Shear flow instabilities in viscoelastic fluids
This dissertation is concerned with the theoretical study of the stability of viscoelastic shear flows. It is divided into two parts: part I studies inertialess coextrusion flows at large Weissenberg number where the instabilities are due to discontinuities in the elastic properties, and part II studies the effect of elasticity on the well-known inertial instabilities of inviscid flows with inflection points. We begin part I with a previously known short-wave instability of Upper Convected Maxwell and Oldroyd–B fluids at zero Reynolds number in Couette flow. We show that if the Weissenberg number is large, the instability persists with the same growth rate when the wavelength is longer than the channel width. Intriguingly, surface tension does not modify the growth rate. Previous explanations of elastic interfacial instabilities based on the jump in normal stress at the interface cannot apply to this instability. These results are confirmed both numerically and with asymptotic methods. We then consider Poiseuille flow and show that a new class of instability exists if the interface is close to the center-line. We analyse the scalings and show that it results from a change in the boundary layer structure of the Couette instability. The growth rates can be large, and the wavespeed can be faster than the base flow advection. We are unable to simplify the equations significantly, and asymptotic results are not available, so we use numerics to verify the results. In studying these instabilities we encounter some others which we mention, but do not analyse in detail. In part II we study the effect of elasticity on the inertial instability of flows with inflection points. We show that the elasticity modifies the development of cat’s eyes. The presence of extensional flow complicates the analysis. Consequently we use the FENE–CR equations.