GENERAL PHYSICS MAGNETOHYDRODYNAMICS

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1 GENERAL PHYSICS MAGNETOHYDRODYNAMICS HALL EFFECT ON MHD MIXED CONVECTIVE FLOW OF A VISCOUS INCOMPRESSIBLE FLUID PAST A VERTICAL POROUS PLATE IMMERSED IN POROUS MEDIUM WITH HEAT SOURCE/SINK BHUPENDRA KUMAR SHARMA ABHAY KUMAR JHA R. C. CHAUDHARY Department of Mathematics University of Rajasthan Jaipur India rcchaudhary@rediffmail.com Received June An analysis is presented with unsteady MHD mixed convective flow of a viscous incompressible fluid past an infinite vertical porous flat plate in the presence of a heat source/sink with Hall effect. The solution is found to be dependent on the governing parameters including the Hartman number the Prandtl number the Grashof number the Schmidt number the Hall parameter rotating parameter and the heat source/sink parameter. Representative results are presented and discussed for velocity temperature concentration species surface heat flux mass flux and skin friction coefficient with the help of graphs and tables. Key words: MHD mass transfer heat source/sink. 1. INTRODUCTION The hydromagnetic convection with heat and mass transfer in porous medium has been studied due to its importance in the design of MHD generators and accelerators in geophysics in design of under ground water energy storage system soil-sciences astrophysics nuclear power reactors and so on. Magnetohydrodynamics is currently undergoing a period of great enlargement and differentiation of subject matter. The interest in these new problems generates from their importance in liquid metals electrolytes and ionized gases. On account of their varied importance these flows have been studied by several authors notable amongst them are Shercliff [1] Ferraro and Plumpton [] and Crammer and Pai [3]. Elbashbeshy [4] studied heat and mass transfer along a vertical plate in the presence of a magnetic field. Hossain and Rees [5] examined the effects of combined buoyancy forces from thermal and mass diffusion by natural convection flow from a vertical wavy surface. Numerical results were obtained to show the evolution of the surface shear stress rate of heat transfer and surface concentration gradient. Recently combined heat and mass transfer in MHD free convection from a vertical surface has been studied by Chen [6]. Rom. Journ. Phys. Vol. 5 Nos. 5 7 P Bucharest 007

2 488 B. K. Sharma A. K. Jha R. C. Chaudhary Further the effect of Hall current on the fluid flow with variable concentration has many applications in MHD power generation in several astrophysical and meteorological studies as well as in plasma flow through MHD power generators. From the point of application model studies on the Hall effect on free and forced convection flows have been made by several investigators. Aboeldahab [7] Datta et al. [8] Acharya et al. [9] and Biswal et al. [10] have studied the Hall effect on the MHD free and forced convection heat and mass transfer over a vertical surface. In the above mentioned studies the heat source/sink effect is ignored. Due to its great applicability to ceramic tiles production problems the study of heat transfer in the presence of a heat source/sink has acquired newer dimensions. A number of analytical studies have been carried made of for various forms of heat generation (Ostrach [11 13] Raptis [14]). Singh [15] analysed MHD free convection and mass transfer flow with heat source and thermal diffusion. The problem investigated here is the study of the Hall Effects on the combined heat and mass transfer unsteady flow which occur due to buoyancy forces caused by thermal diffusion (temperature differences) and mass diffusion (concentration differences) of comparable magnitude past a vertical porous plate which is immersed in porous medium with a constant magnetic field and heat source/sink applied perpendicular to the plate.. FORMULATION Consider the unsteady flow of an electrically conducting fluid past an infinite vertical porous flat plate coinciding with the plane y = 0 such that the x-axis is along the plate and y-axis is normal to it. A uniform magnetic field B 0 is applied in the direction y-axis and the plate is taken as electrically non-conducting. Taking z-axis normal to xy-plane and assuming that the velocity V and the magnetic field H have components (u v w) and (H x H y H z ) respectively the equation of continuity V = 0 and solenoidal relation H = 0 give v = V constant V > 0. (1) 0 0 From Maxwell s electromagnetic field equations y H y = 0. () If the magnetic Reynold number is small induced magnetic field is negligible in comparison with the applied magnetic field so that [16] H = H = 0 and H = B (constant). x z y 0

3 3 Hall effect on MHD mixed convective flow 489 If (J x J y J z ) are the components of electric current density J. The equation of conservation of electric charge J = 0 gives J y = constant. (3) Since the plate is non-conducting J y = 0 at the plate and hence zero everywhere in the flow. Neglecting polarization effect we get E = 0. (4) Hence J = ( J 0 J ) H = (0 B 0) V = ( u V w). (5) x z 0 0 The generalized Ohm s law taking Hall effect into account is given by J ωτ e e 1 ( J H ) E V B p + =σ + + e B 0 eη (6) e where V is the velocity vector ω e is the electron frequency τ e is the electron collision time e is the electron charge η e is the number density of electron p e is the electron pressure σ is the electric conductivity and E is the electric field. In writing equation (6) ion slip thermoelectric effects and polarization effects are neglected. Further it is also assumed that ωτ e e ~ 0 and ωτ i i 1 where ω e ω i are cyclone frequencies of electrons and ions and τ e τ i are collision times of electrons and ions. Equations (5) and (6) yield σb0 Jx = ( mu w) 1+ m σb0 Jz = ( u+ mw) 1+ m where u and w are the x-component and z- component of V and m =ωeτ e is the Hall parameter. The equations of motion energy and concentration governing the flow under the usual Boussinesq approximation are σ B 0 ( u+ mw) * u + v u =ν u + gβ( T T ) + gβ ( C C ) νu t y y ρ (1 + m ) k (7) (8) σ 0 w B ( mu w) + v w =ν w + νw t y y ρ (1 + m ) k (9)

4 490 B. K. Sharma A. K. Jha R. C. Chaudhary 4 K0 ( T T ) ( T T ) ( T T ) + v = + ST ( T ) t y ρc p y + = D ( C C ) ( C C ) ( C C ) v. t y y (10) (11) where g is the acceleration due to gravity T is the temperature of the fluid within the boundary layer C is the specis concentration ρ is the fluid density of boundary layer ν is the kinematic viscosity K 0 is the thermal conductivity c p is the specific heat at constant pressure D is the chemical and molecular diffusivity k is the permeability of porous medium and S is the source sink parameter. In equation (10) the viscous dissipation and ohmic dissipation are neglected and in equation (11) the term due to chemical reaction is assumed to be absent. Now using v = V0 T( y t) T =θ ( y t) and Cyt ( ) C * = C( yt ) Subjecting to the initial and boundary conditions t u( y t) = w( y t) = 0 θ= 0 C = 0 for all y * u(0 t) = w(0 t) = 0 θ (0 t) = aeiωt C* (0 t) = beiωt at y = 0 u( t) w( t) 0 ( t) 0 C* = = θ = ( t) = 0 at y ( t > 0). Introducing the following non-dimensional quantities (1) * 0 0 t u w θ C ν 4ν V0 V0 a b * 4 gβνa gβ νb B0 σν νρcp Pr 3 c V 3 3 K 0 V0 ρv0 0 V ν 0 k k S Sν D 4ν V 0 Vy V T η= = = u = w θ = = C 4 4 G = G = M = = Sc = = =. (13) Equations (8) to (11) are transformed to their corresponding nondimensional forms (dropping the dashes) as u 4 u = 4 u M ( mw + u) + Gθ+ G C u t 1 c η η + m k w 4 w = 4 w + M ( mu w) w t η η 1+ m k (14) (15)

5 5 Hall effect on MHD mixed convective flow 491 θ 4 θ = 4 θ + Sθ t η Pr η C 4 C = 4 C t η Sc η (16) (17) where V 0 is some reference velocity G is the Grashof number for heat transfer Gc is the Grashof number for mass transfer Pr is the Prandtl number Sc is the Schmidt number M is the magnetic field parameter and S is the source/sink parameter. Dropping dashes the modified boundary conditions become t 0 : u( η t) = w( η t) = 0 θ= 0 C = 0 for all η u(0 t) = w(0 t) = 0 θ (0 t) = eiωt C(0 t) = eiωt at η= 0 t > 0 u( t) = w( t) = 0 θ ( t) = 0 C( t) = 0 at η. (18) giving 3. SOLUTION The equations (14) and (15) can be combined using the complex variable M ψ= u+ iw (19) ψ ψ 1 ψ + 1 (1 im) + 1 ψ= 1 Gθ 1 G C. 4 t 4 1 k 4 4 c η η + m (0) Introducing the non-dimensional parameter Ω= 4νω and using equation V 0 (19) the boundary conditions in (18) are transferred to ψ (0 t) =ψ( t) = 0 C(0 t) = eiωt (0 t) eiωt θ = θ( t) = 0 C( t) = 0. Putting θη ( t) = e i Ωt f( η ) in equation (16) we get (1) f ( η+ ) Pr f ( η ) iωpr + SPr = 0 () 4 4 which has to be solved under the boundary conditions f(0) = 1 f( ) = 0. (3) Hence

6 49 B. K. Sharma A. K. Jha R. C. Chaudhary 6 1 Pr Pr + iωpr SPr η f( η ) = e η iω t Pr+ Pr +Ω i Pr SPr ( t) e. θ η = (4) where Separating real and imaginary parts the real part is given by η { } ( Pr R1 cos α η α + ) 1 θr ( η t) = cos Ωt R sin e (5) 1/ S 1/ 4 1 Ω ( ) R1 = Pr [(Pr ) +Ω ] α= tan iωt Putting C( η t) = e g( η ) in equation (17) we get Pr S. (6) g ( η+ ) Sg ( η ) iωsc g( η= ) 0 (7) 4 which has to be solved under the boundary conditions g (0) = 1 g( ) = 0. (8) Hence 1 Sc Sc + iωsc η g( η ) = e (9) η iω t Sc+ Sc +Ω i Sc C( η t) = e. Separating real and imaginary parts the real part is given by η β { } η β Sc R cos + C ( η t) = cos Ωt R sin e (30) r 1/ 1/ 4 1 ( ) R = Sc (Sc +Ω ) β= tan Ω. (31) Sc In order to solve equation (0) substituting ψ= eiωt f( η ) and using boundary conditions F(0) = 0 F( ) = 0 (3) and then separating real and imaginary parts we have η/ 1 η/ η/ η/ 4 9 η/ η/ u A A A A e = [{ cos( η / ) + sin( η/ )} 19 { A cos( Ωt η /) e 1 + A sin( Ωt η/) e } A { A cos( Ωt η / ) e 8 + A sin( Ωt η/ ) e 9 } A ]cosωt [{ A cos( A η/ ) A sin( A η/ )} e 19 (33)

7 7 Hall effect on MHD mixed convective flow η/ η/ η/ 4 9 η/ 6 { A sin( Ωt A η/) e cos( Ωt η/) e 1 } A Ω η Ω η Ω { A sin( t A / ) e 9 A cos( t A / ) e 8 } A ]sin t η/ 1 η/ η/ η/ 4 9 η/ η/ 1 η/ η/ η/ 4 9 8η/ 6 w A A A A e = [{ cos( η / ) + sin( η/ )} 19 { A cos( Ωt η /) e 1 + A sin( Ωt η/) e } A { A cos( Ωt A η / ) e 8 + A sin( Ωt η/ ) e 9 } A ]sinω t+ + [{ A cos( A η/ ) sin( A η/ )} e 19 1 { A sin( Ωt η/) e A cos( Ωt A η/) e } A { A sin( Ωt η/) e cos( Ωt η/) e } A ]cos Ωt. The shearing stress at the wall along x axis is given by (34) τ 1 = u η η= 0 (35) and the shearing stress at the wall along z axis is given by where τ = w η η= 0 = 1/ S +Ω 1/ 4 R = 1/ +Ω 1/ 4 R1 Pr [(Pr ) ] Sc (Sc ) (36) R 3 1/ M Mm = + + k + Ω 1+ m 1+ m 1 Ω 1 Ω ( ) ( ) α= tan Pr S ( ) ( ) β= γ= tan 1 Mm 1 1 M Ω + + k tan Sc A1 = Pr+ R1cos α A R1 sin α = A3 1+ m 1+ m = 1 M k + A 1 + m 4 =Ω Mm 1 + m A = ( A A A ) A6 = (A1A A A4) A = A + A A8 = Sc+ Rcos β / A9 = Rsin β / A = ( A A ) A11 = (A8 A9 A9 A4 ) A1 = ( GA5 cosω t+ GA6 sin Ωt) A5 A13 = ( GA5 sinωt GA6 cos Ω t) A5 A14 = ( GcA10 cosω t+ GcA11 sin Ωt) A6 A15 = ( GcA10 sinωt GcA11 cos Ω t) A6 A 16 = A 10 + A 11 A17 = A1 + A14

8 494 B. K. Sharma A. K. Jha R. C. Chaudhary 8 γ A18 = A13 + A15 A19 = 1+ R3 cos 0 3 sin γ A = R A1 = GA5 A = GA A = G A A G A A 1 A 1 c = c = =. A A DISCUSSION A study of the velocity field variations of temperature and concentration shearing stresses surface heat flux and mass flux in oscillatory hydromagnetic mixed convective flow past an infinite vertical porous plate through porous medium with Hall effect and heat source sink has been carried out in the preceding sections. Approximate solutions are obtained for various flow variables. In order to get physical insight into the problem the velocity temperature concentration fields shear stress rate of heat transfer and rate of mass transfer have been discussed by assigning numerical values of M (Magnetic parameter) G (Grashof number for heat transfer) Sc (Schmidt number) Ω (Frequency parameter). The values of Pr (Prandtl number) are taken as 0.71 and 7.0 that represent air and water at 0 C temperature and one atmosphere pressure respectively. The Grashof number for heat transfer is chosen to be G =.0 and modified Grashof number G c = 5.0 corresponding to a cooled plate (G c > 0) Fig. 1 Variation of velocity component for G =.0 G c = 5.0 and Ωt = π/.

9 9 Hall effect on MHD mixed convective flow 495 while G c = 5.0 corresponding to a heated plate (G c < 0). The values of Hall parameter m = permeability k = 1.0 magnetic parameter M = 5 10 heat source S = 1 (S > 0) and heat sink S = 1 (S < 0) are chosen arbitrarily. The numerical results obtained are illustrated in Figs. 1 to 10 and Tables 1 and. Figs. 1 and depict the variation of the velocity component for G c = 5.0 (cooled Newtonian fluid G c > 0) and G c = 5.0 (heated Newtonian fluid G c < 0) respectively. These velocity profiles are shown for Pr = 0.71 (air) and Pr = 7.0 (water) taking the different values of m M Sc and S. Fig. Variation of velocity component for G =.0 G c = 5.0 and Ωt = π/. Case I: When G c > 0 (i.e. flow on a cooled plate) It is observed that an increase in Hall parameter (m) leads to an decrease in the velocity for both air and water while a reverse effect is observed for the applied magnetic field (M). The velocity is higher for Ammonia (Sc = 0.78 at 5 C temperature and one atmosphere pressure) than Helium (Sc = 0.30 at 5 C temperature and one atmosphere pressure) for both air and water. We notice that the velocity is higher for water than that for air for different parameters. It is also noticed that the velocity is higher for a heat sink (S < 0) than that a heat source for air while the reverse effect is observed for water. Case II: When G c < 0 (i.e. flow on a heated plate) It is noticed that an increase in Hall parameter (m) leads to an increase in the velocity for air while the reverse phenomenon is observed for water. It is

10 496 B. K. Sharma A. K. Jha R. C. Chaudhary 10 found that the values of velocity decrease as the values of M increase for air while there is a rise in the values of velocity in water. The velocity is less for Helium than for Ammonia in air but this behavior is reversed in water. The values of velocity are positive for air while negative for water. A comparative studies of the curves reveal that the values of velocity are more for heat source than heat sink for both air and water. Figs. 3 and 4 represent the variation of the velocity component w for G c = 5.0 and G c = 5.0 respectively. Fig. 3 Variation of velocity component w for G =.0 G c = 5.0 and Ωt = π/. Case I: When G c > 0 (i.e. flow on a cooled plate) An increase in m and M leads to a rise in the velocity for both air and water. The values of velocity are greater for air than that of water. The velocity is greater for Helium than Ammonia for air but the reverse effect is observed for water. It is also noticed that velocity is higher for a sink than that of a source for both air and water. Further it is observed that the velocity increases gradually near the plate and then decreases slowly far away from the plate.

11 11 Hall effect on MHD mixed convective flow 497 Fig. 4 Variation of velocity component w for G =.0 G c = 5.0 and Ωt = π/. Case II: When G c < 0 (i.e. flow on a heated plate) In the case of a heated plate a comparison of velocity profiles shows that velocity increases near the plate and thereafter decreases for air. The velocity decreases due to an increase in m and M for air while the reverse effect is observed for water. For both the cases air and water it is noticed that the velocity is greater for heavier (Ammonia) particles and heat a sink than for lighter particles (Helium) and heat a source. The temperature profiles are shown in Fig. 5. The effects of Pr Ω and S on the temperature profiles are drawn from this figure. An increase in Ω (frequency parameter) leads to a fall in temperature for both air and water. The temperature profiles remain positive for a heat sink and negative for a heat source in both situations (air and water) near the plate and vanish far away from the plate. Fig. 6 gives the concentration for Helium (Sc = 0.30) and Ammonia (Sc = 0.78) and Ω = 1 and 3. We notice that the effect of increasing values of Sc is to decrease

12 13 Hall effect on MHD mixed convective flow 499 the concentration profile in the flow field. Physically the increase of Sc means decrease of molecular diffusivity D. This results in a decrease of concentration boundary layer. Hence the concentration of the species is higher for small values of Sc and lower for larger values of Sc. The values of the concentration increases gradually near the plate and then decrease slowly for away from the plate. Further it is also noticed that the values of the concentration increases at each point with variation of the frequency parameter Ω. Figs. 7 and 8 present the shearing stress τ 1 for G c = 5.0 and G c = 5.0 respectively. Fig. 7 Variation of shearing stress τ 1 for G =.0 G c = 5.0 and Ωt = π/. Case I: When G c > 0 (i.e. flow on a cooled plate) Comparison of shearing stress shows that τ 1 increase near the plate and thereafter remains constant far away from the plate. It is observed that an increase in M increases τ 1 for both air and water whereas the values of τ 1 are higher in Helium than Ammonia for both air and water. We also noticed that τ 1 is higher for a heat sink for air while the reverse phenomenon occurs in the case of water.

13 500 B. K. Sharma A. K. Jha R. C. Chaudhary 14 Fig. 8 Variation of shearing stress τ 1 for G =.0 G c = 5.0 and Ωt = π/. Case II : When G c < 0 (i.e. flow on a heated plate) It is seen that τ 1 decrease near the plate and thereafter remains constant far away from the plate for air while in the case of water the reverse effect is observed. We observed that an increase in M decreases τ 1 for air but in the case of water reverse phenomenon is noticed. The values of τ 1 are higher for Helium for both air and water. Further it is noted that τ 1 is greater for a heat source than a sink in air while the reverse effect is observed. The shearing stress τ for G c = 5.0 and G c = 5.0 has been shown in Figs. 9 and 10 respectively. Case I: When G c > 0 (i.e. flow on a cooled plate) It is observed that an increase in M and Sc decreases τ for both air and water. We also noticed the values of τ are more for a sink than for a source for both cases. Case II: When G c < 0 (i.e. flow on a heated plate) Comparison of shearing stress shown that τ decreases with increasing M and Sc for both cases (air and water). The values of τ are more for a source than

14 50 B. K. Sharma A. K. Jha R. C. Chaudhary 16 sink for both air and water. Further in the vicinity of the plate the values of τ are higher in water than in air and thereafter the reverse effect is observed far away from the plate. Table 1 gives the mass transfer [C(t)] for different cases such as Hydrogen (Sc = 0.) Helium (Sc = 0.30) Water vapour (Sc = 0.60) and Ammonia (Sc = 0.78) at 5 C temperature and one atmosphere pressure. It is observed that C(t) decreases with increasing Ω. The value of C(t) increases with increasing Sc. The data in Table represents heat transfer for Pr S and Ω. It can be seen that with an increase in Ω the values of Q(t) increase in air for a heat source while the reverse effect is observed for heat sink. Further the values of Q(t) decreases with increasing Ω for both heat source and sink in the case of water. It is interesting to note that the values of Q(t) are higher for a heat sink than for a heat source for both air and water. Table 1 Variation of C(t) C(t) Ω Sc Ω Table Variation of Q(t) Q(t) Pr S Acknowledgement. One of the authors Abhay Kumar Jha thank to the UGC New Delhi for awarding a Senior Research Fellowship. REFERENCES 1. J. A. Shercliff A Text Book of Magnetohydrodynamics Pergamon Press London V. C. A. Ferraro C. Plumption An Introduction to Magneto Fluid Mechanics Clarandon Press Oxford K. P. Crammer S. L. Pai Magneto-fluid Dynamic for Engineers and Applied Physicist Mc-Graw Hill book Co. New York 1973.

15 17 Hall effect on MHD mixed convective flow E. M. A. Elbashbeshy Heat and mass transfer along a vertical plate with variable surface temperature and concentration in the pressure of the magnetic field Int. J. Eng. Sc M. A. Hossain D. A. S. Rees Combined heat and mass transfer in natural convection flow from a vertical wavy surface Acta Mech Chien-Hsin-Chen Combined heat and mass transfer in MHD free convection from a vertical surface with Ohmic heating and viscous dissipation Int. J. Eng. Science E. M. Aboeldahab E. M. E. Elbarbary Hall current effect on magnetohydrodynamics free convection flow past a semi-infinite vertical plate with mass transfer Int. J. Eng. Science N. Datta R. N. Jana Oscillatory magnetohydrodynamic flow past a flat plate will Hall effects J. Phys. Soc. Japan M. Acharya G. C. Dash L. P. Singh Hall effect with simultaneous thermal and mass diffusion on unsteady hydromagnetic flow near an accelerated vertical plate Indian J. of Physics B 75B(1) S. Biswal P. K. Sahoo Hall effect on oscillatory hydromagnetic free convective flow of a visco-elastic fluid past an infinite vertical porous flat plate with mass transfer Proc. Nat. Acad. Sci. 69A S. Ostrach Laminar natural convection flow and heat transfer of fluid with and without heat source in channel with wall temperature NACA TN Ibid Combined natural and forced convection laminar flow and heat transfer of fluid with and without heat sources in channels with linearly varying wall temperatures NACA TN Ibid Unstable convection in vertical channels with heating from below including effects of heat source and frictional heating NACA TN A. A. Raptis Free convection and mass transfer effects on the flow past an infinite moving vertical porous plate with constant suction and heat source Astrophy. Space Sci Atul Kumar Singh MHD free convection and mass transfer flow with heat source and thermal diffusion J. of Energy Heat and Mass Transfer T. G. Cowling Magneto-Hydrodynamics Wiley Interscience New York 1957.

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17 Fig. 5 Variation of temperature θr for ωt = π/. Fig. 6 Variation of concentration Cr for Ωt = π/.

18 Fig. 9 Variation of shearing stress τ for G =.0 G c = 5.0 and Ωt = π/. Fig. 10 Variation of shearing stress τ for G =.0 G c = 5.0 and Ωt = π/.