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The effect of deformable porous surface layers on the motion of a sphere in a narrow cylindrical tube

Published online by Cambridge University Press:  26 April 2006

Wen Wang  and
Kim H. Parker
Wen Wang
Affiliation:
Physiological Flow Studies Group, Centre For Biological and Medical Systems, Imperial College of Science, Technology and Medicine, London SW7 2BY, UK
Kim H. Parker
Affiliation:
Physiological Flow Studies Group, Centre For Biological and Medical Systems, Imperial College of Science, Technology and Medicine, London SW7 2BY, UK
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Abstract

The hydrodynamic influence of deformable porous surface layers on the motion of a rigid sphere falling in a narrow cylindrical tube filled with a stationary Newtonian fluid is studied using lubrication theory. The porous layers on both the surface of the tube and the sphere are modelled as binary mixtures of solid and liquid components. The sphere is placed at an arbitrary position in the tube and is free to rotate. Effects of the clearance between the sphere and the tube, the eccentricity of the position of the sphere and the properties of the surface layers on the velocity and rotation of the sphere are studied. It is found that, when the lengthscale on which the velocity varies within the porous layer is much smaller than the clearance, the effects of the porous layer can be represented by an equivalent slip boundary condition, the slip velocity at the boundary being proportional to the local shear rate. The slip velocities have a strong influence on the motion of the sphere when the clearance is small. For a given clearance and slip parameters, both the falling and rotation velocities of the sphere increase with the sphere eccentricity. The shear stresses on the surfaces of both the tube and the sphere are greatly reduced when slip boundary conditions are applied, as is the pressure gradient in the region between the sphere and the tube wall. This work could have some relevance to the creeping motion of blood cells in the microcirculation where the glycocalyx, a polysaccharide-rich layer, covers the external surfaces of both endothelial and red blood cells.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 283 , 25 January 1995 , pp. 287 - 305
Copyright
© 1995 Cambridge University Press

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References

Barry, S. I., Parker, K. H. & Aldis, G. K. 1991 Fluid-flow over a thin deformable porous layer. Z. Angew. Math. Phys. 42, 633–648.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press. Section 4.9, pp. 238–240.
Beavers, G. S. & Joseph, D. D. 1967 Boundary condition at a naturally permeable wall. J. Fluid Mech. 30, 197–207.Google Scholar
Bohlin, T. 1960 On the drag on a rigid sphere moving in a viscous liquid inside a cylindrical tube. Trans. R. Inst. Technol. Stockholm 155, 64.Google Scholar
Bowen, R. M. 1980 Incompressible porous media models by the theory of mixture. Intl J. Engng Sci. 18, 1129–1148.Google Scholar
Brenner, H. & Happel, J. 1958 Slow viscous flow past a sphere in a cylindrical tube. J. Fluid Mech. 4, 195–213.Google Scholar
Brinkman, H. K. 1947 A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. A 1, 27–34.Google Scholar
Bungay, P. M. & Brenner, H. 1973 The motion of a closely-fitting sphere in a fluid-filled tube. Intl J. Multiphase Flow 1, 25–56.Google Scholar
Caro, C. G. 1982 Arterial fluid-mechanics and atherogenesis. Clinical Hemorheol. 2, 131–136.Google Scholar
Chester, W. 1984 The motion of a sphere down a liquid-filled tube. Proc. R. Soc. Lond. A 396, 205–215.Google Scholar
Christopherson, D. G. & Dowson, D. 1959 An example of minimum energy dissipation in viscous flow. Proc. R. Soc. Lond. A 251, 550–564.Google Scholar
Cokelet, G. R., Merrill, E. W., Gilliland, E. R., Shin, H., Britten, A. & Wells, R. E. 1963 The rheology of human blood measurement near and at zero shear rate. Trans. Soc. Rheol. 7, 303–317.Google Scholar
Curry, F. E. 1984 Mechanics and thermodynamics of transcapillary exchange. In Handbook of Physiology, Section 2: The Cardiovascular System. Vol. 4, Microcirculation, part 1, pp. 309–374. American Physiological Society.
Fung, Y. C. & Liu, S. Q. 1993 Elementary mechanics of the endothelium of blood vessels. J. Biomech. Engng 115, 1–12.Google Scholar
Haldenby, K. A., Chappell, D. C., Winlove, C. P., Parker, K. H. & Firth, J. A. 1994 Focal and regional variations in the composition of the glycocalyx of large vessel endothelium. J. Vasc. Biol. 31, 2–9.Google Scholar
Hill, A. E. & Hill, B. S. 1987 Steady-state analysis of ion fluxes in necturus gallbladder epithelialcells. J. Physiol. Lond. 382, 15–34.Google Scholar
Hou, J. S., Holmes, M. H., Lai, W. M. & Mow, V. C. 1989 Boundary conditions at the cartilagesynovial fluid interface for joint lubrication and theoretical verifications. J. Biomech. Engng. 111, 78–87.Google Scholar
Kenyon, D. E. 1976 The theory of an incompressible solid-fluid mixture. Arch. Rat. Mech. Anal. 62, 131–147.Google Scholar
Kenyon, D. E. 1979 A mathematical model of water flux through aortic tissue. Bull. Math. Biol. 41, 79–90.Google Scholar
Kolodziej, J. A. 1988 Influence of the porosity of a porous-medium on the effective viscosity in Brinkman filtration equation. Acta Mechanica 75, 241–254.Google Scholar
Lai, W. M., Hou, J. S. & Mow, V. C. 1991 A triphasic theory for the swelling and deformation behaviours of articular cartilage. J. Biomech. Engng 113, 245–258.Google Scholar
Lanir, Y., Dikstein, S., Hartzshtark, A. & Manny, V. 1990 In vivo indentation of human skin. J. Biomech. Engng 112, 63–69.Google Scholar
Lundgren, T. S. 1972 Slow flow through stationary random beds and suspensions of spheres. J. Fluid Mech. 51, 273–299.Google Scholar
Merrill, E. W., Benis, A. M., Gilliland, E. R., Sherwood, T. K. & Salzman, E. W. 1965 Pressure-flow relations of human blood in hollow fibres at low flow rates. J. Appl. Physiol. 20, 954–967.Google Scholar
Mow, V. C., Kwan, M. K., Lai, W. M. & Holmes, M. H. 1985 A finite deformation theory for nonlinearly permeable soft hydrated biological tissues. In Frontiers in Biomechanics (ed. G. Schmid-Schoenbein, S. L.-Y. Woo & B. W. Zweifach), pp. 153–179. Springer.
Nerem, R. M. & Girard, P. R. 1990 Hemodynamic influences on vascular endothelial biology. Toxic. Pathol. 18, 572–582.Google Scholar
Richardson, S. 1971 A model for the boundary condition of a porous material. Part 2. J. Fluid Mech. 49, 327–336.Google Scholar
Saffman, P. G. 1971 On the boundary condition at the surface of a porous medium. Stud. Appl. Maths 1, 93–101.Google Scholar
Secombe, T. W., Skalak, R., Özkaya, N. & Gross, J. F. 1986 Flow of axisymmetric red blood cells in narrow capillaries. J. Fluid Mech. 163, 405–423.Google Scholar
Silberberg, A. 1991 Polyelectrolytes at the endothelial cell surface. Biophys. Chem. 41, 9–13.Google Scholar
Taylor, G. I. 1971 A model for the boundary condition of a porous material. J. Fluid Mech. 49, 319–326.Google Scholar
Truesdell, C. & Toupin, R. A. 1960 The Classical Field Theories. Handbuch der Physik III/1, Springer.
Wang, H. & Skalak, R. 1969 Viscous flow in a cylindrical tube containing a line of spherical particles. J. Fluid Mech. 38, 75–96.Google Scholar