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

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.