Mathematical model of pulsatile flow of non-Newtonian fluid in tubes of varying cross-sections and its implications to blood flow

Abstract

The effects of rheological behavior of blood and pulsatility on flow through an artery with stenosis have been investigated. Blood has been represented by a non-Newtonian fluid obeying Herschel–Bulkley equation. Using the Reynolds number as the perturbation parameter, a perturbation technique is adopted to solve the resulting quasi-steady non-linear coupled implicit system of differential equations. Analytical expressions for velocity distribution, wall shear stress, volumetric flow rate and the mean flow resistance have been obtained. It is observed that the wall shear stress and flow resistance increase for increasing value of yield stress with other parameters held fixed. One of the remarkable results of the present analysis is not only to bring out the effect of the size of the stenosis but also to study the influence of the shape of the stenosis. The change in the shape of the stenosis brings out a significant change in the value of flow resistance but it has no effect on the variation of wall shear stress except shifting the point (where it attains its maximum value) towards downstream. It is pertinent to point out that pulsatile flow of Newtonian fluid, Bingham plastic fluid and Power-law fluid become particular cases of the present model. The present approach has general validity in comparison with many mathematical models developed by others and may be applied to any mathematical model by taking into account of any type of rheological property of blood. The obtained velocity profiles have been compared with the experimental data and it is observed that blood behaves like a Herschel–Bulkley fluid rather than Power-law and Bingham fluids. Finally, some biorheological applications of the present model have briefly been discussed.