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Effects of Slip Condition on Unsteady MHD Oscillatory Flow of a Viscoelastic Fluid in a Planar Channel

Utpal Jyoti Das

Department of Mathematics, Rajiv Gandhi University, Itanagar-791112, Arunachal Pradesh, INDIA.

Corresponding Addresses:

[email protected]

Research Article

Abstract: An analytical study for unsteady magnetohydrodynamic oscillatory flow of an incompressible viscoelastic fluid in a planar channel filled with saturated porous medium is studied.  It is assumed that the no-slip condition between the wall and the fluid remains no longer valid. The effect of the wall slip on velocity field and shear stress are illustrated graphically and physical aspects of the problem are discussed.

2000 Mathematics Subject Classification: 76A05, 76A10.

Keywords: MHD, visco-elastic, radiation effect, heat transfer, porous channel, slip condition.

1. Introduction

There are many significant differences between fluid flow at macro scale and that at micro/nano scale, e.g., wall-slip phenomena. Effects of slip conditions are very important for some fluids that exhibit wall slip. Fluids exhibiting slip are important in technological applications such as in the polishing of artificial heart valves, internal cavities and polymer melts. Therefore, the better understanding of the slip phenomena is necessary. Navier’s [1] proposed condition assumes that the velocity,  at a solid surface is proportional to the shear stress at the surface,

where  is the slip strength or slip coefficient. If then the general assumed no-slip condition is obtained. If  is finite, fluid slip occurs at the wall but its effect depends upon the length scale of the flow. In the above relation, velocity of the fluid at the plates is linearly proportional to the shear stress at the plate. Also, a non-linear slip boundary conditions can also be imposed.

With a hot-wire anemometer Watanabe et al. [2-3] identify fluid slip at the wall of a strongly hydrophobic duct or pipe. Their velocity profiles are consistent with Navier’s hypothesis. Ruckenstein and Rajora [4] investigated fluid slip in glass capillaries with surfaces made repellent to the flowing liquid. Their experimental results of pressure drop indicate larger slip than that predicted by chemical potential theory, where slip is proportional to the gradient in the chemical potential. Mooney [5] studied the boundary layer flows with partial slip. Many researchers [6–7] had confirmed the phenomena of wall-slip fluids. The effect of slip condition on MHD steady flow in a channel with permeable boundaries has been discussed by Makinde and Osalusi [8]. Mehmood and Ali [9] studied the effect of wall slip on oscillatory flow of a viscous fluid in a planar channel. The stagnation slip flows on a fixed plate and on a moving one were considered by wang [10–11]. Hayat et al. [12] examined the effects of slip boundary conditions on fluid flow in a channel. The non-Newtonian flows with wall slip have been numerically studied in [13- 14].

2. Formulation of the problem

Consider an incompressible flow of a conducting optically thin fluid bounded by two parallel plates separated by a distance a. The channel is assumed to be filled with a saturated porous medium. A uniform magnetic field of strength  is applied perpendicular to the plates. The above plate is heated at constant temperature and the radiation effect is also taken into account. Therefore, the governing equations for this flow in dimensionless form are given by

The dimensionless governing Eqns. together with the appropriate boundary conditions (neglecting the bars for clarity) can be written as

     (1)

        (2)   with

       on

on     (3)

where are Grashoff number, Hartmann number, Radiation parameter, Pe^/clet number, Reynolds number, Darcy   number, porous medium shape factor parameter, visco-elastic parameter, and  the dimensionless slip parameter respectively.

For purely an oscillatory flow, let

         (4)

where is a constant and is the frequency of oscillation.

 Substituting the above expressions into the Eqns. (1) and (2) and using the boundary condition (3), we get

    (5)

     (6)

subject to boundary conditions

       on 

      on              (7)

where dash denotes the differentiation with respect to and .

Solving equation (6), we get

      (8)

Substituting equation (8) in equation (5), and then solving it, we get

           (9)

where      

and

2. Discussion and Conclusion:

The purpose of this study is to bring out the result of physical impact of the slip with various combination of the viscoelastic parameter. For numerical validation of our analytical results, we have taken the real parts of the results throughout. In figures 1-3 we have plotted the velocity against  with slip parameter  for different values of the viscoelastic parameter.  It is observed that by increasing the slip at the wall, the velocity increases at the wall.

Fig 1: Variation of  against with

Fig 2: Variation of  against with

Fig 3:  Variation of  against with

The non-dimensional shear stress at the wall  is given by

The figure 4 exhibits the effects of the shear stress  against time

Fig 4: Variation of  against t with

From the figure 4, it is observed that the shear stress decrease with the increasing values of the slip at the wall.

References

1. C. L. M. H. Navier,  Memoirs de l’Academie Royale des Sciences de l’Institut de France, 1, 414-416 (1823).
2. K. Watanabe, Yanuar, and H. Mizunuma, Slip of Newtonian Fluids at Slid Boundary, JSME International Journal Series B, 41, 525-529 (1998).
3. K. Watanabe,  Yanuar, and H. Udagawa, Drag Reduction of Newtonian Fluid in a Circular Pipe With a Highly Water-Repellant Wall,  J. of Fluid Mechanics, 381, 225-238 (1999).
4. E. Ruckenstein, and P. Rajora, On the No -Slip Boundary Condition of Hydrodynamics, J. of Colloid and Interface Science, 96, 488-491 (1983).
5. M. Mooney, Explicit formulas for slip and fluidity. Journal of Rheology, 2(2), 210–222 (1931).
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8. O.D.Makinde, E. Osalusi, MHD steady flow in a channel with slip at the permeable boundaries, Rom. J. Phys. 51(3-4) 319-328 (2006).
9. A. Mehmood, and A. Ali, The effect of slip condition on unsteady MHD oscillatory flow of a  viscous fluid in a planar channel, Rom. Journ. Phys.,  52(1-2),  85-91 (2007).
10. C. Y. Wang, Flow due to a stretching boundary with partial slip—an exact solution of the Navier-Stokes equations. Chem. Eng. Sci., 57(17), 3745–3747 (2002).
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14. H. I. Andersson, and M. Rousselet, Slip flow over a lubricated rotating disk, Int. J. Heat Fluid Flow, 27(2), 329–335 (2006).
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