Flow of Newtonian fluid in non-uniform tubes with variable wall permeability with application to flow in renal tubules

Abstract

The hydrodynamical aspects of flow through proximal renal tubule have been investigated. Assuming renal fluid as Newtonian fluid, flow through diverging/converging tubes with variable wall permeability has been considered. Solutions have been obtained for approximate Navier-Stokes equations with boundary conditions which include a dynamic condition, i.e., leakage flux at the wall depends on variable wall permeability and transboundary pressure drop. Numerical solutions, using fourth order Runge-Kutta method, and approximate analytic solutions, using perturbation method, have been obtained. A comparison of the numerical solution with approximate analytic solution, shows a good agreement (difference less than 4%) between the two solutions for small values of |e|, a tube non-uniformity parameter. The velocity profiles at different positions along the axis, the axial distribution of wall shear stress, flow rate and leakage flux have been obtained. For a given value of wall permeability, in diverging (converging) tubes the fractional reabsorption FR is more (less) than its corresponding value in uniform tubes. Further, FR increases (decreases) as the wall permeability increases (decreases) as a linear function of axial distance. The results for flow with constant/variable permeability through uniform tubes and for flow through diverging/converging tubes with constant permeability can be obtained as special cases of this analysis. It is shown that by considering the divergent tube model with linear increase of wall permeability along the axis, an improvement of about 20% in total reabsorption can be achieved over the uniform tube model with constant wall permeability. It is concluded that the approach of using a dynamic boundary condition for leakage flux at the wall has an advantage over the method of prescribing the leakage flux at the wall for this physical problem. Using a set of data, relevant to a physiological situation, implications of the results on glomerular tubular balance have been briefly discussed.