GENERAL PHYSICS – THERMAL EFFECTS

RADIATION EFFECT WITH SIMULTANEOUS THERMALAND MASS DIFFUSION IN MHD MIXED CONVECTION FLOW

FROM A VERTICAL SURFACE WITH OHMIC HEATING

R. C. CHAUDHARY , BHUPENDRA KUMAR SHARMAand ABHAY KUMAR JHA

Department of Mathematics, University of RajasthanJaipur-302004 (India)

e-mail: rcchaudhary@rediffmail.com bhupen_1402@yahoo.co.in

Received April 4, 2006

The effects of radiation on heat transfer in MHD mixed convection flow andmass transfer past an infinite vertical plate with Ohmic heating and viscousdissipation have been discussed. Approximate solutions have been derived for thevelocity, temperature field, concentration profiles, skin friction and rate of heattransfer using multi-parameter perturbation technique. The obtained results arediscussed with the help of graphs to observe the effect of various parameter likeSchmidt number (Sc), Prandtl number (Pr), Magnetic parameter (M) and radiationparameter (F), taking two cases viz. Case I: when Gr > 0 (i.e. flow on cooled plate)and Case II: Gr < 0 (i.e. flow on heated plate).

Key words: radiation, magnetic field, heat-mass transfer.

INTRODUCTION

Convection flow driven by temperature and concentration differences hasbeen the objective of extensive research because such processes exists in natureand has engineering applications. The process occurring in nature includesphoto-synthetic mechanism, calm-day evaporation and vaporization of mist andfog, while the engineering application includes the chemical reaction in a reactorchamber consisting of rectangular ducts, chemical vapor deposition on surfacesand cooling of electronic equipment. Heat and mass transfer on flow past avertical plate have been studied by several authors; viz. Somess [1], Soundalgekarand Ganesan [2], Khair and Bejan [3] and Lin and Wu [4] in numerous ways toinclude various physical aspects. Magnetohydrodynamics flows has applicationsin meteorology, solar physics, cosmic fluid dynamics, astrophysics, geophysicsand in the motion of earthes core. In addition to the technological point of view,

Rom. Journ. Phys., Vol. 51, Nos. 7–8 , P. 715–727, Bucharest, 2006

716 R. C. Chaudhary, Bhupendra Kumar Sharma, Abhay Kumar Jha 2

MHD free convection flows have significant applications in the field of stellarand planetary magnetospheres, aeronautics, chemical engineering and electronics.On account of their varied importance, this flows has been studied by severalauthors; worthy to mention among them are Shercliff [5], Ferraro and Plumption[6], Cramer and Pai [7] and Elbashbeshy [8].

In the above mentioned studies the radiation effect is ignored. The effectsof radiation on temperature have become more important industrially. Manyprocesses in engineering areas occur at high temperature and acknowledgeradiation heat transfer become very important for the design of pertinentequipment. Nuclear power plants, gas turbines and the various propulsiondevices for air craft, missiles, satellites and space vehicles are example of suchengineering areas. Hossain and Takhar [9], Raptis and Massals [10] and Hossainand Alim [11] studied the radiation effect on free and forced convection flowspast a vertical plate, including various physical aspects. Aboeldhab [12] studiedthe radiation effect in heat transfer in an electrically conducting fluid atstretching surface. At high operating temperature, radiation effect can be quitesignificant [13]. Heat and mass transfer effects on moving plate in the presenceof thermal radiation have been studied by Muthukumarswamy [14] usingLaplace technique. For the problem of coupled heat and mass transfer in MHDfree convection, the effect of both viscous dissipation and Ohmic heating are notstudied in the above investigations. However, it is more realistic to include thesetwo effects to explore the impact of the magnetic field on the thermal transport inthe boundary layer. With this awareness, the effect of Ohmic heating on theMHD free convection heat transfer has been examined for a Newtonian fluid byHossain [15]. Chen [16] studied the problem of combined heat and mass transferof an electrically conducting fluid in MHD natural convection, adjacent to avertical surface with Ohmic heating.

The propagation of thermal energy through mercury and electrolytic solutionin the presence of magnetic field and radiation has wide range of applications.Hence, our object in the present paper is to study the effect of radiation on heatand mass transfer in mercury (Pr = 0.025) and electrolytic solution (Pr = 1.0)past an infinite porous hot vertical plate in the presence of Ohmic heating andtransverse magnetic field.

FORMULATION OF THE PROBLEM

We consider the mixed convection flow of an incompressible andelectrically conducting viscous fluid such that x*-axis is taken along the plate inupwards direction and y*-axis is normal to it. A transverse constant magneticfield is applied i.e. in the direction of y*-axis. Since the motion is twodimensional and length of the plate is large therefore all the physical variables

3 Radiation effect with simultaneous thermal and mass diffusion in MHD 717

are independent of x*. Let u* and v* be the components of velocity in x* and y*directions, respectively, taken along and perpendicular to the plate. Thegoverning equations of continuity, momentum and energy for a flow of anelectrically conducting fluid along a hot, non-conducting porous vertical plate inthe presence of concentration and radiation is given by

*

*dv 0dy

= (1)

i.e. *0v v (constant)= − (2)

**

*dp

0 pdy

= ⇒ is independent of y* (3)

2 *** * 2 * *

0* *2d uduv g (T T ) B u g (C C )

dy dy∗

∞ ∞μρ = + ρ β − −σ + ρ β − (4)

2 ** 2 * * r* 2 2*p 0* *2 * *

qdT d T duC v B udy dy dy y

⎛ ⎞ ∂ρ = κ + μ − + σ⎜ ⎟ ∂⎝ ⎠

(5)

* 2 **

* *2dC d Cv Ddy dy

= (6)

Here, g is the acceleration due to gravity, T* the temperature of the fluid near theplate, T∞ the free stream temperature, C* concentration, β the coefficient ofthermal expansion, κ the thermal conductivity, p* the pressure, Cp the specificheat of constant pressure, B0 the magnetic field coefficient, μ viscosity of the

fluid, *rq the radiative heat flux, ρ the density, σ the magnetic permeability of

fluid V0 constant suction velocity, ν the kinematic viscosity and D moleculardiffusitivity.

The radiative heat flux is given by [17]

*r **

q4(T T )

y ∞∂ ′= − Ι∂

(7)

where b*0

eI K d ,

T

∞ λ∂′ = λ∂∫ wKλ is the absorption coefficient at the wall and ebλ

is Planck’s function.The boundary conditions are

* *w

* * * *

y 0 : u 0, T T , C C

y : u 0, T T , C C .

∗ ∞

∞ ∞

= = = =

→∞ → → →(8)

718 R. C. Chaudhary, Bhupendra Kumar Sharma, Abhay Kumar Jha 4

Introducing the following non-dimensional quantities

* 2 2*0 022

0 0* *

p

2 20 w

3p w 0

v y Buy , u , Mv v

C T T C CPr , , C

T T C C

v g (T T )E , Gr

C (T T ) v

∞ ∞

ω ∞ ∞

∞

∞

⎫ν σ= = = ⎪ν μ ⎪

⎪μ − − ⎪= θ = = ⎬κ − − ⎪⎪ρ βν −

= = ⎪− μ ⎪⎭

(9)

30

g (C C )Sc , Gm

D v

∗∞ρ β −ν= = and 2

p 0

4 IFC v

′ν=ρ

In the equations (4), (5), (6) and (8), we get

22

2d u du M u Gr Gm C,

dydy+ − = − θ − (10)

222 2

2d d duPr F Pr Pr E Pr E M u 0,

dy dydy⎛ ⎞θ θ+ − θ + + =⎜ ⎟⎝ ⎠

(11)

2

2d C dCSc 0,

dydy+ = (12)

where, Gr = Grashoff number, Pr = Prandtl number,M = Magnetic parameter, F = Radiation parameter,Sc = Schmidt number, E = Eckert number.

The corresponding boundary condition in dimensionless form are reduced to

y 0 : u 0, 1, C 1

y : u 0, 0

= = θ = = ⎫⎬→∞ → θ→ ⎭

(13)

The physical variables u, θ and C can be expanded in the power of Eckertnumber (E). This can be possible physically as E for the flow of anincompressible fluid is always less than unity. It can be interpreted physically asthe flow due to the Joules dissipation is super imposed on the main flow. Hencewe can assume

20 1

20 0

20 1

u(y) u (y) Eu (y) 0(E )

(y) (y) E (y) 0(E )

C(y) C (y) EC (y) 0(E )

⎫= + +⎪

θ = θ + θ + ⎬⎪= + + ⎭

. (14)

5 Radiation effect with simultaneous thermal and mass diffusion in MHD 719

Using equation (14) in equation (10)–(12) and equating the coefficient oflike powers of E, we have

20 0 0 0 0u u M u Gr Gm C ,′′ ′+ − = − θ − (15)

0 0 0Pr F Pr 0,′′ ′θ + θ − θ = (16)

0 0C Sc C 0,′′ ′+ = (17)

21 1 1 1 1u u M u Gr Gm C ,′′ ′+ − = − θ − (18)

2 2 21 1 1 0 0Pr F Pr Pr u Pr M u 0′′ ′ ′θ + θ − θ + + = (19)

1 1C Sc C 0′′ ′+ = (20)

and the corresponding boundary conditions are

0 1 0 1 0 1

0 1 0 1 0 1

y 0 : u 0, u 0, 1, 0, C 1, C 0

y : u 0, u 0, 0, 0, C 0, C 0

= = = θ = θ = = = ⎫⎬→∞ → → θ → θ → → → ⎭

(21)

Solving equations (15) to (20) with the help of (21), we get

4 4 1A y Sc y A y A y0 6 5u A (e e ) A (e e ),− − − −= − + −

1A y0 e ,−θ =

Sc y0C e ,−=

9 1 1 4 4

1 2

A y A y 2A y 2A y A y1 17 10 11 12 13

2 Sc y B y B y14 15 16

v B e B e B e B e B e

B e B e B e ,

− − − − −

− − −

= − + + − +

+ − +

101 1 4

1 2

A yA y 2A y 2A y 2Sc y1 9 3 4 5 6

B y B y7 8

B e B e B e B e B e

B e B e ,

−− − − −

− −

θ = − − + − +

+ −

where

2

1Pr Pr 4 F PrA ,

2+ +=

2

2Pr Pr 4F PrA ,

2+ += −

2

31 1 4MA ,

2− + +=

2

41 1 4MA ,

2+ +=

51 4 1 3

GrA ,(A A )(A A )

=− + 6 2 2

GmA ,Sc Sc M

=− −

2 2 2 27 3 8 15 5A Pr A A , A Pr A A ,= = 2

9 1 4 10 1 45A 2Pr A A A , A A A ,= = +

720 R. C. Chaudhary, Bhupendra Kumar Sharma, Abhay Kumar Jha 6

1 4 2 1B A Sc, B A Sc,= + = + 2 2 2

153

1 1 3

Pr A (A M )B ,

A (2A A )+

=+

2 2 24 5 6

44 1 4 2

Pr (A M )(A A )B ,

(2A A )(2A A )+ +

= −− +

2 25 6 1 4

510 1 10 3

2Pr M (A A )(A A M )B ,

(A A )(A A )+ +

=− −

2 2 26

6 2

Pr A (Sc M )B ,

4Sc 2Sc Pr F Pr

+=

− −

26 5 6 4

7 21 1

2 Pr A (A A )(A Sc M )B ,

B Pr B F Pr+ +

=− −

25 6 1

8 22 2

2 Pr A A (A Sc M )B ,

B Pr B F Pr+

=− − 9 3 4 5 6 7 8B B B B B B B ,= + − + − +

910

1 4 1 3

Gr BB ,

(A A )(A A )=

− +3

111 4 1 3

Gr BB ,

(2A A )(2A A )=

− +

412

4 4 3

Gr BB ,

A (2A A )=

+5

1310 4 10 2

Gr BB ,

(A A )(A A )=

− +

614 2 2

Gr BB ,

4 Sc 2 Sc M=

− −7

15 2 21 1

Gr BB ,

B B M=

− −

816 2 2

2 2

Gr BB ,

B B M=

− − 17 10 11 12 13 14 15 16B B B B B B B B ,= − − + − + −

The skin-friction coefficient at the plate is given by

6 4 5 1 4 4 7 1 10 1 11y 0

4 12 4 13 14 1 15 2 16

u A (Sc A ) A (A A ) E [A B A B 2A By

2A B A B 2 Sc B B B B B ],=

⎛ ⎞∂τ = = − + − − − + +⎜ ⎟∂⎝ ⎠

+ − + − +

and rate of heat transfer in terms of Nusselt number at the plate is given by

1 1 9 1 3 4 4 10 5y 0

6 1 7 2 8

Nu A E [A B 2A B 2A B A By

2Sc B B B B B ]=

⎛ ⎞∂θ= − = + − − + −⎜ ⎟∂⎝ ⎠

− + −

DISCUSSION

A study of velocity field, temperature field, heat transfer, mass transfer andskin friction of the MHD mixed convection flow of a viscous incompressibleelectrically conducting fluid over an infinite vertical porous plate in the presenceof magnetic field has been carried out in the preceding sections, taking radiationeffect into account. We have computed the numerical values of velocity,temperature, skin friction, heat and mass transfer for two cases viz. (i) forcooling of the plate (Gr > 0) and (ii) for heating of the plate (Gr < 0). The values

7 Radiation effect with simultaneous thermal and mass diffusion in MHD 721

of Prandtl number (Pr) are taken 1 and 0.025, which represent electrolyticsolution and mercury at 20°C temperature and 1 atmosphere pressure. The valuesof Eckert number (E) and modified Grashof number (Gm) are taken 0.01 and2.0, respectively. The obtained results are illustrated in Figs. 1 to 9.

Figs. 1 and 2 depict the velocity profiles for cooled Newtonian fluid(Gr > 0) and heating Newtonian fluid (Gr < 0), respectively. These velocityprofiles are drawn for E = 0.01 and Gc = 2.0. From Fig. 1, it is observed that anincrease in M (Magnetic parameter) or Sc (Schmidt number) or F (radiationparameter) leads to a rise in the velocity, for both mercury (Pr = 0.025) andelectrolytic solution (Pr = 1). In the vicinity of the plate, velocity increases andtakes the maximum value and as we move far away from the plate, it goes ondecrease. The effect of magnetic parameter (M) is more dominant as comparedto the other parameter F and Sc. The velocity profiles decreases with increasingPrandtl number. Physically, this is true because the increase in the Prandtlnumber is due to increase in the viscosity of the fluid which makes the fluidthick and hence a decrease in the velocity of the fluid. From Fig. 2, which givesthe velocity profiles for heated Newtonian fluid, we noticed that an increase in Mleads to fall in the velocity for Pr = 1, while, reverse effect is observed forPr = 0.025. The velocity for Ammonia (Sc = 0.78) is less than that of Hydrogen(Sc = 0.22) for both Pr = 1 and Pr = 0.025. An increase in radiation parameter(F) decreases the velocity. The velocity profiles remains positive (≥ 0) for Pr = 1and negative (≤ 0) for Pr = 0.025. Figs. 3 and 4 represent the temperatureprofiles for Gr = 5 and Gr = –5, respectively. The effects of M, Sc and F ontemperature profiles are drawn for Pr = 1 and Pr = 0.025. It is observed that forcooling of the plate(Gr > 0), an increase in magnetic parameter (M) increases thetemperature, while reversal effect is observed for Sc and F, for both Pr = 1 andPr = 0.025. The magnetic field increases the temperature of the fluid inside theboundary layer because of excess heating and consequently decreases in the heatflux. In the case of heating of the plate i.e. for Gr < 0, an increase in M or Sc or Fleads to fall in the temperature. It is also concluded that for electrolytic solution,temperature falls exponentially and for mercury it falls slowly and steadily nearthe plate. The increase of Prandtl number results in the decrease of temperaturedistribution. This is due to the fact that there would be a decrease of thermalboundary layer thickness with the increase of Prandtl number. Fig. 5 gives thespecies concentration for different gases like Hydrogen (Sc = 0.22), Helium(Sc = 0.30), Water-vapour (Sc = 0.60), Oxygen (Sc = 0.66) and Ammonia(Sc = 0.78). It is observed that the concentration at all points in the flow fielddecreases exponentially with y and tends to zero as y → ∞. A comparison ofcurves in the figure shows a decrease in concentration (C) with an increase inSchmidt number. Physically, the increase of Sc means decrease of moleculardiffusivity (D). That results in decrease of concentration boundary layer. Hence,

722 R. C. Chaudhary, Bhupendra Kumar Sharma, Abhay Kumar Jha 8

Fig. 1 – Variation of velocity component for E = 0.01, Gc = 2.0 and Gr = 5.0.

Fig. 2 – Variation of velocity component for E = 0.01, Gc = 2.0 and Gr = –5.0.

9 Radiation effect with simultaneous thermal and mass diffusion in MHD 723

Fig. 3 – Variation of temperature θ for E = 0.01, Gc = 2.0 and Gr = 5.0.

Fig. 4 – Variation of temperature θ for E = 0.01, Gc = 2.0 and Gr = –5.0.

724 R. C. Chaudhary, Bhupendra Kumar Sharma, Abhay Kumar Jha 10

Fig. 5 – Effect of Sc on species concentration.

the concentration of the species is higher for small values of Sc and lower forlarger values of Sc.

Fig. 6 represents the shearing stress (τ) against M, for variation in Pr, Scand F, corresponding to cooling of the plate. It is noticed that shearing stressfalls rapidally near the plate and remains constant far away from the plate forboth Pr = 1 and Pr = 0.025. The shearing stress decreases with increasing Pr forGr > 0. Physically this is true because the increase in the Prandtl number is dueto increase in the viscosity of the fluid which makes the fluid thick and hence adecrease in the velocity of the fluid. An increase in Sc or F decreases theshearing stress (τ). The shearing stress corresponding to heating of the plate(Gr < 0) is given in Fig. 7. The shearing stress for electrolytic solution is morethan that of mercury (Pr = 0.025). The heat transfer coefficient in terms ofNusselt number have been given in Figs. 8 and 9 corresponding to the coolingand the heating of the plate against the magnetic parameter M. It is observed thatan increase in F increases the Nusselt number for both Pr = 1 and Pr = 0.025 andGr > 0 or Gr < 0. Physically increasing the radiation parameter leads to decreasethe boundary layer thickness and to enhance the heat transfer rate. The values ofNusselt number are more for heavier particle (Sc = 0.78) than lighter particle(Sc = 0.22) for cooled plate and both Pr = 0.025 and Pr = 1.0. In the case ofheated plate, an increase in Sc leads to arise in the Nusselt number for Pr = 1.0,while reverse effect is observed for Pr = 0.025. Further, it is concluded that the

11 Radiation effect with simultaneous thermal and mass diffusion in MHD 725

Fig. 6 – Variation of shearing stress τ for E = 0.01, Gc = 2.0 and Gr = 5.0.

Fig. 7 – Variation of shearing stress τ for E = 0.01, Gc = 2.0 and Gr = –5.0.

726 R. C. Chaudhary, Bhupendra Kumar Sharma, Abhay Kumar Jha 12

Fig. 8 – Nusselt number (Nu) for E = 0.01, Gc = 2.0 and Gr = 5.0.

Fig. 9 – Nusselt number (Nu) for E = 0.01, Gc = 2.0 and Gr = –5.0.

rate of heat transfer remains constant for increasing the values of M forPr = 0.025 and falls slowly near the plate for Pr = 1 in both the cases i.e. forcooling and heating of the plate.

13 Radiation effect with simultaneous thermal and mass diffusion in MHD 727

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