Flow of a Newtonian fluid in a non-uniform wavy and permeable tube

Abstract

In this paper, a viscous incompressible fluid flow in a wavy non-uniform rigid tube with permeable wall by taking in to account the influence of slip velocity at the wall is studied. It is assumed that the exchange of fluid across the wall obeys Starling’s hypothesis, that is, the rate of flow per unit area through the wall surface is proportional to the difference between the pressure of the fluid within and outside the wall. The nonlinear governing equations of motion are linearized by perturbation method by assuming δ (ratio of inlet width to wavelength) as a small parameter and the resulting equations are solved by numerical methods. The effects of permeability parameter (α), slope parameter (k), slip coefficient (ξ) and Reynolds number (R_e) on the velocity profiles, pressure and flow rate are presented graphically. Results concerning the velocity, pressure and flow rate, indicate that the slip and permeability parameters influence the flow field significantly. Discussions are made from physiological point of view.

Keywords

• Fluid flow

Kaynakça

1. R. B. Kelman. A Theoretical Note on Exponential Flow in the Proximal Part of the Mammalian Nephron. Bull. of Mathematical Biophysics, vol. 24, 303 - 317, 1962.
2. Robert I. Macey. Pressure Flow Patterns in a Cylinder with Reabsorbing Walls . Bull. of Mathematical Biophysics, 25, 1 - 9, 1963.
3. Robert I. Macey. Hydrodynamics in Renal Tubules. Bull. of Mathematical Biophysics, 27, 117-124, 1965.
4. E. A. Marshall, and E. A. Trowbridge. Flow of a Newtonian Fluid through a Permeable Tube: The Application to the Proximal Renal Tubule. Bull. of Mathematical Biology, vol. 36, 457 - 476, 1974.
5. Paul J. Palatt, Henry Sackin, and Roger I. Tanner. A Hydrodynamical Model of a Permeable Tubule. J. theor. Biol., 44, 287-303, 1974.
6. G. Radhakrisnamacharya, Peeyush Chandra, and M. R. Kaimal. A Hydrodynamical Study of the Flow in Renal Tubules. Bull. of Mathematical Biology, vol. 43, No.2:151-163, 1981.
7. Peeyush Chandra, and J.S.V.R. Krishna Prasad. Low reynolds number flow in tubes of varying cross-section with absorbing walls. Jour. Math. Phy. Sci., vol. 26, No.1, February 1992.
8. P. Chaturani, and T. R. Ranganatha. Flow of Newtonian fluid in non-uniform tubes with variable wall permeability with application to flow in renal tubules. Acta Mechanica, 88, 11-26, 1991.

9. G. S. Beavers, D. D. Joseph. Boundary conditions at a naturally permeable wall. Journal of Fluid Mechanics, 30, 197-207, 1967.
10. P. Dulal, N. Rudraiah, D. Rathna. The effects of slip velocity velocity at a membrane surface on blood flow in the microcirculation. Journal of Mathematical Biology, 26, 705-712, 1988.
11. I. J. Rao, T. Rajagopal. The effect of the slip boundary condition on the flow of fluids in channel. Acta Mechanica, 135, 113-126, 1999.
12. Moustafa Elshahed. Blood flow in capillary under starling hypothesis. Applied mathematics and computation, 149, 431-439, 2004.
13. P. Muthu and Tesfahun Berhane. Mathematical model of flow in renal tubules, Int. J. of Appl. Math. and Mech., 6(20), 94-107, 2010.
14. Syoten Oka and Tadayoshi Murata, A Theoretical Study of the Flow of Blood in a Capillary with Permeable wall, Japanese Journal of Applied Physics, 9(4), p: 345-352, 1974.
15. N. K. Mariamma and S. N. Majhi, Flow of a Newtonian Fluid in a Blood Vessel with Permeable Wall - A Theoretical Model, Computers and Mathematics with Applications, 40, p: 1419-1432, 2000.