Viscosity of Poiseuille flow of a couple stress fluid with applications to blood flow

Abstract

Viscosity of Poiseuille flow of a fluid with couple stress has been studied in this paper. The analytic expression of the viscosity of couple stress fluid has been obtained in the form of modified Bessel’s functions of the order of zero and one. Since it is a complicated function of the couple stress parameters α and η, the numerical values of the relative viscosity ${\mathit{\eta }}_{\mathit{r}}$ have been computed for various values of α and η. These theoretical results are compared with other theoretical and experimental results on blood flow and suspension flows with 40% concentration. For the quick calculation of ${\mathit{\eta }}_{\mathit{r}}$ two simple approximate formulas have been obtained. The values of ${\mathit{\eta }}_{\mathit{r}}$ obtained from the exact formula and the approximate formulas are compared and it is found that they are in good agreement (within 6% error). The variation of ${\mathit{\eta }}_{\mathit{r}}$ (exact and approximate) with α and η is shown graphically which clearly indicates the existence of a discontinuity in the relative viscosity and the velocity at point (α = 0.0 and η = 1.0). The most important conclusions of this analysis are: (i) the conditions for the existence of Fahraeus-Lindquist effect in a tube flow have been obtained; (ii) Up to this date, only velocity profiles have been used to determine the values of α and η (η chosen arbitrarily); here it is shown that by using the experimental velocity profiles and relative viscosity both the couple stress parameters can be determine quite accurately (η no longer chosen arbitrarily). Finally, some biological implications of this theoretical investigation have been indicated.