Evolution of nonlinear instabilities and mathematical analysis of radial flows

Two- and three-dimensional time-dependent numerical simulations using an accurate spectral element method are conducted to investigate the temporal evolution and spatial distribution of instabilities in radial flows. Two models are considered in the axisymmetric situation and three models are considered in the three-dimensional case. Small perturbations are found to decay in an oscillatory manner in both two- and three-dimensional flows for 300 ;Parallel to this work, a novel pressure-based methodology is proposed and implemented to approximate the solution of the partial differential equations that model radial Stokes flows between two parallel disks. Eigenfunction expansions and Green's function representations are combined to obtain directly the pressure distribution for a variety of inflow boundary conditions. This methodology was implemented numerically and validated with exact solutions and numerical results. By exploring the analytical characteristics of the method, we extract additional information on the behavior of the solution in terms of the inflow functions and we prove rigorously the existence and uniqueness of solutions.