Mathematical Model of Fluid Flow and Solute Transport in ‎Converging-diverging Permeable Tubes

This article presents a mathematical model for fluid and solute transport in an ultra-filtered glomerular capillary. Capillaries ‎are assumed to be converging-diverging tubes with permeable boundaries. According to Starling's hypothesis, ultrafiltration is ‎related to the variations in hydrostatic and osmotic pressures in the capillary and Bowman's space and takes place along the ‎length of the capillary. The governing equations of fluid flow for the case of axisymmetric motion of viscous incompressible ‎Newtonian fluid have been considered along with the solute transfer equation. The non-uniform geometry has been mapped ‎into a finite regular computational domain via a coordinate transformation. Correspondingly, the governing equations are ‎transformed to the computational space and solved to get the velocity and pressure values. The solute transfer equation is ‎also solved numerically using a finite difference scheme. The solutions provide the predictions of the axial distribution of ‎hydrostatic pressure and osmotic pressure, velocities, and concentration profiles at various points along the axis and solute ‎clearance quantities along the capillary. The current results are in good agreement with the earlier findings in limiting cases of ‎cylindrical tubes with a constant radius. It is found that there is a significant effect of osmotic pressure on the solute ‎concentration. Using a set of data, the influence of various physiological parameters on the velocity components and solute ‎concentration are presented and discussed through graphs to correlate with physiological situations. The generic structure of ‎the current model also provides an acceptable approach to exploring fluid exchange in organs apart from the glomerular ‎capillary.‎