The nonlinear stability of two-fluid shear flows in the presence of inertia and/or an insoluble surfactant at the interface is studied in this thesis. Asymptotic analysis in the limit of a thin lower layer is performed and a system of coupled weakly nonlinear evolution equations is derived. The system describes the spatiotemporal evolution of the interface and its local surfactant concentration. It contains a nonlocal term which arises by appropriately matching solutions of the linearised Navier-Stokes equations in the thicker layer to the thin layer solution. The problem corresponding to two-dimensional flows is first solved numerically, by implementing highly accurate linearly implicit schemes in time with spectral discretisations in space. Numerical experiments for asymptotically small and finite Reynolds numbers indicate that the solutions are mostly nonlinear travelling waves or time-periodic waves. As the length of the system increases, the dynamics become more complex and include quasi-periodic and chaotic fluctuations. The stability in three-dimensions of the nonlinear travelling waves observed in two-dimensional flows is also examined. The model derived is also shown to be appropriate in describing interfacial wave structures arising in two-fluid Couette flow experiments.
A related two-dimensional dissipative-dispersive partial differential equation is considered in the second part of the thesis. The PDE is similar to the surfactant-free version of the interfacial evolution equation derived in the previous part. A generalisation of that equation with a nonlinearity written in a gradient form provides the well-known two-dimensional Kuramoto-Sivashinsky equation (2D KSE). The 2D KSE has received attention with respect to its mathematical analysis, but numerical solutions are obtained for the first time here. For relatively small domain sizes the solutions are steady states or travelling waves and as the domain becomes increasingly larger, solutions are trapped into a chaotic attractor which is characterised by energy equipartition and symmetry of the energy spectrum.