Mathematical model of solute transport in channels with variable geometry and permeable walls

This paper investigates solute transport dynamics within a steady, viscous, Newtonian fluid flow through channels of varying geometry (convergent-divergent boundaries) and permeable walls. The significant impact of osmotic pressure on solute concentration is particularly highlighted by modeling osmotic pressure as a cubic function of solute concentration. Approximate solutions to the Navier-Stokes equations and solute transport equations, along with the corresponding boundary conditions, have been derived. The effects of varying geometry and various emerging parameters on hydrostatic and osmotic pressures, solute concentration, and solute clearance are illustrated through graphs. Results indicate a linear decline in hydrostatic pressure and a nonlinear rise in osmotic pressure along the channel length, with divergent channels producing higher solute and wall concentrations than uniform or convergent channels. Increased ultrafiltration and absence of osmotic pressure yield greater wall concentrations. Solute clearance rates increase with the wall slope, transmittance coefficient, permeability, and ultrafiltration parameters, underscoring intricate parameter relationships within filtration. A dataset is used to visually present and analyze the influence of different physiological factors on solute concentration, with graphs corresponding to real physiological conditions. For channels with the uniform width, the findings closely align with previous results, demonstrating its applicability to solute exchange in glomerular capillaries as well as to other blood vessels or capillaries.