  The breakdown of finite element (FEM) computations for the flow of an
Oldroyd-B fluid around a cylinder confined between parallel plates, at
Weissenberg numbers Wi = O(1), is shown to arise due to a coil-stretch
transition experienced by polymer molecules traveling along the centerline in
the wake of the cylinder. With increasing Wi, the coil-stretch transition leads
to an unbounded growth in the stress maximum in the cylinder wake. Finite
element computations for a FENE-P fluid reveal that, although polymer molecules
undergo a coil-stretch transition in the cylinder wake, the mean extension of
the molecules saturates to a value close to the fully extended length, leading
to bounded stresses with increasing Wi. The existence of a coil-stretch
transition has been deduced by examining the behavior of ultra-dilute Oldroyd-B
and FENE-P fluids. In this case, the solution along the centerline in the
cylinder wake can be obtained exactly since the velocity field is uncoupled
from the stress and conformation tensor fields. Estimation of the number of
finite elements required to achieve convergence reveals the in-feasibility of
obtaining solutions for the Oldroyd-B model for Wi > 1.
