Z10 =GrG8

(m3 + m5)2 − (m3 + m5)−N1

,

Z11 =GrG9

(m3 + m4)2 − (m3 + m4)−N1

,

Z12 =−GrG10

(m2 + m5)2 − (m2 + m5)−N1,

Z13 =−GrG11

(m2 + m4)2 − (m2 + m4)−N1

,

Z14 =−GrG12

(m4 + Sc)2 − (m4 + Sc)−N1

,

Z15 =−GrG13

(m5 + Sc)2 − (m5 + Sc)−N1

,

Z16 =GrG14

(m1 + m3)2 − (m1 + m3)−N1

,

Z17 =−GrG15

(m1 + m2)2 − (m1 + m2)−N1

,

Z18 =−GrG16

(m1 + Sc)2 − (m1 + Sc)−N1

,

Z20 = −(Z1 + Z2 + Z3 + Z4 + Z5 + Z6 + Z7 + Z8 + Z9

+Z10 + Z11 + Z12 + Z13 + Z14 + Z15 + Z16 + Z17)

Submitted on July 2007.


Uticaj radiajacije i prenosa mase na nestacionarnoMHD slobodno konvekciono tecenje preko zagrejane

uspravne ploce u poroznoj sredini sa viskoznomdisipacijom

Posmatra se nestacionarno dvodimenziono hidromagnetsko laminarno slo-bodno konvekciono tecenje u granicnom sloju nestisljivog Njutnovskogelektroprovodnog i zraceceg fluida preko beskonacne zagrejane uspravneploce. Pri analizi prenosa mase i toplote uzima se u obzir viskozna disi-pacija. Bezdimenzione jednacine problema su resene analiticki koriscenjemdvoclanih harmonijskih i neharmonijskih funkcija. Numericko dobijanjerezultata je izvedeno, te su graficki prikazani rezultati za brzinu, tem-peraturu i profile koncentracije unutar granicnog sloja kao kao i tab-ulirani rezultati za koeficijent “skin-trenja”, Nuseltov i Servudov broj.Primeceno je da sa porastom parametra zracenja brzina i temperaturau granicnom sloju opadaju, dok sa porastom termickog i mesavinskogGrashofovog broja brzina raste.

doi:10.2298/TAM0702135P Math.Subj.Class.: 80A20, 76E25, 76W05

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