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Flow of a non-Newtonian fluid through porous medium with variable suction.

1. INTRODUCTION

The study of flow through porous medium assumed importance because of the interesting applications in the diverse field of science, Engineering and Technology. The practical applications are in the percolation of water through soil, extraction and filteration of oils from wells, the drainage of water, irrigation and sanitary engineering and also in the inter disciplinary fields such as biomedical engineering, the lung alveolar is an example that finds applications in an animal body. The classical Darcy's law Musakat (1, 2), states that the pressure gradient pushes the fluid against the body forces exerted by the medium which can be expressed as [??] = - (k/[mu])[nabla]P with usual notation.

The classical Darcys law gives good result in the situations where the flow is uni directional or at low speed. In general specific discharge in the medium need not be always low. As the specific discharge increases the convective forces get developed and the internal stress generates in the fluid due to viscous nature and produces distortions in the velocity field. Modifications for the classical Darcy's law are considered by Beavers and Joseph (3), Saffman (4) and others. A generalized Darcy's law is proposed by Brinkman (5).

[rho] d[bar.v]/dt = Div [S.sub.ij] - ([mu]/k)[??]

where [S.sub.ij] is the Stress Tensar of the fluid, [rho] is the density, [bar.v] is velocity of the fluid, k is permeability coefficient.

Yamamoto and Iwamura (6), Narasimha Charyulu (7,8), Narasimha Charyulu and Pattabi Rama Charyulu (9), Narasimha Charyulu and Sunder Ram (10) and several investigation adopted the generalized law proposed by Brinkman.

A number of investigators have studied the unsteady flow of second order fluids (Ting (11), Erdogen (12), Lighthill (13) etc.).

The oscillatory flows earlier investigated by Stuart (14), Messina (15), etc. Soundalgekar and Das Gupta (16) discussed the problem of oscillating visco elastic fluid past an infinite porous plate with variable suction. In the present problem the oscillatory flow of second order non-Newtonian fluid is examined with variable suction and the free stream velocity oscillates in time about a non zero constant mean value. The effect of permeability parameter on the velocity is studied and graphically the results are presented.

2. Formulation of the problem:

Consider the flow of a second order Non-Newtonian fluid through highly porous medium with permeability k, bounded by a semiinfinite porous plate. Let the coordinate system O(x, y, z) is taken such that x-axis, lies parallel to the length of the plate and y-axis perpendicular to the infinite plate. The velocity of the fluid is give by [bar.V](u, v, 0). U(t) represents the free-stream velocity parallel to the plate. The velocity components are independent of x.

The continuity equation of the motion of the fluid [nabla].[bar.V] = 0 gives

[partial derivative]v / [partial derivative]y = 0 (2.2)

implies v is independent of y

The Governing equations of motion are given by [A.K. Johri et al. (17)].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.3)

where p is modified pressure

The free-stream velocity satisfies

dU/dt = 1 / [rho] [partial derivative][bar.p]/[partial derivative]x - v U/K (2.4)

Where p is density of the fluid, V is coefficient of viscosity, v = [mu]/[rho] is kinematic viscosity of the fluid, k is permeability of the porous medium.

U (t) is free-stream velocity

The free stream velocity and suction velocity are taken to be

U(t) = [U.sub.o] (1+ [member of] [e.sup.int]) (2.5)

V(t) = -[V.sub.0](1 + A [member of] [e.sup.int]) (2.6)

where [member of] [less than or equal to] 1, A [member of] [less than or equal to] 1 and A is constant.

Where [U.sub.0] and [V.sub.0] are mean free-stream and mean suction velocities respectively, n is frequency of oscillations.

The boundary conditions for the problem are

u = 0 at y = 0 u [right arrow] U(t) as y [right arrow] [infinity] (2.7)

Eliminating pressure term from (2.3) and (2.4) we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.8)

By using the following non-dimensional quantities

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The non-dimensional form of the equation (2.8) after removing (*) will be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.9)

The corresponding free-stream velocity and suction velocity will be

U = 1 + [member of] [e.sup.int] (2.10)

V = -(1 + A [member of] [e.sup.int]) (2.11)

Together with the boundary condition

u = 0 at y = 0 ] u = U as y [right arrow] [infinity] (2.12)

3. Solution of the problem:

Following the method of Light-Hill (13), the velocity component u(y,t) = f(y) + [member of] . g(y)[e.sup.int] is substituted in the equation (2.9) and separating harmonic and nonhormonic terms, we get.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.1)

[d.sup.2]f/[dy.sup.2] + df / dy - 1 / k f = -1/k (3.2)

The boundary conditions become

f = g = 0 at y = 0 f [right arrow] 1, g [right arrow] 1 as y [right arrow] [infinity] (3.3)

The solution of (3.2) satisfying (3.3) is given by

f(y) = 1 - [e.sup.-Ry] (3.4)

where

R = 1 + [square root of (4/k + 1)] / 2

The solution of (3.1) in view of (3.3) and (3.4) is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.5)

Hence, the non-dimensional stream-wise velocity component in the boundary layer is found to be

u(y,t) = 1 -[e.sup.-Ry] + [member of] ([S.sub.r] cos nt - [S.sub.i] sin nt) (3.6)

Where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The non-dimensional form of the skin friction at the porous plate is given by

[[tau].sub.0] = [R + [member of] [absolute value of B] cos(nt + [theta])]([mu] + in[beta]) (3.7)

Where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Results and discussion:

The effect of permeability of the porous medium and the effect of non Newtonian parameter [beta] on the velocity of the fluid is examined. The graphs are drawn taking the fluctuvating parts [S.sub.r] and [S.sub.i] of the velocity for different values of k and [beta]. As the Non-newtonian parameters ([beta]) increases [S.sub.r] and [S.sub.i] increasing (Fig. 3, Fig. 4). Increase in the permeability parameter shows decrease in [S.sub.r] but increase in [S.sub.i] (Fig. 5, 6). As frequency of oscillations is increasing the value of [S.sub.r] and [S.sub.i] are also increases (Fig. 1,2). The increasing permeability parameter increases the amplitude through which the Skinfriction increases (Fig. 7).

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

References

[1] Musakat M. Flow of Homogenous fluid through porous medium, Mc Graw Hill, Inc., New York, 1937.

[2] Musakat, M. Physical Principles of oil production, Mc Graw-Hill, Inc, New York, 1949.

[3] Beavers S.G. and Joseph D.D., Boundary conditions at natural permeable wall, Int. J. FluidMech. 30 (1967), 197-207.

[4] Saffman P.G. On the boundary conditions at the surface of porous medium. Stud. Appl. Math, 50 (1970), 93-101.

[5] Brinkman H.C., The calculation of viscous force exerted by a flowing fluid on a dense swarf of particle, J. Appl. Sci. Res., 27(A1) (1947), 27-34.

[6] Yamamoto K and Iwamura N, Flow with convective acceleration through a porous medium. J. Eng. Math. 10(1) (1976), 41-54.

[7] Narasimha Charyulu, V. Magneto Hydrodynamic flow through a straight porous tube of arbitrary cross section, Indian J. Math., 39 (1997), 267-274.

[8] Narasimha Charyulu V. Oscillatory flow of Newtonian fluid through porous medium under magnetic field, Inter. J. Eng. Math. Theo. Appl, 1 (2007), 7784

[9] Narasimha Charyulu V and Pattabhi Rama Charyulu, N. Ch. Steady flow through a porous region beteen two cylinders, IISC 60(2) (1978), 37-42

[10] Narasimha Charyulu V and Sunder Ram M, Laminar flow of an incompressible micropolar fluid between two parallel plates with porous lining. Int. J. Appl. Math Mech, 6(14) (2010), 81-92.

[11] Ting T.W., Certain non steady flows of second order fluids, Arch. Rational Mech 14 (1963), 1-26.

[12] Erdogen M.E., On un-steady motion of second order fluid over a plane wall, Inter J. Non-linear Mech. 38(7) (2003) 1045-1051.

[13] Light Hill M.J. The response of laminar skin friction and heat transfer to fluctuvations in the steram velocity, Proceedings of Royal Society of London Series A 224 (1954) 1-23.

[14] Stuart J.T. A solution of the Navier-stokes and energy equations illustrating the response of the skin friction and temperature of an infinite plate thermometer to fluctuations in stream velocity. Proc. R. Soc., 231A, 116-130.

[15] Messiha, S.A.S. Laminar boundary layer in oscillating flow along an infinite flat plate with variable suction. Proc. Camb. Phil. Soc., 62, 329-337.

[16] Soundalgekar, V.M. and Das Gupta A.K. Oscillatory flow of an viscoelastic fluid past and infinite porous plate with variable suction. Acta. Ciencia Indica, 3(3) (1977) 274-278.

[17] Johari A.K. and Sharma, J.S. Fluctuating flow of a viscoelastic fluid past a porous plate in a Rotating medium with an applied magnetic field. Indian J. Pure Appl. Math., 13(9) (1982), 1098-1107

Narasimha Charyulu. V. and Shiva Shanker K.

Research Centre of Mathematics, Kakatiya Institute of Technology & Science, Warangal, Andhra Pradesh, India. E-mail vn_charyulu@yahoo.com and sskache@gmail.com
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Author:V., Narasimha Charyulu.; K., Shiva Shanker
Publication:International Journal of Computational and Applied Mathematics
Article Type:Report
Geographic Code:9INDI
Date:Jul 1, 2012
Words:1583
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