Instability of fluid flows, including boundary slip

Abstract

We investigate the onset of instability in a variety of fluid models, and present results and details of their computation in each case. The fluid models we consider are: convection in the setting of the Navier-Stokes equations with boundary slip; Poiseuille-type solutions of the Navier-Stokes equations, again with boundary slip; Poiseuillfe-type solutions of the Green-Naghdi and dipolar fluid equations. In Chapter two we examine the onset of thermal convection in a thin fluid layer, with slip boundary conditions at the top and bottom surfaces of the layer. We show that non-zero boundary conditions do not affect the classical steady state solution, and the principle of exchange of stabilities still applies. It is seen that boundary slip reduces the critical Rayleigh number at which convection begins, below that found in the setting of no-slip boundary conditions. The next two chapters concern the transition to turbulence of pressure driven flow in a microchannel, at the boundaries of which the fluid obeys slip boundary conditions. In Chapter three we perform linear and nonlinear stability analyses for this flow, and show that we do not have exchange of stabilities for such flows. In Chapter four we perform a linear stability analysis for channel flow in the case when the fluid viscosity is a function of temperature. We show that for pressure driven flow in the plane, boundary slip stabilizes the flow. In Chapter five we develop a model of thread-annular flow, in which we believe boundary slip to be an important part. As well as our development of the model, we present previously unpublished results on the linear stability of thread-annular flow to non-axisymmetric disturbances. Some surprising behaviour is observed in the neutral curves, including behaviour missed by the computations of previous authors. Finally, we use Chapter six to discuss two alternative fluid models: the Green- Naghdi equations and the dipolar equations. We find Poiseuille flow type solutions in both of these settings, and perform linear stability analyses. These fluid models are systems of fourth order differential equations, and we show that the fourth order derivative terms dominate the stability of the flow.