| The viscoelastic character of polymer solutions and melts gives rise to instabilities not seen in the flows of Newtonian liquids. In this thesis, we computationally study four such instabilities.; The first instability we discuss is melt fracture, which takes the form of gross distortions of the polymer surface during extrusion. This instability is linked to multiplicity in the slip curve. We show here that when the dependence of slip velocity on pressure is taken into account, multiplicity in the slip law does not necessarily imply a multi-valued flow curve or melt fracture.; Next, we study the “filament-stretching” instability, which takes the form of non-axisymmetric deviations of the free surface of a polymeric liquid bridge being extended between two parallel plates. We model the portion of the filament near the endplates as an elastic membrane enclosing an incompressible fluid and show that this is unstable to non-axisymmetric disturbances.; The third instability we discuss is the purely elastic instability in Dean flow. This instability is linked to elastic instabilities in more complicated and industrially important coating flows with curved streamlines. We show how the addition of a small secondary axial flow in a steady or periodic fashion can significantly delay the onset of the instability.; Recent experimental observations by Groisman and Steinberg ( Phys. Rev. Lett. 78(8), 1460–1463, 1997) and Baumert and Muller (Phys. Fluids, 9(3), 566–586, 1999) have shown the formation of spatially isolated, stationary, axisymmetric patterns in the nonlinear regime of circular Couette flow, termed “diwhirls” or “flame patterns.” Modeling these patterns is complicated by the absence of a stationary bifurcation in isothermal circular Couette flow. We show here how these solutions may be accessed by numerical continuation from stationary bifurcations in Couette-Dean flows. Although the solutions we compute are unstable, they show qualitative and quantitative similarities to the experimentally observed structures. We also use the results from our computations to propose a fully nonlinear self-sustaining mechanism for these patterns. |
