The partial differential equations governing two-dimensional thermosolutal convection in a Boussinesq fluid with free boundary conditions have been solved numerically in a regime where oscillatory solutions can be found. A systematic study of the transition from nonlinear periodic oscillations to temporal chaos has revealed sequences of period-doubling bifurcations. Overstability occurs if the ratio of the solutal to the thermal diffusivity τ < 1 and the solutal Rayleigh number RS is sufficiently large. Solutions have been obtained for two representative values of τ. For τ = 0.316, RS = 104, symmetrical oscillations undergo a bifurcation to asymmetry, followed by a cascade of period-doubling bifurcations leading to aperiodicity, as the thermal Rayleigh number RT is increased. At higher values of RT, the bifurcation sequence is repeated in reverse, restoring simple periodic solutions. As RT is further increased more period-doubling cascades, followed by chaos, can be identified. Within the chaotic regions there are narrow periodic windows, and multiple branches of oscillatory solutions coexist. Eventually the oscillatory branch ends and only steady solutions can be found. The development of chaos has been investigated for τ = 0.1 by varying RT for several different values of RS. When RS is sufficiently small there are periodic solutions whose period becomes infinite at the end of the oscillatory branch. As RS is increased, chaos appears in the neighbourhood of these heteroclinic orbits. At higher values of RS, chaos is found for a broader range in RT. A truncated fifth-order model suggests that the appearance of chaos is associated with heteroclinic bifurcations.

A model problem is analysed to study the microscopic flow near the surface of two-dimensional porous media. In the idealized problem we consider axial flow through infinite and semi-infinite lattices of cylindrical inclusions. The effect of lattice geometry and inclusion shape on the permeability and surface flow are examined. Calculations show that the definition of a slip coefficient for a porous medium is meaningful only for extremely dilute systems. Brinkman's equation gives reasonable predictions for the rate of decay of the mean velocity for certain simple geometries, but fails for to predict the correct behaviour for media anisotropic in the plane normal to the flow direction.

The time-dependent settling of a dilute monodisperse suspension in a centrifugal force field is considered. The settling takes place between two axisymmetric narrowly spaced conical disks that are rotating rapidly. Clear fluid and suspension are assumed to behave as Newtonian fluids of different densities. The viscosities are, for simplicity, assumed to be the same. All effects of the sediment are neglected. The fluid motion is assumed to be almost parallel with the disks and is computed by using lubrication theory. This leads to a nonlinear hyperbolic equation of first order for the location of the interface between clear fluid and suspension. Local multivaluedness of the solution is removed by inserting shocks. Such a shock is a model for a small region where the slope of the interface, as scaled in the lubrication approximation, is large. Two problems are solved: batch settling and a case where the suspension is pumped into a conical channel that is initially filled with clear fluid. In the batch-settling case, the solution is quite similar to that computed by Herbolzheimer & Acrivos for settling due to a constant gravity field in a narrow tilted channel. For large values of the Taylor number, it is found that the blocking of radial flow outside the Ekman layers leads to a somewhat slower separation process than expected. In the filling problem, the character of the solution is distinctly different for Taylor numbers of order unity and large values of this parameter.

Thermal convection in a saturated porous medium contained between two undulating fixed boundaries of mean horizontal disposition is considered when the layer is heated from below. In an analytic study, the amplitudes of the two-dimensional undulations are assumed to be small compared with the mean depth, and the wavelength is taken to be close to the critical wavelength for the onset of Lapwood convection. For values of the mean Darcy-Rayleigh number Ra below the Lapwood critical value Rac an analytical formula is found for the mean Nusselt number. As Ra→Rac, convection driven by baroclinic effects induced by boundary variations is greatly amplified by convective instabilities. The natures of the resultant bifurcations are examined when the configuration is varicose and also non-varicose. Consideration is given to both longitudinal and transverse modes and to the effects of detuning. The effects of finite amplitude and larger Rayleigh number are examined, for the varicose configuration, in a numerical study of two-dimensional convection. Periodic solutions are found and the existence of the flows delimited in the parameter space of Ra and the boundary amplitude a.