Stability of Flows Over Stretching Surfaces

Abstract

Extrusion flows are common in industry and have received considerable attention in the literature. Work in this area began with the experimental results of Trouton [111], who developed an empirical formula relating the applied tension to the stretching rate. Trouton’s model has since been validated by numerous authors through the application of asymptotic arguments based on small thickness-to-length ratios in the sheet (or fibre) being drawn ([51],[84],[75],[98]). In addition to the work on modelling the drawn sheet or fibre, the boundary layers induced by such flows have also been extensively studied. A notable early contribution is the work of Crane [23], who found exact analytical solutions of the steady Navier-Stokes equations under the assumptions of a flat sheet undergoing a linear rate of stretching. Until recently, Crane’s flow has been thought to be linearly stable. However, recent analysis by Griffiths et al. [39] shows that this flow is actually susceptible to travelling wave instabilities in the form of Tollmien-Schlichting (TS) waves, possibly leading to defects in industrial extrusion processes. Despite these advances, a significant gap remains between the models used to describe the induced boundary layer and the physics of the underlying industrial extrusion processes. In this thesis, we address several of these issues by independently accounting for large temperature gradients and the curvature of the sheet. We demonstrate that failing to incorporate these additional physical effects leads to poor quantitative descriptions of the basic flow profiles in the boundary layer. We also explore the implications of temperature dependence on flow stability, using a highly accurate numerical spectral method and complementary large Reynolds number asymptotic analysis. Additionally, we show that in the isothermal case, non-modal instability mechanisms present a more likely transition scenario, with significant energy amplification occurring at Reynolds numbers that are orders of magnitude smaller than the critical values reported in [39]. This was achieved using an adjoint based power iteration method, first introduced by Corbett and Bottaro [22]. Preliminary numerical investigations using the parabolised stability equations (PSE) indicate that non-parallel effects are destabilising in the isothermal case. While we have not performed a full parametric analysis to account for non-parallel effects in Crane’s flow, such an approach could be readily adapted to model the stability of our new basic flow solutions that account for the curvature of the sheet.