  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.
