Nonlinear dynamics aspects of subcritical transitions and singular flows in viscoelastic fluids

Abstract

Recently, there has been a renewed interest in theoretical aspects of flows of viscoelastic fluids (such as dilute polymer solutions). This thesis addresses two distinct issues related to such flows. Motivated by the possible occurrence of subcritical (finite-amplitude) instabilities in parallel flows - instabilities that cannot be captured by the usual linear stability analyses - I present and evaluate a method to describe these subcritical transitions by means of a direct expansion in the amplitude of the linearly least stable mode. A second issue is the behaviour of viscoelastic fluids in steady elongational flow. Here, singular solutions have recently been found for flows involving a stagnation point. These solutions appear to be the mathematical structures underlying the birefringent strands that have been observed experimentally in these flows. In this thesis, explicit approximate solutions are found for idealized extensional flow geometries and simple constitutive equations. Asymptotic results are derived for the width of the strand and other typical parameters. It appears that non-analytical solutions are a general feature of elongational viscoelastic flows, which should also occur for more realistic flows and models.