Unsteady Couette Flow through a Porous Medium in a Rotating System

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1 Ope Joural of Fluid Dyamics, 0,, Published Olie December 0 ( Usteady Couette Flow through a Porous Medium i a Rotatig System Maitree Jaa, Saata Das, Rabidra Nath Jaa Departmet of Applied Mathematics, Vidyasagar Uiversity, Midapore, Idia Departmet of Mathematics, Uiversity of Gour Baga, Eglish Bazar, Idia jaa67@yahoo.co.i Received October 9, 0; revised November 3, 0; accepted December 5, 0 ABSTRACT A ivestigatio has bee made o a usteady Couette flow of a viscous icompressible fluid through a porous medium i a rotatig system. The solutio of the goverig equatios has bee obtaied by the use of Laplace trasform techique. It is foud that the primary velocity decreases ad the magitude of the secodary velocity icreases with a icrease i rotatio parameter. The fluid velocity compoets are decelerated by a icrease of Reyolds umber. A icrease i porosity parameter leads to icrease the primary velocity ad the magitude of the secodary velocity. It is also foud that the solutio for small time coverges more rapidly tha the geeral solutio. The asymptotic behavior of the solutio is aalyzed for small as well as large values of rotatio parameter ad Reyolds umber. It is observed that a thi boudary layer is formed ear the movig plate of the chael ad the thickesses of the boudary layer icreases with a icrease i porosity parameter. Keywords: Couette Flow; Rotatio Parameter; Reyolds Number; Porous Medium; Rotatig System ad Boudary Layer. Itroductio The flow betwee two parallel plates is a classical problem that has may applicatios i accelerators, aerodyamic heatig, electrostatic precipitatio, polymer techology, petroleum idustry, purificatio of crude oil, fluid droplets ad sprays. Such a flow model is of great iterest, ot oly for its theoretical sigificace, but also for its wide applicatios to geophysics ad egieerig. A lot of research work cocerig the flow betwee two parallel plates studied i a rotatig system have appeared, for eample, Batchelor [], Gaapathy [3], Gupta [4] ad Mazumder [5]. The flows through porous medium are very much prevalet i ature ad therefore, the study of such flows has become of pricipal iterest i may scietific ad egieerig applicatios. This type of flows has show their great importace i petroleum egieerig to study the movemets of atural gas, oil ad water through the oil reservoirs; i chemical egieerig for the filtratio ad water purificatio processes. Further, to study the udergroud water resources ad seepage of water i river beds oe eed the kowledge of the fluid flow through porous medium. Therefore, there are umber of practical uses of the fluid flow through porous media. Rotatio has a immese importace i various pheomea such as i cosmical fluid dyamics, meteor- ology, geophysical fluid dyamics, gaseous ad uclear reactors ad may egieerig applicatios, that is why, the study of Couette flow through porous medium i a rotatig system ehaces a iterest to the researchers due to its applicatios i the aforesaid area. Such a study has a greater importace i the desig of turbies ad turbo mechaics, i estimatig the flight path of rotatig wheels ad spi-stabilized missiles. A large umber of ivestigatios has bee made o the flow through a porous medium i a rotatig system. I geeral, most of solutios for usteady flows of viscous fluids are i a series form. These series may be rapidly coverget for large values of the time but slowly coverget for small values of the time or vice versa. Sometimes, it ca be difficult to obtai the solutio for small values of the time but it ca be easy to obtai it for large values of the time ad the opposite ca also be true. Vidyaidhi ad Nigam [6] studied the chael flow betwee rotatig parallel plates uder costat pressure gradiet. Jaa ad Dutta [7] studied the steady Couette flow of a viscous icompressible fluid betwee two ifiite parallel plates, oe statioary ad the other movig with uiform velocity, i a rotatig frame of referece. Sigh ad Sharma [8] have preseted the three dimesioal Couette flow through porous media. A periodic solutio of oscillatory Couette flow through a porous medium i rotatig sys- Copyright 0 SciRes.

2 50 M. JANA ET AL. tem has bee obtaied by Sigh et al. [9]. Guria et al. [0] have described the usteady Couette flow i a rotatig system. Das et al. [0] have studied the usteady Couette flow with a oscillatory velocity of oe of the plates i a rotatig system. The usteady MHD Couette flow i a rotatig system has bee ivestigated by Das et al. []. Attia [3] has studied the effect of porosity o usteady Couette flow with heat trasfer i the presece of uiform suctio ad ijectio. Israel-Cookey et al. [4] have preseted the MHD oscillatory Couette flow of a radiatig viscous fluid i a porous medium with periodic wall temperature. The usteady hydromagetic Couette flow through a porous medium i a rotatig system have bee preseted Prasad ad Kumar [5]. Das et al. [6] have studied the Couette flow through porous medium i a rotatig system. I the preset paper, we have studied the usteady Couette flow betwee two ifiite horizotal parallel plates i a porous medium i a rotatig system whe oe of the plate movig with uiform velocity ad the other oe held at rest. The fluid ad plates are i a state of rigid body rotatio with uiform agular velocity. The solutios for the velocity distributios as well as shear stresses have bee obtaied for small time as well as for large time by the Laplace trasform techique. It is foud that the primary velocity u decreases ad the magitude of the secodary velocity v icreases with a i- crease i rotatio parameter K. The primary velocity u ad the magitude of the secodary velocity v decrease with a icrease i Reyolds umber Re. A icrease i porosity parameter leads to icrease i the values of both the primary velocity u ad the magitude of the secodary velocity v. It is also foud that the solutio for small time coverges more rapidly tha the geeral solutio. For the steady state solutio, the asymptotic behavior of the solutio is aalysed for small as well as large values of rotatio parameter Reyolds umber Re. It is observed that a thi boudary layer is formed ear the statioary plate ad the thickesses of the boudary layer icreases with a icrease i porosity parameter.. Mathematical Formulatio ad Its Solutio K ad Cosider the usteady flow of a viscous icompressible fluid betwee two ifiite parallel porous plates embedded i a porous medium. The plates are separated by a distace h. The fluid ad chael rotate i uiso about a ais ormal to the plaes of the plates with a uiform agular velocity. Choose a Cartesia co-ordiate system with -ais alog the lower statioary plate i the directio of the flow, the y-ais is ormal to the plates ad the z-ais perpedicular to y-plae (see Figure ). Flow withi the chael is iduced due to the motio of the upper plate at y h parallel to itself i -directio with a uiform velocity u 0. Iitially, at time t 0, the fluid as well as the plates of the chael are assumed to be at rest. At time t 0 the upper plate at y h starts movig with uiform velocity u 0 alog -directio i its ow plae while the lower plate at y 0 is kept fied. The velocity compoets are uvw,, relative to a frame of referece rotatig with the fluid. Sice plates of the chael are ifiitely log alog ad y directios, all physical quatities will be fuctios of z ad t oly. v The equatio of cotiuity gives 0 which o ite- y gratio yields v costat v0, v0 0 for suctio ad v0 0 for the blowig at the plate. The -, y- ad z-compoets of Navier-Stokes equatio are u u u v0 w u, t y y k () p 0 v0, () y k w v w w 0 u, t y y k w (3) ad are respectively the fluid desity, the kiematic viscosity ad k the permeability of the porous medium. The iitial ad boudary coditios are u w0, vv0, for 0 yh, t 0, u w0, vv0, at y 0 for t 0, (4) u u, w0, vv, at y hfort Itroducig the o-dimesioal variables y u v t, u, w, (5) h u u h 0 0 Figure. Geometry of the problem. Copyright 0 SciRes.

3 M. JANA ET AL. 5 Equatios ( ) ad (3) become Takig the Laplace trasform, Equatio (8) becomes u u u Re K w u, (6) d q dq Re s ik q 0, d d w w w Re K u w, h K is the rotatio parameter ad vh 0 ku 0 Re the Reyolds umber ad the po- rosity parameter. Combig Equatios (6) ad (7), we have q q q Re i K q q, (7) (8) q u iw adi. (9) The iitial ad boudary coditios for q, are q,0 0 for 0 ad 0, (0) q 0, 0 ad q 0, for 0. () 0 () q q, e s d. (3) The boudary coditios for q, s are q0, s 0 ad q, s. (4) s The solutio of Equatio () subject to the boudary coditios (4) is q, s Re sih s a s sih s a e, a = 4 Re i. (5) K (6) The iverse Laplace s trasform of Equatio (5) is sih i π e Re q, siπ e sih i π i (7) Re 4 Re π i,, 4 K. 4 4 O separatig ito a real ad imagiary parts, we get Re C π siπ e S C π 4 u S S C π cos si e, π Re C S S C π siπ e S C π 4 w π cos π si e, (8) (9) (0) C C S sihcos, coshsi, () S sihcos, coshsi. The solutio give by Equatios (9) ad (0) eists for both Re < 0 (correspodig to v0 0 for the blowig at the plates) ad Re > 0 (correspodig to v 0 0 for the suctio at the plates). Solutios for Small Time Followig Carslaw ad Jaegar [7], for small time, the solutio of () subject to the boudary coditios (4) is obtaied by Laplace trasform techique i the followig form Copyright 0 SciRes.

4 5 M. JANA ET AL. ik Re k00 q, e e ik 4 c d j erfc j erfc Re c d cj erfc dj erfc 8 c m, d m, erfc erfc, j j d 0 erfc jerfc erfc d, j erfc. The solutio () ca be writte as ik Re q, e e k 0 r 0,, 4,6,, T r k 0 r c r d j erfc j erfc () ik 4 Tr, (3) Re r c r d cj erfc dj erfc, 4 (4) r 0,, 4,6, O separatig ito a real ad imagiary parts, we get the velocity distributios for the primary ad the secodary flow as u, cos, si e, Re e P K Q K (5) Re w Q K P K e, e, cos, si P T0 T K 4 4 K T6 4K Q, K 4T 4 T4 6K 6 3 8K 4 4 T6 4, T 4 (6) (7) Equatios (5) ad (6) describe the fluid velocities for small time. 3. Results ad Discussio To study the effects of rotatio, Reyolds umber ad porosity parameter o the velocity distributios we have preseted the o-dimesioal velocity compoets u ad w agaist i Figures -6 for several values of the rotatio parameter K, Reyolds umber Re, porosity parameter ad time. It is see from Figure that the primary velocity u decreases ad the magitude of the secodary velocity w icreases with a icrease i rotatio umber K. Figure 3 reveals that both the primary velocity u ad the magitude of the secodary velocity w decreases with icrease i Reyolds umber Re. It is observed from Figure 4 that both the primary velocity u ad the magitude of the secodary velocity w icrease with a icrease i porosity parameter. The presece of porous medium produces a resistig force i the flow field. So, the resistace i the Figure. Velocities u ad w for differet K whe Re =, σ = 0. ad τ = 0.. Copyright 0 SciRes.

5 M. JANA ET AL. 53 Fi gure 3. Velocities u ad w for differet Re oly whe K =, σ = 0. ad τ = 0.. Figure 4. Velocities u ad w for differet σ whe K =, Re = ad τ = 0.. flow field decreases as the porosity parameter icreases. This idicates that porosity of the me dium has a acceleratig ifluece o the flow field. Thus we ca cotrol the velocity field by itroducig porous medium i a rotatig system. It is observed from Figure 5 that both the primary velocity u ad the magitude of the secodary velocity w icreases with a icrease i times. For small values of time, we have draw the velocity compoets u ad w o usig the solutio give by Equatios (5) ad (6) ad the geeral solutio give by Equatios (9) ad (0) i Figures 6 ad 7. It is see that the solutio for small time give by Equatios (5) ad (6) coverges more rapidly tha the geeral solutio give by (9) ad (0). Hece, we coclude that for small times, the umerical values of the velocity compoets ca be evaluated from Equatios (5) ad (6) istead of Equatios (9) ad (0). The o-dimesioal shear stresses at the statioary plate 0 due to the primary ad the secodary flows are give by q i y e Re 0 sih i π i π i i π e (8) O separatig ito a real ad imagiary parts, we get the shear stress compoets due to the primary ad secodary flows at the statioary plate 0 as Copyright 0 SciRes.

6 54 M. JANA ET AL. Figure 5. Velocities u ad w for differet time τ whe K =, Re = ad σ = 0.. Figure 6. Velocity u for geeral solutio ad solutio for small time whe K =, Re = ad σ = 0.. Figure 7. Velocity w for geeral solutio ad solutio for small time whe K =, Re = ad σ = 0.. Copyright 0 SciRes.

7 M. JANA ET AL. 55 Re sih cos cosh si π e e y cosh cos π 4 π π cos si e Re sih cos cosh si π cosh cos π 4 π cos π si e (9) (30) The umerical values of the o-dimesioal shear stresses ad y at the plate 0 are preseted i Figures 8-0 agaist Reyolds umber Re for several values of K, ad. It is see from Figure 8 that both the absolute value of the shear stresses ad y icrease with icrease i rotatio parameter K while the absolute value of the shear stresses icreases ad that of the she ar stress y decreases with a icrease i Re. Rey olds umber Re is the ratio of iertial forces to viscous forces ad quatifies the relative importace of these two types of forces for give flow coditios. The viscous forces are domiat at low Re yolds um- bers while the iertial forces are domiat at high Reythe physical poit of view. Figures 9 ad 0 show that the absolute olds umbers. It is a good agreemet with value of the shear stress icreases while the absolute value of the shear stress y icreases with a icrease i either porosity parameter or time. For small times, the o-dimesioal shear stresses due to the primary ad secodary flows at the statioary plate 0 are give by q i y 0 i K e ik 4 Yr, 0 r 0,,,3,, Y m r r j erfc m0 m 8 r m Re j erfc j erfc. 4 Re r m (3) (3) O separatig ito a real ad imagiary parts, we get the shear stress compoets due to the primary ad secodary flows as Re e C0, cosk D0, sik e, Re y e D0, cosk C0, sik e, 4 C0, Y 4 Y 4K 4 4 Y 4 K Y5 4K D0, K 4Y 4 Y3 6K 6 3 8K 4 4 Y5. 3 (33) (34) (35) For small time, the umerical values of the shear stress compoets calculated from Equatios (9), (30), (33) ad (34) are give i Tables ad for several values of Re ad. It is observed that for small time the shear stresses calculated from Equatios (33) ad (34) give better result tha that calculated from Equatios (9) ad (30). Hece, we coclude for small times shear stress compoets should be evaluated from Equatios (33) ad (34) istead of Equatios (9) ad (30). We shall ow discuss the asymptotic behavior of the solutios (5) ad (6) for small ad large values of K ad Re, whe the motio is i steady state. I the steady state, Equatio (7) becomes sih i q,. (36) sih i Case ): Whe K ad Re. Whe Re is large ad K is of small order of magitude, the flow becomes boudary layer type. For the boudary layer flow ear the upper plate, itroducig the boudary layer coordiate, we obtai the velocity distributios from (36) as Re u e cos, (37) Copyright 0 SciRes.

8 56 M. JANA ET AL. Figure 8. Shear stresses τ ad τ y for differet K whe σ = 0. ad τ = 0.. Figure 9. Shear stresses τ ad τ y for differet σ whe K = ad τ = 0.. Figure 0. Shear stresses τ ad τ y for differet τ whe K = ad σ = 0.. Copyright 0 SciRes.

9 M. JANA ET AL. 57 Table. Shear stress 0τ due to primary flow whe K = ad σ = 0.. Re\ Geeral solutio Solutio for small times Table. Shear stress 0 τ y due to secodary flow whe K = ad σ = 0.. Re\ Geeral solutio Solutio for small times Re w e si, (38) Re 4 4K,. Re Re (39) It is evidet from Equatios (37) ad (38) that there eists a sigle-deck boudary layer of thickess of order Re O ear the movig plate of the chael is give by (39). The thickess of this boudary layer decreases with a icrease i porosity parameter sice decreases with icrease i. Case ): Whe K ad Re. I this case, the velocity distributios are obtaied from the Equatios (36) as Re u e cos, (40) Re w e si, (4), K 4K. (4) 4 Re Equatios (40) ad (4) show that there eists a sigledecker boudary layer of thickess of order Re O adjacet to the movig plate of the chael is give by (4). The thickesses of the layer decreases with a icrease i Reyolds umber Re while it icreases with icrease i porosity parameter. 4. Coclusio [] G. K. Batchelor, A Itroductio to Fluid Dyamics, Cambridge Uiver sity Press, Cambridge, 967. [] G. S. Seth, R. N. Jaa ad M. K. Maiti, Usteady Hydromagetic Couette Flow i a Rotatig System, Iter- The usteady Couette flow of a viscous icompressible fluid through a porous medium i a rotatig system has bee ivestigated. It is foud that the primary velocity decreases ad the magitude of the secodary velocity icreases with a icrease i rotatio parameter. The fluid velocity compoets decrease with a icrease i Reyolds umber. A icrease i the porosity of the medium both the primary ad the secodary velocities icrease. That is, the porosity of the medium has a acceleratig ifluece o the flow field. I tur, it ca cotrol the velocity field by itroducig po rous medium i a rotatig system. It is also foud that the solutio for small time coverges more rapidly tha the geeral solutio. Fo r steady state, the asymptotic behavior of the solutio is aalyzed for small as well as large values of rotatio parameter ad Reyolds umber. It is observed that a thi bou dary layer is form ed ear the movig plate of the chael ad the thickesses of the layer icreases with a icrease i porosity parameter. REFERENCES Copyright 0 SciRes.

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