The interaction between shear and buoyancy effects for Bénard convection in plane Couette flow is studied by performing direct numerical simulations. At moderate Rayleigh number (≈10000−50000), shear tends to organize the flow into quasi-two-dimensional rolls parallel to the mean flow and can enhance heat transfer, while at higher Rayleigh number (>150000), shear tends to disrupt the formation of convective plumes and can reduce heat transfer. A significant temporal oscillation in the local Nusselt number was consistently observed at high Rayleigh numbers, a factor that may contribute to the scatter seen in experimental data. This effect, plus the time-varying reversal of the mean temperature gradient in the middle of the channel, is consistent with a flow model in which the dynamics of large-scale, quasi-two-dimensional, counter-rotating vortical cells are alternately driven by buoyancy and inertial effects. An analysis of the energy balance in the flow shows that the conservative pressure diffusion term, which has been frequently neglected in turbulence models, plays a very important dynamical role in the flow evolution and should be more carefully modelled. Most of the turbulent energy production due to mean shear is generated in the boundary layers, while the buoyant production occurs mainly in the relatively uniform convective core. The simulations and the laboratory experiments of Deardorff & Willis (1967) are in very reasonable qualitative agreement, suggesting that the basic dynamics of the flow are being accurately simulated.

A method for calculating the effect of the curvature of a solid wall bounding a viscous fluid upon the quasi-static Stokes force F and torque T experienced by a spherical particle performing arbitrarily directed translational and rotational motions in close proximity to the wall is given. The results presented herein are valid for values of the ratio κ = a/d (a = sphere radius, d = shortest perpendicular distance from the sphere centre to the wall) over the entire range 0 [les ] κ [les ] 1, provided that β = a/R0 [Lt ] 1 and, simultaneously, d/R0 ≡ β/κ [Lt ] 1 (R0 = characteristic radius of curvature of the wall). Unlike existing wall-effect theories, our results are valud for κ = O(1). It is shown that to the first-order in β (and, concomitantly, in d/R0, wall curvature effects upon F and T depend linearly upon two scalar principal curvature coefficients of the wall at the foot of the shortest normal to the wall from the sphere centre. This single-particle analysis is used to resolve a ‘paradox’ relating to macroscopic slip boundary conditions prevailing at a wall bounding a dilute ferrofluid suspension undergoing rotation relative to a magnetic field.

The flow in a two-dimensional curved channel driven by an azimuthal pressure gradient can become linearly unstable owing to axisymmetric perturbations and/or non-axisymmetric perturbations depending on the curvature of the channel and the Reynolds number. For a particular small value of curvature, the critical Reynolds number for both these perturbations becomes identical. In the neighbourhood of this curvature value and critical Reynolds number, non-linear interactions occur between these perturbations. The Stuart-Watson approach is used to derive two coupled Landau equations for the amplitudes of these perturbations. The stability of the various possible states of these perturbations is shown through bifurcation diagrams. Emphasis is given to those cases that have relevance to external flows.