Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-06-07T00:31:27.162Z Has data issue: false hasContentIssue false
  • English
  • Français

Axisymmetric pressure-driven flow of rigid pellets through a cylindrical tube lined with a deformable porous wall layer

Published online by Cambridge University Press:  26 April 2006

E. R. Damiano ,
B. R. Duling ,
K. Ley  and
T. C. Skalak
E. R. Damiano
Affiliation:
Departments of Biomedical Engineering and Physiology, University of Virginia, Charlottesville, VA 22908, USA
B. R. Duling
Affiliation:
Departments of Biomedical Engineering and Physiology, University of Virginia, Charlottesville, VA 22908, USA
K. Ley
Affiliation:
Departments of Biomedical Engineering and Physiology, University of Virginia, Charlottesville, VA 22908, USA
T. C. Skalak
Affiliation:
Departments of Biomedical Engineering and Physiology, University of Virginia, Charlottesville, VA 22908, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

A closed-form analytic solution for the motion of axisymmetric rigid pellets suspended in a Newtonian fluid and driven under a pressure gradient through a rigid impermeable cylindrical tube lined with a porous deformable biphasic wall layer is derived using mixture and lubrication theories. The analysis details the velocity distributions in the lubrication and wall layers as well as the solid-phase displacement field in the wall layer. Expressions for the shear stress and pressure gradient are obtained throughout the lubrication and wall layers. Results are presented in terms of resistance, volume flow, and driving pressure relative to smooth-walled tubes for cases both with and without rigid spheres flowing in the free lumen. The analysis is motivated by its possible relevance to the rheology of blood in the microcirculation wherein the endothelial-cell glycocalyx – a carbohydrate-rich coat of macromolecules consisting of proteoglycans and glycoproteins expressed on the luminal surface of the capillary wall – might exhibit similar behaviour to the wall layer modelled here. Estimates of the permeability of the glycocalyx are taken from experimental data for fibrinogen gels formed in vitro. In a tube without pellets lined with a porous wall layer having a thickness which is 15% of the tube radius and having a permeability in the range of fibrinogen gels, approximately a 70% greater pressure drop is required to achieve the same volume flow as would occur in an equivalent smooth-walled tube without a wall layer. If, in the presence of this same wall layer, a rigid spherical pellet is introduced which is 99.5% of the free-lumen radius, the apparent viscosity increases by as much as a factor of four with a concomitant reduction in tube hematocrit of about 10% relative to the corresponding values in an equivalent smooth-walled tube having the same sphere-to-tube diameter ratio without a wall layer.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 314 , 10 May 1996 , pp. 163 - 189
Copyright
© 1996 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barnard, A. C. L., Lopez, L. & Hellums, J. D. 1968 Basic theory of blood flow in capillaries. Microvasc. Res. 1, 23–34.Google Scholar
Beavers, G. S. & Joseph, D. D. 1967 Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30, 197–207.Google Scholar
Blombäck, B., Carlsson, K., Hessel, B., Liljeborg, A., Procyk, R. & Åslund, N. 1989 Native fibrin gel networks observed by 3D microscopy, permeation and turbidity. Biochim. Biophys. Acta 997, 96–110.Google Scholar
Blombäck, B. & Okada, M. 1982 Fibrin gel structure and clotting time. Thromb. Res. 25, 51–70.Google Scholar
Chien, S., Usami, S., Taylor, H. M., Lundberg, J. L. & Gregersen, M. I. 1966 Effects of hematocrit and plasma proteins on human blood rheology at low shear rates. J. Appl. Physiol. 21, 81–87.Google Scholar
Copley, A. L. 1974 Hemorheological aspects of the endothelium-plasma interface. Microvasc. Res. 8, 192–212.Google Scholar
Desjardins, C. & Duling, B. R. 1990 Heparinase treatment suggests a role for the endothelial cell glycocalyx in regulation of capillary hematocrit. Am. J. Physiol. 258, H647–H654.Google Scholar
Fitz-Gerald, J. M. 1969 Mechanics of red-cell motion through very narrow capillaries. Proc. R. Soc. Lond. B 174, 193–227.Google Scholar
Gretz, J. E. 1995 The role of capillary anatomical and functional dimensions in capillary tube hematocrit variability. PhD thesis, University of Virginia, Charlottesville, VA.
Hou, J. S., Holmes, M. H., Lai, W. M. & Mow, V. C. 1989 Boundary conditions at the cartilage-synovial fluid interface for joint lubrication and theoretical verifications. J. Biomech. Engng 111, 78–87.Google Scholar
Hou, J. S., Holmes, M. H., Lai, W. M. & Mow, V. C. 1990 Squeeze film lubrication for articular cartilage with synovial fluid. In Biomechanics of Diarthrodial Joints (ed. V. C. Mow, A. Ratcliffe & S. L-Y. Woo), pp. 347–367. Springer.
King, K., Procyk, R., Chien, S., Blombäck, B. & Copley, A. L. 1988 The effect of factor XIII and fibronectin on the viscoelasticity of fibrinogen surface layers. Thromb. Res. 49, 139–144.Google Scholar
Klitzman, B. & Duling, B. R. 1979 Microvascular hematocrit and red cell flow in resting and contracting striated muscle. Am. J. Physiol. 237, H481–H490.Google Scholar
Krindel, P. & Silberberg, A. 1979 Flow through gel-walled tubes. J. Colloid Interface Sci. 71, 39–50.Google Scholar
Lahav, J., Eliezer, N. & Silberberg, A. 1973 Gel-walled cylindrical channels as models for the microcirculation: Dynamics of flow. Biorheology 10, 595–604.CrossRefGoogle Scholar
Lai, W. M. & Mow, V. C. 1980 Drag-induced compression of articular cartilage during a permeation experiment. Biorheology 17, 111–123.Google Scholar
Levick, J. R. 1987 Flow through interstitium and other fibrous matrices. Q. J. Exp. Physiol. 72, 409–438.Google Scholar
Lighthill, M. J. 1968 Pressure-forcing of tightly fitting pellets along fluid-filled tubes. J. Fluid Mech. 34, 113–143.Google Scholar
Mow, V. C., Kuei, S. C., Lai, W. M. & Armstrong, C. G. 1980 Biphasic creep and stress relaxation of articular cartilage in compression: Theory and experiments. J. Biomech. Engng 102, 73–84.Google Scholar
Özkaya, N. 1986 Viscous flow of particles in tubes: Lubrication theory and finite element models. PhD thesis, Columbia University, New York, NY.
Pries, A. R., Neuhaus, D. & Gaehtgens, P. 1992 Blood viscosity in tube flow: dependence on diameter. Am. J. Physiol. 263, H1770–H1778.Google Scholar
Pries, A. R., Secomb, T. W., Gaehtgens, P. & Gross, J. F. 1990 Blood flow in microvascular networks: experiments and simulation. Circ. Res. 67, 826–834.Google Scholar
Pries, A. R., Secomb, T. W., Geßner, T., Sperandio, M. B., Gross, J. F. & Gaehtgens, P. 1994 Resistance to blood flow in microvessels in vivo. Circ. Res. 75, 904–915.Google Scholar
Procyk, R. & King, K. 1990 The elastic modulus of fibrin clots and fibrinogen gels: the effect of fibronectin and dithiothreitol. Biopolymers 29, 559–565.CrossRefGoogle Scholar
Secomb, T. W. 1991 Red blood cell mechanics and capillary blood rheology. Cell Biophys. 18, 231–251.Google Scholar
Secomb, T. W. 1995 Mechanics of blood flow in the microcirculation. In Biological Fluid Dynamics (ed. C. P. Ellington & T. J. Pedley), pp. 305–321. Company of Biologists, Cambridge.
Secomb, 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
Skalak, R. & Özkaya, N. 1987 Models of erythrocyte and leukocyte flow in capillaries. In Physiological Fluid Dynamics II (ed. L. S. Srinath & M. Singh), pp. 1–10. Tata McGraw-Hill, New Delhi.
Skalak, R., Özkaya, N. & Skalak, T. C. 1989 Biofluid mechanics. Ann. Rev. Fluid Mech. 21, 167–204.Google Scholar
Tözeren, H. & Skalak, R. 1978 The steady flow of closely fitting incompressible elastic spheres in a tube. J. Fluid Mech. 87, 1–16.Google Scholar
Truesdell, C. & Toupin, R. 1960 The classical field theories. In Handbuch der Physik (ed. S. Flügge), pp. 226–793. Springer.
Wang, W. & Parker, K. H. 1995 The effect of deformable porous surface layers on the motion of a sphere in a narrow cylindrical tube. J. Fluid Mech. 283, 287–305.Google Scholar
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