Mathematical analysis of viscosity and reabsorption on urine flow through a straight narrow tube

This study investigates the effect of variable viscosity (exponential and linear) and constant reabsorption for the urine flow through a narrow tube. The inertial free flow of viscous fluid has been governed by the momentum and mass conservation through the cross-section of axisymmetric tube. The governing partial differential equations have been simplified with the help of stream function and stress components with exponential and linear variable viscosity. The resulting partial differential equations have been solved by the inverse method and give the explicit expressions for velocity, pressure, shear stress, flux and leakage of flow. It has been observed that flow in transverse direction increases with the increase in reabsorption velocity at wall, whereas horizontal flow, shear stress and volume flow rate become slow with the increase in uniform reabsorption velocity. Effect of viscosity is significant near the walls of tube because the axial velocity accelerates by increasing viscosity parameter due to the pressure gradient near the center of tube but it decelerates near the walls of tube due to surface friction. Also, the special case of variable viscosity is discussed by assuming the linear type of viscosity. The derived data for the velocity and flow rate have been used to measure the fractional reabsorption in proximal tube with varying viscosity near the wall.