Parameter effects on shear stress of Johnson–Segalman fluid in Poiseuille flow

Abstract

Exact solutions of shear stress versus velocity gradient and the numerical solutions of streamwise velocity distribution in radial direction of a Johnson–Segalman fluid in a circular pipe are obtained. The effects of material parameters, Weissenberg number, ratio of viscosities and slip parameter, on shear stress and streamwise velocity have been considered to investigate the discontinuous velocity derivatives and stick-slip phenomenon at the wall. We find that there is a non-monotonic relationship between the shear stress and rate of shear for certain values of the material parameters and consequently, the velocity profile has discontinuous derivatives. Also the non-monotonic region is shortening with the increasing of ratio of viscosities, and is lengthening with the increasing of slip parameter. For the flow with larger slip parameter, the initially driving pressure gradient has to be larger for the appearance of spurt phenomenon. Moreover, the variational range of material parameters is given for the appearance of a non-monotonic relationship between the shear stress and the rate of shear. Finally, we have shown the exact expression of critical pressure gradient and also have given the conditions where spurt phenomena occur.