Kragujevac J. Sci. 34 (2012) UDC 532.5: :537.63

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1 5 Kragjevac J. Sci. 34 () 5-. UDC 53.5: 536.4: UNSTEADY MHD FLOW AND HEAT TRANSFER BETWEEN PARALLEL POROUS PLATES WITH EXPONENTIAL DECAYING PRESSURE GRADIENT Hazem A. Attia and Mostafa A. M. Abdeen Department of Engineering Mathematics and Physics Faclty of Engineering Fayom University El-Fayom 6354 Egypt Department of Engineering Mathematics and Physics Faclty of Engineering Cairo University Giza Egypt (Received May 5 ) ABSTRACT. The nsteady magnetohydrodynamic flow of an electrically condcting viscos incompressible flid bonded by two parallel non-condcting poros plates is stdied with heat transfer. An external niform magnetic field and a niform sction and injection are applied perpendiclar to the plates while the flid motion is sbjected to an exponential decaying pressre gradient. The two plates are kept at different bt constant temperatres while the Jole and viscos dissipations are inclded in the energy eqation. The effect of the magnetic field and the niform sction and injection on both the velocity and temperatre distribtions is examined. INTRODUCTION The magnetohydrodynamic flow between two parallel plates known as Hartmann flow is a classical problem that has many applications in magnetohydrodynamic (MHD) power generators MHD pmps accelerators aerodynamic heating electrostatic precipitation polymer technology petrolem indstry prification of crde oil and flid droplets and sprays. HARTMANN and LAZARUS [] stdied the inflence of a transverse niform magnetic field on the flow of a condcting flid between two infinite parallel stationary and inslated plates. Then a lot of research work concerning the Hartmann flow has been obtained nder different physical effects [-]. In the present stdy the nsteady magnetohydrodynamic flow and heat transfer of an incompressible viscos electrically condcting flid between two infinite non-condcting horizontal poros plates are stdied. The flid is acted pon by an exponential decaying pressre gradient a niform sction and injection and a niform magnetic field perpendiclar to the plates. The indced magnetic field is neglected by assming a very small magnetic Reynolds nmber [4 5]. The two plates are maintained at two different bt constant temperatres. This configration is a good approximation of some practical sitations sch as heat exchangers flow meters and pipes that connect system components. The cooling of these devices can be achieved by tilizing a poros srface throgh which a coolant either a liqid

2 6 or gas is forced. Therefore the reslts obtained here are important for the design of the wall and the cooling arrangements of these devices. The eqations of motion are solved analytically sing the Laplace transform method while the energy eqation is solved nmerically taking the Jole and the viscos dissipations into consideration. The effect of the magnetic field the Hall crrent the ion slip and the sction and injection on both the velocity and temperatre distribtions is stdied. DESCRIPTION OF THE PROBLEM The two non-condcting plates are located at the y±h planes and extend from x- to and z- to. The lower and pper plates are kept at the two constant temperatres T and T respectively where T >T. The flid flows between the two plates nder the inflence of an exponential decaying pressre gradient dp/dx in the x-direction and a niform sction from above and injection from below which are applied at t. The whole system is sbjected to a niform magnetic field B o in the positive y-direction. This is the total magnetic field acting on the flid since the indced magnetic field is neglected. From the geometry of the problem it is evident that / x / z for all qantities apart from the pressre gradient dp/dx which is assmed constant. The velocity vector of the flid is v( y t) ( y t) i + v o j with the initial and bondary conditions at t and at y±h for t>. The temperatre T(yt) at any point in the flid satisfies both the initial and bondary conditions TT at t TT at y+h and TT at y-h for t>. The flid flow is governed by the momentm eqation ρ + ρv t o dp + µ σb dx o () where ρ µ and σ are respectively the density the coefficient of viscosity and the electrical condctivity of the flid. To find the temperatre distribtion inside the flid we se the energy eqation [] T T T ρ c + ρcvo k + µ + σb t o () where c and k are respectively the specific heat capacity and the thermal condctivity of the flid. The second and third terms on the right-hand side represent the viscos and Jole dissipations respectively. The problem is simplified by writing the eqations in the non-dimensional form. The characteristic length is taken to be h and the characteristic time is ρh / µ while the characteristic velocity is µ / ρh. We define the following non-dimensional qantities x xˆ yˆ h y h zˆ z h ˆ ρh Pˆ µ Pρh µ t tµ ρh

3 7 S ρvoh / µ is the sction parameter Pr µc / k is the Prandtl nmber / σ Bo h / µ Ec µ ρ ch ( T T ) is the Eckert nmber Ha where Ha is the Hartmann nmber In terms of the above non-dimensional variables and parameters the basic Eqs. ()-() are written as (the "hats" will be dropped for convenience) + S t dp + Ha dx (3) T t T + S T Pr + Ec + EcHa (4) The initial and bondary conditions for the velocity become t y ± t > (5) and the initial and bondary conditions for the temperatre are given by t : T t > : T y + T y. (6) Analytical soltion of the eqations of motion Eqation (3) is the eqation of motion which if solved give the velocity field as fnctions of space and time. Eqation (3) is a linear inhomogeneos partial differential eqation which can be solved analytically sing the Laplace transform (LT) method nder the initial and bondary conditions given by Eq. (5). Taking the LT of Eq. (3) gives d U ( y du ( y S K( U ( y F( (7) dy dy where U(yL((yt)) -F( is the LT of the pressre gradient and soltion of Eq. (7) with y as an independent variable is given as K ( Ha + s. The U ( y F( sinh( S / ) sinh( qy) cosh( S / ) cosh( qy) + exp( Sy / ) K sinh( q) cosh( q) where q S / 4 + K. Using the complex inversion formla and the reside theorem [] the inverse transform of U(y is determined as ( y t) C n PN + α PN + α ( ( exp( PN xt) exp( αt) ) + ( exp( PN xt) exp( αt) )

4 8 3 + ( exp( PN ) exp( )) 4 3xt αt + ( exp( PN4xt) exp( αt) ) ) (8) PN3 + α PN4 + α where dp dx C exp( αt) PN PN NN PN 3 PN4 NN / / Ha NN 3 + PN 3 4 Ha NN 3 + PN NN4 Ha + PN3 Ha NN 4 + PN 4 NN π ( n ) S NN π ( n.5) S n / 4 / 4 NN3 π ( ) ( n ) exp( Sy / ) sinh( S / ) sin( π ( n ) y) n+ NN4 π ( ) ( n.5) exp( Sy / ) cosh( S / ) cos( π ( n.5) y) Nmerical Soltion of the Energy Eqation The exact soltion of the eqation of motion given by Eq. (8) determines the velocity field for different vales of the parameters Ha and S. The vales of the velocity components when sbstitted in the right-hand side of the inhomogeneos energy eqation (4) make it too difficlt to solve analytically. he energy eqation is to be solved nmerically with the initial and bondary conditions given by Eq. (6) sing finite differences [3]. The Crank-Nicolson implicit method is applied. The finite difference eqations are written at the mid-point of the comptational cell and the different terms are replaced by their second-order central difference approximations in the y-direction. The diffsion term is replaced by the average of the central differences at two sccessive time levels. The viscos and Jole dissipation terms are evalated sing the velocity components and their derivatives in the y-direction which are obtained from the exact soltion. Finally the block tri-diagonal system is solved sing Thomas' algorithm. Unlike the velocity the temperatre distribtion depends on C. All calclations have been carried ot for C α Pr and Ec..

5 9 RESULTS AND DISCUSSION Figre presents the velocity and temperatre distribtions as fnctions of y for different vales of the time starting from t to the steady state. Figres a and b are evalated for Ha and S. The velocity crves are asymmetric abot the y plane becase of the sction as shown in Fig. a. It is observed that the velocity component decreases monotonically with time althogh the temperatre T increases monotonically with t y (a) t.5 t t.5 T y (b) t.5 t t Fig.. - Time development of the profile of: (a) ; and (b) T (Ha and S)

6 t (a) Ha Ha Ha3 T t (b) Ha Ha Ha3 Fig.. - Effect of Ha on the time variation of: (a) at y; (b) T at y. (S) Figre shows the effect of the Hartmann nmber Ha on the time development of the velocity and temperatre T at the centre of the channel (y). In this figre S (sction sppressed). It is clear from Fig. 3a that increasing the parameter Ha decreases and its steady state time. This is de to increasing the magnetic damping force on. Figre b and Table indicate that increasing Ha increases T at small times bt decreases it at large times. This can be attribted to the fact that for small times is small and an increase in Ha increases the Jole dissipation which is proportional to Ha and therefore the temperatre increases. For large times increasing Ha decreases and in trn decreases the Jole and viscos dissipations and conseqently decreases T. This acconts for crossing the crves of T with time for varios vales of Ha. Table. - Time variation of the temperatre at y for varios vales Ha (S). T t. t.4 T.6 t.8 t t. t.4 t.6 t.8 t Ha Ha Ha

7 t (a) S S S T t (b) S S S Fig Effect of S on the time variation of: (a) at y; (b) T at y. (Ha) Figre 3 shows the effect of the sction parameter on the time development of the velocity and temperatre T at the centre of the channel (y). In this figre Ha (hydrodynamic case). In Fig. 3a it is observed that increasing the sction decreases the velocity at the center and its steady state time de to the convection of flid from regions in the lower half to the center which has higher flid speed. In Fig. 3b the temperatre at the center is affected more by the convection term which pmps the flid from the cold lower half towards the centre. CONCLUSION The nsteady Hartmann flow of a condcting flid nder the inflence of an applied niform magnetic field and an exponential decaying pressre gradient has been stdied in the presence of niform sction and injection. The effect of the magnetic field and the sction and injection velocity on both the velocity and temperatre distribtions has been investigated. It is fond that the magnetic field has a marked effect on the velocity distribtion more than its

8 effect on the temperatre distribtion. On the other hand the sction and injection velocity has a more apparent effect on the temperatre distribtion than on the velocity distribtion. It is of interest to see that the effect of the magnetic field on the temperatre at the center of the channel depends on time. For small time increasing the magnetic field increases the temperatre however for large time increasing the magnetic field decreases the temperatre. References: [] HARTMANN J. and LAZARUS F. (937): Kgl. Danske Videnskab. Selskab Mat.-Fys. Medd. 5 (67). [] TAO I. N. (96): Magnetohydrodynamic effects on the formation of Coette flow. J. of Aerospace Sci [3] ALPHER R. A. (96): Heat transfer in magnetohydrodynamic flow between parallel plates. Int. J. Heat and Mass Transfer [4] SUTTON G. W. and SHERMAN A. (965): Engineering Magnetohydrodynamics. McGraw-Hill Book Co. [5] CRAMER K. R. and PAI S.-I. (973): Magnetoflid dynamics for engineers and applied physicists. McGraw-Hill Book Co. [6] NIGAM S. D. and SINGH S.N. (96): Qart. J. Mech. Appl. Math [7] TANI I. (96): Steady motion of condcting flids in channels nder transverse magnetic fields with consideration of Hall effect. J. of Aerospace Sci [8] SOUNDALGEKAR V. M. VIGHNESAM N. V. and TAKHAR H. S. (979): Hall and Ion-slip effects in MHD Coettee flow with heat transfer. IEEE Trans. Plasma Sci. PS-7 (3) 78. [9] SOUNDALGEKAR V. M. and UPLEKAR A. G. (986): IEEE Trans. Plasma Sci. PS-4 (5) 579. [] ATTIA H. A. (999): Transient MHD flow and heat transfer between two parallel plates with temperatre dependent viscosity. Mech. Res. Comm. 6 () 5. [] SCHLICHTING H.: Bondary layer theory. McGraw-Hill Book Co. (986). [] SEGEL M. R. (986): Theory and problems of Laplace transform. McGraw-Hill Book Co. [3] AMES W. F. (977): Nmerical soltions of partial differential eqations nd ed. Academic Press New York.