VISCOUS FLUID FLOW IN AN INCLINED CHANNEL WITH DEFORMABLE POROUS MEDIUM

International Journal of Mechanical Engineering and Technology (IJMET) Volume 9, Issue, January 08, pp. 970 979 Article ID: IJMET_09_0_04 Available online at http://www.iaeme.com/ijmet/issues.asp?jtype=ijmet&vtype=9&itype= ISSN Print: 0976-6340 and ISSN Online: 0976-6359 IAEME Publication Scopus Indexed VISCOUS FLUID FLOW IN AN INCLINED CHANNEL WITH DEFORMABLE POROUS MEDIUM S. Sreenadh, G. Gopi Krishna Department of Mathematics, Sri Venkateswara University, Tirupati, India V. Manoj Kumar Uppuluri Department of Mathematics, Vignan s Foundation for science, Technology and Research, Guntur, A.P, India A.N.S.Srinivas Department of Mathematics, School of Advanced Sciences, VIT University, Vellore, Tamilnadu, India ABSTRACT: In the present study, the viscous fluid flow in an inclined channel through deformable porous media bounded by moving parallel plates is investigated. The coupled phenomenon of fluid flow and deformation of porous materials is a problem of prime importance in biological applications. The present problem is modeled by considering the coupled phenomenon under the assumption that bounding walls of the channel are moving with different velocities U andu. The analytical solution is obtained for coupled governing equations. The expressions for the velocity field and solid displacement are derived. The influence of the volume fraction, the drag on the flow velocity and displacement are analyzed graphically. It is observed that the displacement increases with increasing drag, whereas the opposite behavior is noticed in the case of velocity. The effect of various pertinent parameters on the flow quantities are discussed through graphs and tables. Keywords: Viscous flow, Porous media, Deformable boundaries, Inclined channel; Shear stress. Cite this Article: S. Sreenadh, G. Gopi Krishna, V. Manoj Kumar Uppuluri and A.N.S.Srinivas, Viscous fluid flow in an inclined channel with deformable porous medium, International Journal of Mechanical Engineering and Technology 9(), 08. pp. 970 979. http://www.iaeme.com/ijmet/issues.asp?jtype=ijmet&vtype=9&itype= http://www.iaeme.com/ijmet/index.asp 970 editor@iaeme.com

S. Sreenadh, G. Gopi Krishna, V. Manoj Kumar Uppuluri and A.N.S.Srinivas. INTRODUCTION: The porous medium is characterized by a partitioning of the total volume in to solid matrix and pore space. Porous media is a composition of the solid phase which is referred as solid skeleton and closed and open pores which may fill with a pore fluid. The solid and fluid phases usually have a relative velocity with each other, and they also have different material properties. Thus, there is an interaction between these phases. The complete prediction of behavior of the solid material deformation interacting with a fluid flow is achieved by a detailed study of porous media. The coupled phenomenon of fluid flow and deformation of porous materials is a problem of prime importance in geomechanics, biomechanics, biological soft tissue modeling including cartilage, skin myocardium and arterial walls. The fluid flow through porous media arises frequently in different forms with several fields of science and technology. Flow through porous media can be observed in the process of filtration, petroleum industry etc. The knowledge of porous media is very useful in the scientific applications such as extraction of energy from the geothermal regions, filtration of solids from liquids and oil form the underground reservoirs/ river basins. The theory of deformable porous media is more adequate to understand the phenomenon such as hemodynamic effects of the endothelial Glycocalyx. In view of these significant applications, the theory of mixtures is introduced by Biot [] and others to describe the flow through deformable porous media. Kenyon [] has studied a mathematical model of water flux through aortic tissue. Jayaraman [3] discussed water transport in the arterial wall. Mow and Holmes [4] contributed important theory for the study of rectilinear cartilages and biological tissue mechanics. Klanchar et al. [5] investigated the models for water flow through arterial tissue. Oomens et al. [6] discussed the mixture approach to describe the mechanics of skin. Yang et al. [7] studied the possible role of poroelasticity in the apparent viscoelastic behavior of passive cardiac muscle. Huyghe et al. [8] analyzed a two-phase finite element model of the diastolic left ventricle. Barry et al. [9], Ranganatha and Siddagamma [0] and Sreenadh et al. [-4] presented some mathematical models involving flow through deformable porous media. Most of the research works was concentrated on fluid flow through undeformable porous media. So it is interesting to study the flow characteristics in deformable porous medium which is useful to understand many industrial and biological applications. In view of the above studies, the flow of a Newtonian fluid in an inclined channel through deformable porous media is investigated. The bounding walls of the channel are moving with different velocities U andu. The fluid velocity and displacement of the solid matrix are obtained. The effects of various physical parameters on the flow quantities are discussed through graphs and tables.. MATHEMATICAL FORMULATION: Consider a steady, fully developed flow through an inclined channel with solid walls at y hand y h through a deformable porous media as shown in Figure. The solid walls are moving with velocities U and U respectively. The fluid velocity and solid displacement p are v,0,0 and u,0,0 respectively. A pressure gradient P is applied, producing an x axially directed flow in the channel. http://www.iaeme.com/ijmet/index.asp 97 editor@iaeme.com

Viscous fluid flow in an inclined channel with deformable porous medium Figure Physical Model In view of the assumptions mentioned above, the equations of motion in the deformable porous media are (Sreenadh et. al.,[-4]). u p g sin K v 0 y x v p a g sin K v 0 y x The boundary conditions are y h v U u :, 0 :, 0 y h v U u The non-dimensional quantities are * y * u * v * ph * x * U * U Uh U y, u, v, p, x, U, U, Re, Fr h U U U h U U gh a a a In view of the above dimensionless quantities, after neglecting the stars (*), the equations () (4) take the following form du dy dv dy Re Pv sin (4) Fr Re v P sin (5) Fr dp Where P dx The parameter is a measure of the viscous drag of the outside fluid relative to drag in the porous medium. The boundary conditions are y v U u :, 0 :, 0 y v U u 3. SOLUTION OF THE PROBLEM: Equations (4) - (5) are coupled with differential equations that can be solved by using the boundary conditions (6). The solid displacement and fluid velocities in the free flow region and deformable porous layer are obtained as below, () () (3) (6) http://www.iaeme.com/ijmet/index.asp 97 editor@iaeme.com

S. Sreenadh, G. Gopi Krishna, V. Manoj Kumar Uppuluri and A.N.S.Srinivas y y ay 3 ay 4 Ae Be y u( y) a a Cy D (7) y y v( y) Ae Be a a (8) Where, a P, a Re sin Fr Ue Ue B e e a a e e A 3 4 a a e e D B e e a a, a Re P, a 4 sin Fr 3 A B, C e e e e, 4. SHEAR STRESS: The non-dimensional form of shear stress in the free flow region in is given by dv (9) dy 5. MASS FLUX: The dimensionless mass flow rate M per unit width of the channel in the free flow region 0 y is given by: M 0 vdy M A( e ) B( e ) ( a a) 6. RESULTS AND DISCUSSIONS: In the present study, viscous flow in an inclined channel with deformable porous media is investigated. The results are discussed for physical parameters such as the volume fraction of the fluid, drag, lower plate velocityu and upper plate velocityu. The effects of various physical parameters on flow velocity and displacement are discussed through graphs. In this study, P, 0.6,, U, U, Fr and Re are used for numerical computation. These values are kept as common in the entire study except for varied values as displayed in all the graphs. The variation of solid displacement u with y is calculated from equation (7) for different values of, U, U, are presented in Figures to 5 respectively. It is observed from Figure that the solid displacement increases with growing values of viscous drag. Figure 3 and 4 illustrates that increasing lower and upper plate velocities enhances the solid displacement. From Figure 5, it is found that for different values of, the solid displacement increases. The (0) http://www.iaeme.com/ijmet/index.asp 973 editor@iaeme.com

Viscous fluid flow in an inclined channel with deformable porous medium opposite behavior is observed with the case of volume fraction. That is from Figure 6 it is clear that the solid displacement reduces with increasing volume fraction. The variation of fluid flow velocity v with y is calculated from equation (8) for different values of, U, U, and P are shown in Figures 7 to respectively. From Figure 7, it is noticed that the velocity of the fluid increases with increasing values of volume fraction. The increasing values of lower and upper plate velocities increases the fluid velocity represented in Figures 8 and 9 respectively. Further from Figures 0 and, it is found that the growing values of and P causes the increase in fluid velocity respectively. The variation of fluid flow velocity for different value of is represented in Figure. It is observed that the fluid velocity decreases with increasing drag. Since the increasing drag produces the resistive force which slowdowns the fluid motion. Figure Displacement for different Figure 3 Displacement profile for different values of values of U Figure 4 Displacement profile for different Figure 5 Displacement profile for different values of U values of http://www.iaeme.com/ijmet/index.asp 974 editor@iaeme.com

S. Sreenadh, G. Gopi Krishna, V. Manoj Kumar Uppuluri and A.N.S.Srinivas Figure 6 Displacement profile for different Figure7 Velocity profile for different values of values of Figure 8 Velocity profile for different Figure 9 Velocity profile for different values of U values of U Figure 0 Velocity profile for different Figure Velocity profile for different values of values of P http://www.iaeme.com/ijmet/index.asp 975 editor@iaeme.com

Viscous fluid flow in an inclined channel with deformable porous medium Figure Velocity profile for different values of The shear stress and for different physical parameters,,u andu are calculated from equation (9) and is illustrated in table I. It is evident from Table that the shear stress at the lower plate increases with the increase in drag and upper plate velocity U, whereas, opposite behavior is observed in lower plate U and. The shear stress at the upper plate enhances with the increase in volume fraction and upper plate velocityu, whereas, the opposite behavior is observed in the case of andu. The variation of mass flux M for different values of volume fraction of the fluid, drag, lower plate velocityu and upper plate velocityu are calculated in equation (0) which is tabulated in Table. It is observed that the mass flux increases with increase in the volume fraction of the fluid, whereas, opposite behavior is observed in the case of drag. Table Variation of and at the lower wall and upper wall y and y with for different values of,,u andu for fixed values of P,Re, Fr and 3. U U 0. -0.0503 0.0335 0.4-0.06 0.35 0.6-0.3549 0.366 0.8-0.507 0.338 0.6-0.3549 0.366 0.6.5 0.033-0.056 0.6 0.3354-0.36 0.6.5 0.6008-0.4006 0.6-0.3549 0.366 0.6-0.6306-0.4549 0.6 3-0.9064 -.465 0.6 4 -.8 -.8380 0.6-0.3549 0.366 0.6 0.684 0.404 0.6 3.797 0.604 0.6 4.7570 0.788 http://www.iaeme.com/ijmet/index.asp 976 editor@iaeme.com

S. Sreenadh, G. Gopi Krishna, V. Manoj Kumar Uppuluri and A.N.S.Srinivas Table : Variation of Mass flux with for different values of,,u andu for fixed values of P, Re, Fr and 3. U U 0..057 0. - -0.080 0. 0 0.8660 0.4.0634 0.4 - -0.603 0.4 0 0.937 0.6. 0.6 - -0.6 0.6 0 0.964 0.8.588 0.8 - -0.0649 0.8 0.0090 0.6.5 0.999 0.6.5 - -0.394 0.6.5 0 0.874 0.6 0.9007 0.6 - -0.580 0.6 0 0.809 0.6.5 0.867 0.6.5 0 0.7445 0.6.5 0 0 0.959 7. CONCLUSIONS: In this article, the flow of a Newtonian fluid in an inclined channel with deformable porous media is investigated. The results are analyzed for different values of the pertinent parameters, namely, drag, the volume fraction of the fluid, upper plate velocity and lower plate velocity. Some of the important observations are summarized as follows. The solid displacement in the deformable porous layer increases with increasing drag, U, U and. The increase in volume fraction of the fluid, decreases the solid displacement. The fluid velocity in the deformable porous layer increases with increasing, U, U, and P, whereas opposite behavior is observed with increase in. The findings of the present problem will be useful in understanding the synovial fluid flow behavior in the cartilage of knee joint (modeled as a deformable porous layer). M http://www.iaeme.com/ijmet/index.asp 977 editor@iaeme.com

Viscous fluid flow in an inclined channel with deformable porous medium Nomenclature: a U f K u U P U v Apparent viscosity of the fluid in the porous material. Average velocity. Coefficient of viscosity. Drag coefficient. Displacement in x -direction. Lame constant. lower plate velocity Shear stress at the lower wall. Shear stress at upper wall. The volume fraction of the fluid. Typical pressure gradient. upper plate velocity Viscous drag. Velocity of the fluid in the deformable porous layer. REFERENCES: [] M. A. Biot, General theory of three-dimensional consolidation, J. Appl. Phys. (94) 55-64. [] D. E. Kenyon, A mathematical model of water flux through aortic tissue, Bull. Math. Biol. 4(979) 79-90. [3] G. Jayaraman, Water transport in the arterial wall-a theoretical study, J. Biomech. 6 (983) 833-840. [4] V.C. Mow, M.H. Holmes, M. Lai, Fluid transport and mechanical properties of articular cartilage: a review, J. Biomechanics. 7 (984) 377-394. [5] M. Klanchar, J.M. Tarbell, Modeling water flow through arterial tissue, Bull. Math. Biol. 49 (987) 65-669. [6] C.W.J. Oomens, D.H. Van Campen, H.J. Grootenboer, A mixture approach to the mechanics of skin, J. Biomech. 0 (987) 877-885. [7] M. Yang, and L.A. Taber, The possible role of poroelasticity in the apparent viscoelastic behavior of passive cardiac muscle, J. Biomech. 4 (99) 587-597. [8] J.M. Huyghe, D.H. Van Campen, T. Arts, R.M. Heethaar, A two-phase finite element model of the diastolic left ventricle, J. Biomech.4 (997) 57-538. [9] S.I. Barry, K.H. Parker, and G.K. Aldis, Fluid flow over a thin deformable porous layer, Journal of Applied Mathematics and Physics (ZAMP). 4 (99) 633-648. [0] T.R. Ranganatha, and N.G. Siddagamma, Flow of Newtonian fluid a channel with deformable porous walls, Proc. of National Conference on Advances in fluid mechanics. (004) 49-57. [] S. Sreenadh, M. Krishnamurthy, E. Sudhakara, G. Gopi Krishn, Couette flow over a deformable permeable bed, International Journal of Innovative Research in Science& Engineering. (04) 347-307. [] S Sreenadh M. Krishnamurthy, E. Sudhakara, G. GopiKrishna, D. Venkateswarlu Naidu, MHD free surface flow of a Jeffery fluid over a deformable porous layer, Global Journal of Pure and Applied Mathematics. (05) 3889-3903. http://www.iaeme.com/ijmet/index.asp 978 editor@iaeme.com

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