Oscillatory Mixed Convention in a Porous Medium

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1 Journl of Applied Fluid Mechnics, Vol. 7, No., pp. -5,. Avilble online t ISSN 75-57, EISSN Oscilltory Mixed Convention in Porous Medium M. Jn, S. L. Mji, S. Ds, R. N. Jn nd S. K. Ghosh Deprtment of Applied Mthemtics, Vidysgr University, Midnpore 7, West Bengl, Indi Deprtment of Mthemtics, University of Gour Bng, Mld 7, West Bengl, Indi Deprtment of Mthemtics, Nrjole Rj College, Nrjole, Midnpore 7, West Bengl, Indi Corresponding Author Emil: g - swpn@yhoo.com (Received April 6, ; ccepted Februry, ) ABSRAC Oscillting mixed convection on the fully developed flow between two infinitely long verticl wlls heted symmetriclly in porous medium, hs been studied. It experiences g -jitter force ssocited with microgrvity in the field of spce science with reference to crystl growth in spce. Our present study cn be exposed to rel life sitution of microgrvity field. he time vrying grvity field genertes oscilltory free convection velocity field. his is combined with the forced oscillting flow driven by pressure grdient. Keywords: g -jitter forces, Porous medium, Drcy number, Criticl Grshof number, Free convection nd forced convection. NOMENCLAURE defined in eqution (5) P dimensionless pressure D kl / F ul i Rel( Fe ) Drcy number Dimensionless velocity / component rel prt of i Fe r / t temperture rtio prmeter time temperture t the entrnce region gt () g -jitter or time vrying grvity field fluid temperture g mgnitude of g -jitter, Gr plte temperture t the cold wll y = nd the hot wll y L gl ( )/ Grshof number u velocity component in x direction Gr criticl vlue of Grshof number u i complex quntity u k permebility of porous medium x, y, z velocity scle dimensionless velocity Crtesin co-ordintes L M width of the chnnel L dp/ dx / dimensionless pressure grdient = / viscosity rtio Coefficient of therml expnsion n Frequency prmeter yl / non-dimensionlized width of the chnnel p pressure grdient in x -direction = / temperture defined in eqution () fluid density dimensionless

2 M. Jn et l. / JAFM, Vol. 7, No., pp. -5,. viscosity of the fluid, sher stresses t the cold wll. INRODUCION effective viscosity of the fluid sturted by porous medium kinemtic viscosity An excellence of study of porous medium flow is the subject motivted by severl importnt pplictions in science nd engineering. Due to its brod rnge of pplictions in science nd industry, this field hs gined extensive ttention ltely. In brod sense, the study of porous medium embrces fluid nd therml sciences, geotherml, petroleum nd combustion engineering. Over the pst decdes, severl studies hve been tken into ccount of porous medium flow with reference to Nield nd Bejn (6), Bejn (97), Kviny (995), Vfi nd Kim (995) nd Inghm nd Pop (). All these uthors hve been studied in prospective wy to determine its flow behvior of severl spects of fluid engineering,therml science nd combustion engineering. However severl studies hve been tken into ccount of porous medium flow. A literture survey revels to study of Debruge nd Hn (97), Du nd Bilgen(99), Hossin nd Wilson (), Guri et l. (9), Beg et l. (), Chmkh (997), Stpthy et l. (998), Chmkh et l. (), Rptis nd Perdikis () nd Jn et l.(). Our present study dels with mixed convection with symmetric heting of the wll embedded in porous medium with reference to study of Aung nd Worku (986). Aung nd Worku (986) studied mixed convection flow in the bsence of Drcy number. he study of mixed convection flow in the presence of mgnetic field hs been developed by Ghosh nd Nndi (), Ghosh et l.() nd Guri et l.(7). A recent study hs been developed with g -jitter force to exert its influence of mixed convection flow with symmetric heting of the wll in the presence of mgnetic field. his hs been studied by Bo Pn nd Ben Q (998). However, trnsverse mgnetic field effect on g - jitter driven flow ssocited with oscillting pressure grdient in chnnel hs been studied by Lehoczky et l. (99), Antr nd Nuotio - Antr (99) nd Nelson (99). Resently, trnsient pproch to rditive het trnsfer free covection flow with rmped wll temperture hs been nlyzed by Ptr et l. (). he im of the present investigtion is to del with the study of n oscillting mixed convection driven by g - jitter fores ssocited with microgrvity field with decisive importnce to highly permeble medium. A g - jitter driven flow in cvity bers direct relevnce to crystl growth in spce so tht the net mss flow rte of the system is zero. he better model cn be considered by representing the flow system of long prllel plte chnnel with significnt effect of pressure grdient. he prcticl ppliction of unstediness nd g - jitter force is of gret importnce to microgrvity field in tking into ccount of spce fluid system design. In relistic sitution, the co- reltion of unstediness nd g - jitter force becomes relevent to time vrying grvity field driven by time hrmonic g - jitter components with frequency of oscilltion nd the flow t the entrnce nl / nd the hot wll dimensionless frequency of oscilltion lso oscilltes becuse of n pplied pressure grdient. he representtion of g -jitter forces is the time vrying grvity field to generte oscilltory free convection velocity field. his is combined with the forced oscillting flow driven by pressure grdient. he importnce of study of this problem hs mny pplictions of geotherml, combustion engineering nd strospce science. We hve considered g -jitter forces driven by oscillting mixed convection of viscous incompressible fluid flow in porous medium between two infinitely long verticl wlls. Here we considered the Brinkmn eqution subject to the limit k tht gives Drcy flow nd the limit k gives the viscouss flow, k being the permebility of the porous medium. It is found tht the velocity decreses ner the chnnel wlls while it increses t the middle of the chnnel with n increse in Drcy number D. he criticl vlue of the buoyncy force Gr for which there is no flow reversl ner the cold wll nd it decreses with n increse in Drcy number D.. MAHEMAICAL FORMULAION AND IS SOLUION Consider the mixed convection of viscous incompressible fluid flow in porous medium between two infinitely long verticl wlls seprted by distnce L. he fluid is t temperture nd flows due to temperture nd pressure grdient in between two prllel wlls. he wlls re infinitely long long x -xis nd z -direction with different wll tempertures [see Fig.]. We consider the fully developed unstedy flow in porous medium. We ssume tht the g-jitter field under considertion is sptilly constnt nd otherwise vries with time hrmoniclly. Fig.. Geometry of the problem

3 M. Jn et l. / JAFM, Vol. 7, No., pp. -5,. Further, the flow t the entrnce of the chnnel lso oscilltes hrmoniclly due to n pplied pressure grdient. An oscilltory free convection velocity will generte becuse of the vrition of the grvity field. hus, the problem becomes free nd forced convection oscillting flow in the presence of pressure grdient. he eqution of motion under the usul Boussinesq pproximtion is u p d u = g( t) ( ) u, t x dy k () where k the permebility of the porous medium, the therml expnsion co-efficient, the fluid viscosity, the effective viscosity of the fluid sturted by porous medium nd the fluid density. he energy eqution is d =. dy he velocity nd temperture boundry conditions re () u t y nd u t y L, () t y nd = t y L. () Assuming u u ( y) e, g( t) g e, p P( x) e, (5) i nt i nt i nt the Eq. () becomes dp d u i n u = g ( ) u. dx dy k Introducing non-dimensionl vribles y L u L =,, F, Eqs. (6) nd Eq. () become df i F = ( Gr ), d MD d =, d gl ( ) where Gr is the Grshof number, = nl the frequency prmeter, = L dp dx the non-dimentionl pressure grdient, M = the k viscosity rtio nd D = the Drcy number. It my L be noted tht the limit D gives cler fluid nd (6) (7) (8) (9) the limit D gives the unmitigted Drcy flow. he corresponding boundry conditions for F( ) nd ( ) re F() = = F (), () () = r, () =, () where r is temperture rtio prmeter. he solution of the Eq. (9) subject to the boundry conditions Eq. () is ( ) = ( r ) r. () On the use of Eq. (), the Eq. (8) becomes i F [( Gr r) Gr( r) ]. d M D df () he solution of the Eq. () subject to the boundry conditions Eq. () is sinh sinh ( ) F( ) ( Gr r ) sinh sinh sinh Gr ( r ), sinh where i i, MD, =. ( M D ) M D () (5) he pressure grdient is determined by requiring tht the flow stisfies the following condition, F( ) d =. On the use of the Eq. () in the bove eqution, the mss conservtion eqution for s sinh Gr( r ). sinh ( cosh ) (6) If the rte of mss flow is zero, the pressure pressure grdient reduces to Gr ( r ). (7) On the use of Eq. (6), the Eq. () becomes sinh F( ) = ( ) Gr r sinh (cosh ) sinh sinh ( ) sinh Gr( r ). sinh sinh sinh It is interesting to note tht the velocity distribution given by Eq. (8) for Gr (the pure forced (8) 5

4 M. Jn et l. / JAFM, Vol. 7, No., pp. -5,. convection) t ny vlues of r nd for r = for ny vlues of Gr re identicl nd the solution for F( ) given by Eq. (8), reduces to F( ) sinh sinh ( cosh ) sinh sinh ( ) sinh sinh (9) In free convection process, substituting into the solution Eq. (), we hve sinh sinh ( ) ( ) = Gr F r sinh sinh sinh ( r ). sinh () In the limit D (for cler fluid), nd r = the Eq. () becomes F ( ) = Gr. () In the vicinity of the centrl section of the chnnel, the velocity is given by Eq. () reduces to F( ) = Gr. () 8 We shll now discuss some limiting cses:. Soluton D >> nd << he velocity distribution nd the pressure grdient re given by Eq. () nd Eq. (6) respectively, become F Gr r 6 i 6 5 D Gr r nd i Gr r. D In the limit D, the Eq. () becomes F Gr r i = Gr r () () (5) he velocity distribution given by Eq. () is dependent on Grshof number, Drcy number nd frequency of oscilltion. For r = ( uniform temperture vrition long the wll) the velocity distribution is closely resemblnce to forced convection flow due to bsence of Grshof number. Hence the velocity distribution is independent of Grshof number but dependent on Drcy number nd frequency of oscilltion. he expression for pressure grdient given by Eq. () plys significnt role with reference to temperture rtio prmeter r so tht the effect of buoyncy force becomes predominnt for the constnt rte of flow. he expression for the velocity distribution given by Eq. (5) leds to the cler fluid nd the buoyncy force ccelertes the entire flow sitution on the velocity field.. Soluton for n oscillting mixed convection with Drcy number D (cler fluid) he Eq. () together with the expression Eq.(6) yields F sinh = i i i i sinh i cosh i sinh i sinh i sinh i sinh i sinh i sinh i Gr r. sinh i sinh i If frequency of the oscilltions is very lrge, the bove Eq. (6) becomes i i i ( ) i F( ) e e i ( ) i Gr r e e. i At the limit of, we hve (7) i i ( ) lim F( ) = e e. (8) If the frequency of oscilltions is very lrge the expression for the velocity given by Eq. (7) is n exponentil in nture with the frequency of oscilltions nd Grshof number plys n importnt role in determining the flow behvior. In this sitution, the frequency of oscilltions survives exponentilly on the velocity field with significnt effect of buoyncy force. he Eq. (8) shows tht thin stokes lyer form ner the wlls for lrge frequency of the oscillting flows. he thickness of the lyer is (6) O / 6

5 M. Jn et l. / JAFM, Vol. 7, No., pp. -5,. corresponding to decrese with n increse in frequency of oscilltions.. RESULS AND DISCUSSION he quntity plotted long the verticl xis is i Img( Fe ) Rel( F)cos Img( F) sin, which represents the mesurble vlue for driving i force g Im g ( ge ) g sin nd uo ( ) u sin, where Img stnds for imginry prt. It is noted tht form Fig. tht t round nd the velocity profile oscilttes long the width of the chnnel but for other times the velocity profile is pproximtely prbolic, s expcted for the symetriclly heted wlls r nd for forecd convection in chnnel. It is evident fron Fig. tht the profiles re skewed ner the cold wll s expected for higher buoyncy while the velocity profiles re of oscilltory chrcter ner the cold wll due to symmetric heting of the wll wheres the velocity profiles oscillte t round nd long the width of the chnnel. unifrom temperture distribution (symmetriclly heted wll) when r. In the cse of symmetric heting of the wll the velocities increse t the cold wll while the velocities decrese t the hot wll with increse in r wheres t ech vlue of r, every profile coincides with the mid plne of the centrl section of the chnnel. Fig.. Velocity profile for with r, Gr, D.5,, M Fig.. Velocity profile for r with D =.5, Gr =, =, =, M It is noticed from Fig. 5 tht the effect of Drcy number leds to increse the velocities ner the cold wll s well s ner the hot wll while the effect of Drcy number becomes predominnt over the centrl region. In the centrl region, the velocities re mrkedly incresed with increse in Drcy number. his sitution revels tht the velocities depend on permebility of the porous medium to boost up the velocities in the centrl region for moderte vlues of Drcy number. Fig. 6 revels tht the velocity profile becomes prbolic when the buoyncy force is bsent Gr. Fig.. Velocity profile for with r., Gr, D.5,, M Fig. demonstrtes tht t higher buoyncy Gr the profile becomes prbolic t r. his indictes tht the flow is chrcterised by Fig.5. Velocity profile for D with r =., Gr =, =, =, M It is seen from Fig. 6 tht the velocity increses ner the hot wll with increse in Grshof number Gr while flow reversl occurs ner the cold wll with increse in 7

6 M. Jn et l. / JAFM, Vol. 7, No., pp. -5,. Grshof number Gr with reference to higher buonyncy Gr. he nture of velocity profile on Fig. 6 is closely resemblnce to the study of Aung nd Worku (986). of the porous medium plys n importnt roll in determining the nture of sher stress when the frequency of oscilltion is tken into ccount. In turn the effects of sher stresses become relevnt to the cse of n inverse permebility of the porous medium propelled by the viscosity of the fluid so tht sher stresses increse with increse in frequency prmeter. It is noticed form Fig. 8 tht the sher stress t the wll nd the mgnitude of the sher stress t the wll decreses with n increse in Drcy number D. hus it rrives t n interesting conclusion tht the permebility of the porous medium leds to slow down the prticle so tht sher stresses decrese with increse on Drcy number. Further, for fixed vlue of nd D, stedily increses while mgnitude of stedily decreses with n increse in temperture rtio prmeter r. Fig.6. Velocity profile for Gr with r.,, D.5,, M he sher stresses t the cold wll nd the hot wll re respectively given by df = Rel d where = e i nd df i = Rel e, d = df d sinh ( cosh ) Gr( r ) ( cosh ) sinh, (9) Fig.7. Sher stresses nd t the wlls =, = for with D =.5, Gr =, df d sinh (cosh ) Gr( r ) sinh ( cosh ) () where sinh sinh ( cosh ) () nd is given by Eq. (5). he numericl vlues of the sher stresses nd t the wll nd re shown in the Figs.7-8 ginst r for severl vlues of nd D when M. Figure 7 shows tht the sher stress t the wll nd the mgnitude of the sher stress t the wll increses with n increse in frequency prmeter. It is remrked here tht the permebility Fig. 8. Sher stresses nd t the wlls =, = for D with Gr,, 8

7 M. Jn et l. / JAFM, Vol. 7, No., pp. -5,. he criticl vlue of Gr for which there is no flow reversl ner the cold wll = to obtined from df d = =, which is ( cosh ) Gr Rel, ( r ) ( cosh ) sinh () where is given by Eq. (5) nd the corresponding vlue of the criticl Grshof number t the hot wll is given by Gr Gr. he vlues of the criticl Grshof number Gr re entered in ble for different vlues of D nd r. It is seen from ble tht the criticl Grshof number Gr t the wll increses with increse in the temperture rtio prmeter r wheres it decreses with increse in the Drcy number D. Although buoyncy force ccelertes the fluid prticle to become flow reversl t higher buoyncy [see Fig. 6]. Criticl Grshof number is obtined where no flow reversl occurs t the cold wll but tendency of incresing remins unltered. ble Vrition of Gr r \ D he vlues of the pressure grdient re entered in ble for different vlues of D nd r. It is seen from ble tht the pressure grdient decreses with increse in both the temperture rtio prmeter r nd Drcy number D. ble Vrition of r \ D CONCLUSION Oscilltory mixed convection viscous incompressible flow in porous medium between two infinitely long verticl wlls is considered. It is found tht the velocity profile is lmost prbolic in nture for the smll vlue of buoyncy force Gr while the skewness is chrcterised by the higher buoyncy nd t lrge vlue of Gr there rise flow reversl ner the cold wll. It is found tht the velocity profile increses ner the cold wll of the chnnel while it decreses ner the hot wll of tht chnnel with increse in temperture rtio prmeter r. In the presence of Drcy number, the mgnitude of the sher stresses t the wll increses while tht of the sher stress t the wll = decreses with increse in r. he criticl Grshof number, for which there is no flow reversl ner the wll nd it decreses with increse in Drcy number. REFERENCES Aung, W. nd G. Worku (986). heory of fully developed, combined convection including flow reversl. Journl of Het rnsfer, 8, Beg, O.A., S.K. Ghosh nd M. Nrhri (). Mthemticl modeling of oscilltory MHD Couette flow in rotting highly permeble medium permeted by n oblique mgnetic field. Chemicl Engineering nd Communictions, 98, 5-5. Bejn, A.(99). Convection het trnsfer, Weiley, New York. Chmk, A.J. (997). Solr rdition ssisted nturl convection in uniform porous medium supported by verticl flt plte. ASME J. Het rnsfer, 9, Chmk, A.J., H.S.khr nd O.A. Beg (). Rditive free convective non-newtonin fluid flow pst wedge embedded in porous medium. Int. J. Fluid Mechnics Reserch,,-5. Debruge, L.L. nd L.S. Hn (97). Het trnfer in chnnel with porous wll for turbine cooling ppliction. J. Het rnsfer, 9, Du, Z.-G. nd E.Bilgen (99). Nturl convection in verticl cvities with internl het generting porous medium. Wärme-und Stoffübertr, 7, Ghosh, S.K. nd D.K. Nndi (). Mgnetohydrodynmic fully developed combined convection flow between verticl pltes heted symmetriclly. J. echnicl Physics,, Ghosh, S.K., I. Pop nd D.K. Nndi (). MHD fully developed mixed convection flow with symmetric heting of the wll. Int. J. Applied Mechnics nd Engineering, 7, -8. Guri, M., B.K. Ds, R.N. Jn nd S.K. Ghosh (7). Effects of wll conductnce on MHD fully developed flow with symmetric heting of the wll. Int. J. Fluid Mechnics Reserch,, 5-5. Guri, M., R.N. Jn, S.K. Ghosh nd I. Pop (9). hree dimensionl free convection flow in verticl chnnel filled with porous medium. Journl of Porous Medi,, Hossin, M.A. nd M. Wilson (). Nturl convection flow in fluid-sturted porous medium enclosed by non-isotherml wlls with 9

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