This thesis considers exact solutions of Newtonian flows linear in one coordinate. In advection-diffusion systems, this flow structure enables the reduction of the governing equations to ordinary differential equations, without discarding nonlinear terms. These provide insight into key features of the flows.
We consider steady Boussinesq flow in a weakly curved channel driven by linearly varying temperature gradient in the transverse direction. We seek a solution as a series expansion in G, a parameter proportional to the Grashof number and the square root of the curvature, which reveals a real singularity at some value G = Gc and anticipates multiple solution branches. Analytical continuation extends the convergence circle of the series and unveils non-trivial solutions when our forcing parameter G = 0. Numerical path-continuation extends the bifurcation diagram and gives additional insight into these newly found “unforced” solutions. Two types of solution are found, in one the velocity remains coupled with temperature and the other is purely hydrodynamic. These constitute previously unreported solutions, not only of the Dean equations, but also of the Navier–Stokes in an annulus of arbitrary curvature. Dynamo action is investigated in these circumstances. Magnetic field instabilities with the same spatial structure are sought. The kinematic eigenvalue problem is found to have two growing modes for moderate values of the magnetic Reynolds number, Rm. As Rm → ∞ it is shown that the modes are governed by layers on the outer wall. As the field grows, saturated solutions to the nonlinear dynamo problem are found and the bifurcation structure is investigated.
The final problem we investigate is a generalisation to 3−dimensions of the steady, incompressible Falkner-Skan boundary layer equations past a flat plate. The outer flow is an exact solution of the Navier-Stokes equations. Two distinct self-similar solutions are found numerically and their stability is investigated.