A mathematical model of blood flow in a permeable channel: application to flat plate dialyzer

Abstract

This article presents the theoretical study of the physiological phenomenon brought up by filtration of a non-Newtonian Casson fluid between two parallel permeable membranes. This situation corresponds to blood filtration process in a flat plate hemodialyzer (FPH). Seepage of fluid across the membrane is considered in accordance with the Darcy’s law and the equations of motion governing the flow are modeled. Using the low Reynolds number and long membrane length assumption, equations of motion are solved exactly. Equations describing velocity and pressure filed and various flow variables are derived and effects of wall slip parameter, wall filtration coefficient and the yield stress are presented graphically. A strong influence of these parameters is observed on the flow in an FPH. Theoretical values of the membrane filtration coefficient and mean pressure drop in an FPH are calculated and they are found to be in close agreement with the corresponding available empirical and experimental values in the literature. For certain limiting range of physical parameters, derived solutions reveal that the axial flow rate of Casson fluid in an FPH decays at an exponential rate. This is a physically valid and widely admitted result, used by several researchers in studying the blood filtration process in renal tubules of mammalian kidneys. Since the presented solutions in this article are reduced to their corresponding Newtonian fluid flow solutions between permeable membrane, therefore, it is concluded that a wide range of applications in physiology and engineering can be covered up by the present investigation.