[Sreenadh, 3(2): February, 2014] ISSN: Impact Factor: 1.852

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1 IJESRT INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOG Flow o a Jerey Flid between Finite Deormable Poros Layers S.Sreenadh *, A.Parandhama, E.Sdhakara 3, M. Krishna Mrthy 4 *,3,4 Department o Mathematics, Sri Venkateswara University, Tirpati- 575, India Department o Mathematics, Sree Vianikethan Engineering College, Tirpati, India prosreenadh@gmail.com Abstract The low o a Jerey lid between a thin deormable poros layers is investigated. The governing eqations are solved in the ree low and poros low regions. The expressions or the velocity ield and deormation are obtained. When λ, the reslts agree with the corresponds ones o Barry et al (99). The eects o s Jerey parameter, the pressre gradient, φ and φ on the low velocity and deormation are discssed. It is ond that the velocity increases with the increase in the non-newtonian Jerey parameter λ. This st is also relevant to iltration technology, soil mechanics and to other biological problems sch as the mechanics o articlar cartilage. Keywords: Jerey Flid, Finite Deormable. Introdction Viscos low throgh and past poros media has important applications in engineering and medicine. Most o the research works available deal with low throgh rigid poros media. Bt when a biolid lows in a physiological system, there will be an interaction between ree low and tisse (deormable) regions. Ths the st on ree low in a deormable poros layer is necessitated. Frther most o biological lids are observed to be non-newtonian and these lids may be modelled with a simple elegant Jerey model. The st o deormation in poros materials with copled lid movement was initiated by Terzaghi [] and later contined by Biot [,3,4] into a sccessl theory o soil consolidation and acostic propagation. Atkin and Craine [5], Bowen [6] and Bedord and Drmheller [7] made important works on the theory o mixtres. Mow et al. [8] developed a similar theory or the st o biological tisse mechanics. Using this theory arterial wall permeability is discssed by Jayaraman [9]. The same theory was also applied by Mow et al. [], Holmes and Mow [] or the st o articlar cartilages. Mch o this analysis, has been on one dimensional or prely radial compression withot consideration o the inlence o shear stresses on the deormable poros media. The movement o bio-lids in a physiological system has to be investigated thoroghly in order to solve diagnostic problems that arise in a living bo. Some o the bio-lids like blood are observed to behave like non-newtonian lids. Since there is no niversal model to describe all non-newtonian lids in physiological systems, several models are proposed to explain the behavior o these bio-lids. Hayat and Ali [] investigated the peristaltic motion o a Jerey lid nder the eect o a magnetic ield. Elshehaway [3] has stdied peristaltic transport in an asymmetric channel throgh a poros medim. Inlence o partial slip on the peristaltic low in a poros medim and a mathematical model o peristalsis in tbes throgh a poros medim is investigated by Hayat et al. [4]. Vajravel et al. [5] investigated the peristaltic low and heat transer in a vertical poros annls with long wavelength approximation. Kothandapani and Srinivas [6] made a st on the peristaltic transport o a Jerey lid nder the eect o magnetic ield in an asymmetric channel. Hayat and Ali [7] investigated the peristaltic motion o a Jerey lid nder the eect o a magnetic ield. Among several non-newtonian models proposed or physiological lids, Jerey model is signiicant becase Newtonian lid model can be dedced rom this as a special case by taking λ =. Frther it is speclated that the physiological lids sch as blood exhibit Newtonian and non-newtonian behaviors dring circlation in a living bo. Vajravel et al [8] stdied the inlence o heat transer on peristaltic transport o a Jerey lid in a vertical poros stratm. Krishna Kmari et al [9] stdied the eect o magnetic ield on the peristaltic pmping o a Jerey lid in an inclined channel.

2 [Sreenadh, 3(): Febrary, 4] Motivated by these stdies, the stea low o a Jerey lid between deormable poros layers is investigated. The lid velocity in the ree and poros regions is obtained. The expression or the displacement in the poros layer is also obtained. The eects o varios physical parameters φ, G, λ, η and d, on the velocity and displacement are discssed throgh graphs. Formlation o the Problem The geometry consists o a stea, lly developed Jerey lid low throgh a symmetrical channel with solid walls at y = ± h and a poros layer o thickness L attached to both walls as shown in Fig.. By symmetry only hal o the channel y [, h] is considered. The low region between the plates is divided into two layers. The low region between the lower plate y = and the interace y = h L is termed as ree low region whereas the low region between the interace y = h L and the pper plate y = h is designated as deormable poros region. The lid velocities in the ree low and deormable are q and v. The displacement in the deormable poros region. The lid velocity in the ree low region and poros low region are assmed to be (q,, ) and (v,,) respectively. The displacement de to the deormation o the solid matix is taken as (,,). A pressre gradient p = G is applied, prodcing an axially directed low. x De to the assmption o an ininite channel, there is no x dependence in any o the terms except the pressre. Fig. Physical Model. With the assmptions mentioned above, the eqations o motion in the ree low and deormable poros regions are (ollowing Barry []) () () µ φ y s = G K v, µ a v = φ G + K v + λ y (3) µ q p = λ y x + ISSN: where µ is the apparent viscosity o the lid a in the poros material, λ is Jerey parameter, G is the pressre gradient and φ is the viscos parameter. We note that dot denotes dierentiation with respect to time. Non-Dimensionalization o the Flow Qantities It is convenient to introdce the ollowing nondimensional qantities. h G h G h G L y = hyˆ, ˆ =, v = vˆ ˆ, q = q, ε =, µ µ µ h where G is a typical pressre gradient. In view o the above dimensionless qantities, the eqations () (3) take the ollowing orm. The hats ( ) are neglected here ater. (4) (5) (6) d φ s = G v d v ( ) + η + λ v = φ Gη ( + λ ) d q = G( + λ ) where Kh, ˆ G µ dp = G =, η =, G =. µ G µ dx The parameter is a measre o the viscos drag o the otside lid relativee to drag in the poros medim. The parameter η is the ratio o the blk lid viscosity to the apparent lid viscosity in the poros layer. The bondary conditions are at y = : = v = (7a) at y = : dq = (7b) a

3 at y = ε : q = φ v (7c) dq dv = ηφ d = s φ (7d) The irst two eqations represent the no slip condition or low at the solid bondary and the symmetry condition along the centre o the channel. Eqation (7c) eqates the lid velocity at the interace with the volme averaged velocity o the poros layer. The inal two eqations arise rom the conservation o axial momentm across the lid-poros layer interace and the assmption that the proportion o the total stress in the poros layer borne by each component is proportional to its volme raction Soltion o the Problem Eqations (4) to (6) are copled dierential eqations that can be solved by sing the bondary conditions (7). The displacement and velocities in ree low region and poros regions are obtained as 3 y ( my) y s y = c3 y + c4 c c e φ G + φ G m 6 (8) ( my) φ G v = c + ce + y (9) y q = c y + c G( + λ ) () 5 6 where s m ( m) φ G φ G m = η( + λ ), m = φ Gη ( + λ ), m =, a =, a = e, a 3 =, m m 6 ( m) φ G m( ε ) G( + λ )( ε ) G( φ ) ( ε ) a4 = e, a5 =, a6 = φ, a7 = a6 e, a8 =, m( ε ) φ G e a9 = + G( + λ )( ε ) φ η, a = ( ε ), a =, a3 = a5 a9a4, c = a3, m s G( ε ) a = φ G( ε ) + a a9, a4 = a3 a6 + a9 a7 a8, a5 = a aa 3, a = a a a a + a a, c = a, c = a, c = a Reslts and Discssions In this paper, the stea low o a Jerey lid between thin, deormable poros layers is investigated. When the Jerey parameter λ, the reslts (8) to () redce to the corresponding displacement and lid velocity o Barry et al. (99). The soltions or the lid velocity and displacement are evalated nmerically or dierent vales o physical parameters sch as Jerey parameter λ, the pressre gradient G, the viscos parameter φ, the viscos drag parameter and the viscosity parameter η. The variation o velocity proile o ree low region q and deormable poros layer v with y is calclated, rom eqations (8) to (), or dierent vales o φ and is shown in Fig.. or ixed G =, λ =.5, =, ε =. and η =.5. We observe that the velocities q increase with the increase inφ. The variation o velocity proile o ree low region q and deormable poros layer v with y is calclated, rom eqations (8) to (), or dierent vales o pressre gradient G and is shown in Fig. 3,or ixed φ =.5, λ =.5, =, ε =. and η =.5. Here we observe that the velocities q and v increase with the increase in pressre gradient G. The variation o velocity proile o ree low region q and deormable poros layer v with y is calclated, rom eqations (8) to (), or dierent vales o Jerey parameter λ and is shown in Fig. 4, or ixed, φ = =, ε =. and η =.5. Here we observe that the velocities q and v increase with the increase in Jerey parameter λ. The variation o velocity proile o ree low region q and deormable poros layer v with y is calclated, rom eqations (8) to (), or dierent vales o ratio o blk lid viscosity parameter η and is shown in Fig. 5, or ixed λ = G =, φ =.5, =, ε =.. It is.5, G =.5, ond that the velocity q and v increase with the increase in the o ratio o blk lid viscosity parameter η. The variation o displacement with y is calclated, rom eqations (9) to (), or dierent vales o Jerey parameter λ and is shown in Fig. 6,or ixed G =,, φ = η =.5 ε =.. Here we observe that the displacement increases with the increase in Jerey parameter λ. The variation o displacement with y is calclated, rom eqations (8) to (), or dierent vales o pressre gradient G and is shown in Fig. 7, or ixed λ =.5, =, φ =.5, η =.5 ε =.. =.5,

4 Here we observe that the displacement increases with the increase in pressre gradient G. The variation o displacement with y is calclated, rom eqations (8) to (), or dierent vales o viscos drag parameter and is shown in Fig. 8, or ixed λ = G =, φ =.5, η =.5 ε =.. Here.5, we observe that the displacement increases with the increase viscos drag parameter. The variation o displacement with y is calclated, rom eqations (8) to (), or dierent vales o ratio o blk lid viscosity parameter η and is shown in Fig. 9, or ixed λ =.5, G =, φ =.5,, = ε =.. Here we observe that the displacement increases with the increase in o ratio o blk lid viscosity parameterη. q (o r) v φ =.85 φ =.85 φ =.7 φ =.7 φ =.5 φ =.5 q (or) v Fig 3. Velocity proile o ree low region q (y=-.8) and deormable poros layerv(y=.8-) or dierent vales o G λ =.5 λ =.5 λ = λ = λ =.5 λ =.5 G =.5 G =.5 G = G = G =.5 G =.5.. q (or) v Fig. Velocity proile o ree low region q (y=-.8) and deormable poros layer v(y=.8-) or dierent vales o φ Fig 4. Velocity proile o ree low region q (y=-.8) and deormable poros layer v(y=.8-) or dierent vales o λ.

5 .7.6 η =.5 η =.5 η = η = η = η =.6.4. G =. G =.5 G =.5. q (or) v Fig 5. Velocity proile o ree low region q (y=-.8) and deormable poros layer v(y=.8-) or dierent vales o η..3.5 λ =.5 λ = λ = Fig 7. Displacement proile in the deormable poros layer or dierent vales o G =.5 = = Fig 6.Displacement proile in the deormable poros layer or dierent vales o λ Fig 8. Displacement proile in the deormable poros layer or dierent vales o.

6 η =.5 η =.7 η = Fig 9. Displacement proile in the deormable poros layer or dierent vales o η. Acknowledgement One o the athors Pro. S.Sreenadh expresses thanks to UGC or providing inancial spport throgh the Major Research Project to ndertake this work. Reerences [] Terzaghi, K., Erdbamechanik a BodenphysikalischenGrndlagen. Deticke (95). [] Biot, M.A., General theory o three-dimensional consolidation. J. Appl. Phys., (94), [3] Biot, M.A., Theory o elasticity and consolidation or poros anisotropic solid. J. Appl. Phys., 6(955), [4] Biot, M.A., Mechanics o deormation and acostic propagation in poros media. J. Appl. Phys., 7 (956). [5] Atkin, R.J. and Craine, R.E.,Continm theories o mixtres: Basic theory and historical development. Qart. J. Appl. Math., 9 (976), [6] Bowen, R.M., Incompressible poros media models by the theory o mixtres. Int. J. Engng. Sci., 8(98), [7] Bedord, A. and Drmheller, D. S., Recent advances, theory o immiscible and strctred mixtres. Int. J. Engng. Sci., (983), [8] Mow, M.H. Holmes and Lai, M., lid transport and mechanical properties o articlar cartilage: a review. J. Biomechanics, 7 (984) [9] Jayaraman, G., Water transport in the arterial wall- A theoretical st. J. Biomechanics, 6 (983), [] Mow,V.C., Kwan, M.K.,Lai,W.M. and Holmes,M.H., A inite deormation theory or non linearly permeable sot hydrated biological tisses. Frontiers in Biomechanics, G.Schmid-Schoenbein, S. L.. Woo and B. w. zweiach (eds), Springer-Verlag, New ork, (985) [] Holmes,M.H. and Mow,V.C., The nonlinear characteristics o sot gels and hydrated connective tisses in ltrailtration. J. Biomechanics, 3, (99), [] Hayat, T and Ali, N., Physica A Statistical Mechanics and its Applications, 37 (6), [3] Elshehaway, E.F., Eldabe, N.T., Elghazy, E.M. and Ebaid, A., Peristaltic transport in an asymmetric channel throgh a poros medim", Appl. Math. Compt., 8, (6), 4 5. [4] Hayat,T., Khan, S.B. and Khan, M., The inlence o Hall crrent on the rotating Oscillating lows o an Oldroyd-B lid in a poros medim. Nonlinear Dyn.,47, (7), [5] Vajravel K, Radhakrishnamacharya, G and Radhakrishnamrty V., Peristaltic low and heat transer in a vertical poros annls, with long wavelength approximation. Int J NonlinearMech, 4 (7), [6] Kothandapani, M. andsrinivas,s., On the inlence o wall properties in the MHD peristaltic transport with heat transer and poros medim, Phys A. 37 (8) [7] Hayat T, Ali N. Peristaltic motion o a Jerey lid nder the eect o a magnetic ield in a tbe. Commn Non-linear Sci Nmer Siml, 3 (8), [8] K. Vajravel, S. Sreenadh and P. Lakshminarayana. The inlence o heat transer on peristaltic transport o a Jerey lid in a vertical poros stratm. CommnNonlinear Sci NmerSimlat,6 () [9] Krishna Kmari,S.V.H.N., Ramana Mrthy,M.V., Ravi Kmar,.V.K. andsreenadh,s., Peristaltic pmping o a Jerey lid nder the eect o a magnetic ield in an inclined channel, Appl.Math. Sciences, 5(),

7 [] Barry, S.I., Parker, K.H. and Aldis, G.K. Flid low over a thin deormable poros layer. Jornal o Applied Mathematics and Physics (ZAMP), 4, (99),