Unsteady motion past a sphere translating steadily in wormlike micellar solutions: A numerical analysis

Abstract

This study numerically investigates the ow characteristics past a solid and smooth sphere translating steadily along the axis of a cylindrical tube filled with wormlike micellar solutions in the creeping ow regime. The two-species VCM (Vasquez-Cook- McKinley) and single-species Giesekus constitutive models are used to characterize the rheological behaviour of micellar solutions. Once the Weissenberg number exceeds a critical value, an unsteady motion downstream of the sphere is observed in the case of two-species model. We provide the evidence that this unsteady motion downstream of the sphere is caused due to the sudden rupture of long and stretched micelles in this region, resulting from an increase in the extensional ow strength. The corresponding single-species Giesekus model for the wormlike micellar solution, with no breakage and reformation, predicts a steady ow field under otherwise identical conditions. Therefore, it further ascertains the evidence presented herein for the onset of this unsteady motion. Furthermore, we find that the onset of this unsteady motion downstream of the sphere is delayed as the sphere to tube diameter ratio decreases. A similar kind of unsteady motion has also been observed in earlier several experiments for the problem of a sphere sedimenting in a tube filled wormlike micellar solutions. We find a remarkable qualitative similarity in the ow characteristics between the present numerical results on the steadily translating sphere and prior experimental results on the falling sphere.