Mathematics

The behaviour of many real fluids is well described by Navier-Stokes theory which is based on the assumption of a Newtonian constitutive equation. Specifically, the extra-stress tensor can be expressed as a linear, isotropic function of the velocity gradients. Many common fluids such as water and air can be assumed to be Newtonian. However, rheologically complex fluids such as polymer solutions, soaps, blood, paints, shampoo are not adequately described by a Newtonian constitutive equation. Viscoelastic fluids are examples of non-Newtonian fluids, they exhibit both viscous and elastic properties when undergoing deformation. The aim of my PhD research was to understand the stability behavior of such fluids in boundary layers. First, we consider the flow of second order Rivlin-Ericksen fluids over a semi-infinite wedge/corner. In the two-dimensional case, a linear stability analysis leads to a modified OrrSommerfeld equation that is solved numerically using a Chebyshev collocation method. We consider both two-dimensional and three-dimensional configurations and present the non-Newtonian effects on the stability characteristics. Later, we expand the analysis to include bypass transition scenarios to better study the variations of perturbation energy over short time periods. A preliminary study of the boundary layer flow of rheologically more complex fluids is performed.