Creeping Flows Through Permeable Ducts: Application to Renal Tubule

Abstract

The primary objective of this thesis is to investigate the flow of various physiological and biological fluids in a small diameter permeable tube under different flow conditions as an application to the flow in the renal tubules of kidney. The problem of creeping motion of an incompressible fluid in a small diameter permeable tube is being presented. The fluid absorption at the tube wall is taken as a function of wall permeability and the pressure gradient across the tube wall. Hydro dynamical flow equations and the appropriate boundary conditions for the flow of Newtonian fluid and some non-Newtonian fluids in a permeable tube are developed. Assuming the tube radius to be very small as compared to its length, exact/approximate solutions of these equations are obtained. Expressions describing variations in the axial volumetric flow rate, mean pressure difference, wall leakage flux, wall shear stress and the fractional re-absorption are derived. Graphs are sketched in order to describe effects of various parameters of interest on different flow variables and implications of the results are discussed briefly. The obtained solutions are applied to the flow problem in the renal proximal tubule of kidneys and theoretical values of the wall permeability parameter, inlet pressure, mean pressure drop and fractional re-absorption in the proximal tubule are obtained. It is found that along with other quantities of interest, the mean pressure drop and the fractional reabsorption in proximal tubule have significant variations as the underlying fluid model and/or flow conditions are varied. For specific values of parameters, the expression of the axial flow rate reveals that volumetric flow rate in the renal tubule decreases exponentially as a function of the axial distance. This is a biologically significant, well accepted and frequently reported result in the literature. When certain parameters are set equal to zero, the derived solutions coincide well with the existing solutions for creeping flow of a linearly viscous (Newtonian) fluid in a small diameter permeable tube.