ON THE EFFECTIVENESS OF POROSITY ON UNSTEADY COUETTE FLOW AND HEAT TRANSFER BETWEEN PARALLEL POROUS PLATES WITH EXPONENTIAL DECAYING PRESSURE GRADIENT

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1 17 Kragujevac J. Sci. 8 (006) ON THE EFFECTIVENESS OF POROSITY ON UNSTEADY COUETTE FLOW AND HEAT TRANSFER BETWEEN PARALLEL POROUS PLATES WITH EXPONENTIAL DECAYING PRESSURE GRADIENT Hazem Ali Attia Department f Mathematics, Cllege f Science, King Saud University (Al-Qasseem Branch), P.O. Bx 37, Buraidah 81999, KSA ah1113@yah.cm (Received December 9, 005) ABSTRACT. The unsteady Cuette flw thrugh a prus medium f a viscus, incmpressible fluid bunded by tw parallel prus plates is studied with heat transfer. A unifrm suctin and injectin are applied perpendicular t the plates while the fluid mtin is subjected t an expnential decaying pressure gradient. The tw plates are kept at different but cnstant temperatures while the viscus dissipatin is included in the energy equatin. The effect f the prsity and the unifrm suctin and injectin n bth the velcity and temperature distributins is examined. 1. INTRODUCTION The flw between tw parallel plates is a classical prblem that has many applicatins in acceleratrs, aerdynamic heating, electrstatic precipitatin, plymer technlgy, petrleum industry, purificatin f crude il, fluid drplets and sprays, magnethydrdynamic (MHD) pwer generatrs and MHD pumps. Hartmann and Lazarus [1] studied the influence f a transverse unifrm magnetic field n the flw f a cnducting fluid between tw infinite parallel, statinary, and insulated plates. Then, a lt f research wrk cncerning the Hartmann flw has been btained under different physical effects [-10]. In the present study, the unsteady Cuette flw and heat transfer in a prus medium f an incmpressible, viscus, fluid between tw infinite hrizntal prus plates are studied. The fluid is acted upn by an expnential decaying pressure gradient, and a unifrm suctin and injectin perpendicular t the plates. The upper plate is mving with a cnstant velcity while the lwer plate is kept statinary. The flw in the prus media deals with the analysis in which the differential equatin gverning the fluid mtin is based n the Darcy s law which accunts fr the drag exerted by the prus medium [11-13]. The tw plates are maintained at tw different but cnstant temperatures. This

2 18 cnfiguratin is a gd apprximatin f sme practical situatins such as heat exchangers, flw meters, and pipes that cnnect system cmpnents. The cling f these devices can be achieved by utilizing a prus surface thrugh which a clant, either a liquid r gas, is frced. Therefre, the results btained here are imprtant fr the design f the wall and the cling arrangements f these devices. The gverning equatins are slved numerically taking the viscus dissipatin int cnsideratin. The effect f the prsity and the suctin and injectin n bth the velcity and temperature distributins is studied.. DESCRIPTION OF THE PROBLEM The tw nn-cnducting plates are lcated at the y=±h planes and extend frm x=- t and z=- t. The lwer and upper plates are kept at the tw cnstant temperatures T 1 and T, respectively, where T >T 1. The fluid flws between the tw plates under the influence f an expnential decaying pressure gradient dp/dx in the x- directin, and a unifrm suctin frm abve and injectin frm belw which are applied at t=0. The upper plate is mving with a cnstant velcity U while the lwer plate is kept statinary. The flw is thrugh a prus medium where the Darcy mdel is assumed [13]. Frm the gemetry f the prblem, it is evident that / x= / z=0 fr all quantities. The velcity vectr f the fluid is r r r v( y, t) = u( y, t) i + v j with the initial and bundary cnditins u=0 at t 0, and u=0 at y=-h, and u=u at y=h fr t>0. The temperature T(y,t) at any pint in the fluid satisfies bth the initial and bundary cnditins T=T 1 at t 0, T=T at y=+h, and T=T 1 at y=-h fr t>0. The fluid flw is gverned by the mmentum equatin ρ + ρv = dp u µ + µ u dx K (1) where ρ and µ are, respectively, the density and the cefficient f viscsity and K is the Darcy permeability [11-13]. T find the temperature distributin inside the fluid we use the energy equatin [14] T ρ c + ρcv = k + µ, () where c and k are, respectively, the specific heat capacity and the thermal cnductivity f the fluid. The secnd term n the right-hand side represents the viscus dissipatin. The prblem is simplified by writing the equatins in the nn-dimensinal frm. We define the fllwing nn-dimensinal quantities x y z u ˆ, ˆ, ˆ, ˆ, ˆ P tu x = y = z = u = P =, t =, h h h U U h ρ Re = ρ hu / µ, is the Reynlds number, S = v /, U is the suctin parameter,

3 19 Pr = µc / k is the Prandtl number, Ec = U c( T T ) is the Eckert number, / 1 M = h / K is the prsity parameter. In terms f the abve nn-dimensinal variables and parameters, the basic Eqs. (1)- () are written as (the "hats" will be drpped fr cnvenience) + S = dp dx + 1 Re u M u, Re (3) + S = 1 Re Pr T Ec +, Re (4) The initial and bundary cnditins fr the velcity becme t 0 : u = 0, t > 0 : u = 0, y = 1, u = 1, y = 1 (5) and the initial and bundary cnditins fr the temperature are given by t 0 : T = 0, t > 0 : T = 1, y = + 1, T = 0, y = 1. (6) where the pressure gradient is assumed in the frm αt dp / dx = Ce. 3. NUMERICAL SOLUTION OF THE GOVERNING EQUATIONS Equatins (3) and (4) are slved numerically using finite differences [15] under the initial and bundary cnditins (5) and (6) t determine the velcity and temperature distributins fr different values f the parameters M and S. The Crank-Niclsn implicit methd is applied. The finite difference equatins are written at the mid-pint f the cmputatinal cell and the different terms are replaced by their secnd-rder central difference apprximatins in the y-directin. The diffusin term is replaced by the average f the central differences at tw successive time levels. Finally, the blck tri-diagnal system is slved using Thmas' algrithm. All calculatins have been carried ut fr C=- 5, α=1, Pr=1 and Ec= RESULTS AND DISCUSSION Figure 1 presents the velcity and temperature distributins as functins f y fr different values f the time starting frm t=0 t the steady state. Figures 1a and 1b are evaluated fr M=1 and S=1. It is bserved that the velcity cmpnent u and temperature T d nt reach the steady state mntnically. They increase with time up t a maximum value and then decrease as time develps up till their steady state values. Figure shws the effect f the prsity parameter M n the time develpment f the velcity u and temperature T at the centre f the channel (y=0). In this figure, S=0 (suctin suppressed). It is clear frm Fig. a that increasing the parameter M decreases u and its steady state time. This is due t increasing the prsity damping frce n u. Figure b indicates that increasing M decreases T and its steady state time. This can be attributed

4 0 t the fact that increasing M decreases u and, in turn, decreases the viscus dissipatin which decreases T. Figure 3 shws the effect f the suctin parameter n the time develpment f the velcity u and temperature T at the centre f the channel (y=0). In this figure, M=0. In Fig. 3a, it is bserved that increasing the suctin decreases the velcity u at the center and its steady state time due t the cnvectin f fluid frm regins in the lwer half t the center, which has higher fluid speed. In Fig. 3b, the temperature at the center is affected mre by the cnvectin term, which pumps the fluid frm the cld lwer half twards the centre. 5. CONCLUSION The unsteady Cuette flw thrugh a prus medium f a viscus incmpressible fluid has been studied in the presence f unifrm suctin and injectin. The effect f the prsity and the suctin and injectin velcity n the velcity and temperature distributins has been investigated. It is fund that bth the prsity and suctin r injectin velcity has a marked effect n bth the velcity and temperature distributins. References [1] J. Hartmann and F. Lazarus (1937): Kgl. Danske Videnskab. Selskab, Mat.-Fys. Medd. 15 (6,7) [] I.N. Ta (1960): J. f Aerspace Sci. 7, 334 [3] R.A. Alpher (1961): Int. J. Heat and Mass Transfer. 3, 108 [4] G.W. Suttn and A. Sherman (1965): Engineering Magnethydrdynamics. McGraw-Hill Bk C. [5] K. Cramer and S.-I. Pai (1973): Magnetfluid dynamics fr engineers and applied physicists. McGraw-Hill Bk C. [6] S.D. Nigam and S.N. Singh (1960): Quart. J. Mech. Appl. Math. 13, 85 [7] I. Tani (196): J. f Aerspace Sci. 9, 87 [8] V.M. Sundalgekar, N.V. Vighnesam, and H.S. Takhar (1979): IEEE Trans. Plasma Sci. PS-7 (3), 178 [9] V.M. Sundalgekar and A.G. Uplekar (1986): IEEE Trans. Plasma Sci. PS-14 (5), 579 [10] H.A. Attia (1999): Mech. Res. Cmm. 6(1), 115 [11] D.D. Jseph, D.A. Nield, and G. Papaniclau (198): Water Resurces Research. 18 (4), 1049 [1] D.B. Ingham and I. Pp (00): Transprt phenmena in prus media. Pergamn, Oxfrd. [13] A.R.A. Khaled and K. Vafai (003): Int. J. Heat Mass Transf. 46, 4989 [14] H. Schlichting (1968): Bundary layer thery. McGraw-Hill Bk C.

5 [15] W.F. Ames (1977): Numerical slutins f partial differential equatins. nd ed., Academic Press, New Yrk. 1