Consider the flow of an incompressible fluid between two infinite concentric circular cylinders. The outer cylinder is at rest whilst the angular velocity of the inner cylinder has a steady part and also a harmonically oscillating component. We examine the situation where, for a suitable choice of parameters, two types of vortex instability can occur simultaneously; first a short-wavelength mode which is essentially trapped in a thin ‘Stokes’ layer near the inner cylinder and, secondly, a long-wavelength mode which fills the whole region between the cylinders. We investigate the problem in which two short-wavelength vortices and one long-wavelength vortex coexist and are such that each pair interacts to drive the third. Additionally, the short-wavelength disturbances are nonlinear in their own right. Coupled amplitude equations for the three modes are derived and their solution discussed.

This form of interaction may also take place in a boundary layer. Such a situation is more complex than that under consideration here as it would be necessary to take into account the growth of the boundary layer. However, this simplified problem gives an insight into the behaviour of the more difficult situation.

We describe a mathematical model of double-diffusive (thermosolutal) convection in a saturated porous layer, when the solubility of the solute depends on the temperature, and the porosity and permeability of the porous medium evolve through dissolution and precipitation. We present the results of linear and weakly nonlinear stability analyses and explore the longer-term development of the system numerically. When the solutal concentration gradient is destabilising, the dynamics are somewhat similar to those previously found for single-species convection (Ritchie & Pritchard, J. Fluid Mech., vol. 673, 2011, pp. 286–317), including the occurrence of subcritical instabilities driven by a reaction–diffusion mechanism. However, when the solutal concentration gradient is stabilising and the thermal gradient is destabilising, novel dynamics emerge. These include a vertical segregation of circulation cells and porosity perturbations near the onset of convection, and over longer time scales the formation of a low-permeability region in the middle of the layer, pierced by occasional high-permeability channels. Under these conditions, convection may die away to nearly zero for extended periods before resuming vigorously in localised regions at later times.

Ordinary two-dimensional turbulence corresponds to a Hamiltonian dynamics that conserves energy and the vorticity on fluid particles. This paper considers coupled systems of two-dimensional turbulence with three distinct governing dynamics. One is a Hamiltonian dynamics that conserves the vorticity on fluid particles and a quantity analogous to the energy that causes the system members to develop a strong correlation in velocity. The other two dynamics considered are non-Hamiltonian. One conserves the vorticity on particles but has no conservation law analogous to energy conservation; the other conserves energy and enstrophy but it does not conserve the vorticity on fluid particles. The coupled Hamiltonian system behaves like two-dimensional turbulence, even to the extent of forming isolated coherent vortices. The other two dynamics behave very differently, but the behaviours of all four dynamics are accurately predicted by the methods of equilibrium statistical mechanics.

The structure of a steady axisymmetric thermal plume rising through a very viscous fluid with strongly temperature-dependent viscosity of the form $\mu \propto \exp (-\gamma T)$ is investigated. An analytic asymptotic solution is derived for the fast-flowing core of the plume, which predicts that the excess centreline temperature decays exponentially as $\exp \{ - 12 \pi \kappa z/(\gamma A) \}$, where $\kappa $ is the thermal diffusivity, $z$ the height and $A$ the vertical heat flux. This rate of decay, which is found to be in good agreement with numerical simulations of the boundary-layer equations, is three times faster than that predicted by the oft-quoted model of Olson, Schubert and Anderson (J. Geophys. Res., vol. 98 (B4), 1993, pp. 6829–6844).

Journal of Fluid Mechanics, vol. 360 (1998), pp. 213–228

As pointed out to us by Mr T. Heath, the following printing errors can be quite misleading when using the formulas in the paper to obtain eigenfrequencies and damping rates to compare with experiments:

in (A 13) 1 should read −1 on the right-hand side;

in (A 22) and (A 26) Ω20 should read Ω−20;

in (A 25) the factor Ω40 must be omitted on the right-hand side.

When revising again the printed version of the paper, we discovered several additional misprints:

A factor C was omitted in the first two integrals in the expression for J2, immediately following equation (2.9).

The sign of the second expression for I1 in (2.23) should be changed.

The expression (W0Wz +3WW0z)z=0 should read 2(W0Wz +WW0z)z=0 in equation (2.24).

The expression W0(1, z)W0z(1, z) in (2.26) should read W0(r, 0)W0z(r, 0).

None of the misprints above affect the results of the paper, which were obtained with the correct expressions.

The boundary-element method is used to solve Stokes equations for periodic arrays of force-free and torque-free rigid particles. Simple cubic arrays of spheres, spheroids, cubes, and clusters of spheres are subjected to a bulk simple shearing flow. The effective volume-averaged stress tensor for the suspension and the detailed velocity and stress fields throughout the Newtonian suspending fluid are calculated. We find that even crude meshes give very good volume-averaged results, but fine meshes are required to track local minima and maxima in the stress field. For simple cubic arrays of spheres, the boundary-element results are in excellent agreement with the analytical viscosity predictions of Nunan & Keller (1984). Even at the highest concentration of solids studied, no significant normal stress differences were observed, in agreement with Nunan & Keller's results (1984). Up to moderate concentrations of particles, the volume-averaged properties of the suspension display only a weak dependence on the particle geometry. Suspensions of spheroids and cubes behave approximately as suspensions of spheres on the average despite large differences in the local micromechanics of stress and velocity fields. Simple cubic arrays of clusters of spheres tend to behave on a macroscopic level as a cubic array of spheres whose effective volume fraction is about 150% of the total volume fraction of the spheres in the clusters.

In the published article Monaghan et al. (2009) one of the authors' names was misspelled and should have read Catherine Mériaux.

The Press apologises to Dr Mériaux, her coauthors and the readers for this error.

Laboratory measurements on the instability of axisymmetric capillary surfaces pinned to the corners of annular grooves of rectangular section rotating at constant angular velocity Ω have been conducted. In stable configurations the fluid contact lines remain pinned to the corners of the groove with contact angles θ1,2 relative to the inner and outer vertical walls. Using water as the test fluid in narrow grooves of nearly constant width, the critical frequency Ωc for instability generally decreases with increasing overfill volume ΔV and mean groove radius. Numerical integration of the describing equation gives the shape of the rotating meniscus as a function of five independent parameters. In the range of contact angles θ1, 2 < π, a comparison of experimental results with numerically computed meniscus profiles suggests three mechanisms for contact line movement based on the effective static advancing (θA) and receding (θR) contact angles for liquid pinned to a sharp corner. Measurements of critical frequencies over a wide range of overfill volumes in six different grooves are in favourable agreement with composite regime diagrams for the critical static meniscus configuration. An interesting feature of this system is the existence of a range of overfill volumes inaccessible to experiments conducted by fixing the overfill volume on a stationary disk and subsequently elevating the disk rotation until contact line movement is observed. Numerical studies showing the effects of Bond number, groove curvature and contact angle hysteresis are presented.

The linear stability of plane Couette flow is investigated when the plates are horizontal, and the fluid is stably stratified with a cubic basic density profile. The disturbances are treated as inviscid and diffusion of the density field is neglected. Previous studies have shown that this density profile can develop multiple neutral curves, despite the stable stratification, and the fact that plane Couette flow of homogeneous fluid is stable. It is shown that when the neutral curves are plotted with wave angle on one axis, and location of the density inflexion point on the other axis, they produce a self-similar fractal pattern. The repetition on smaller and smaller scales occurs in the limit when the waves are highly oblique, i.e. longitudinal vortices almost aligned with the flow; the corresponding limit for two-dimensional waves is that of strong buoyancy/weak flow. The fractal set of neutral curves also represents a fractal of bifurcation points at which nonlinear solutions can be continued from the trivial state, and these may be helpful for understanding turbulent states. This may be the first example of a fractal generated by a linear ordinary differential equation.

The effect of combustion or heating on a slightly non-uniform flow in a tube of constant radius is explored by examining a simplified model, in which a density decrease occurs across a plane actuator-disk normal to the axis of the tube. The flow is considered to be steady, axi-symmetric, incompressible and inviscid and to originate at an injector-plate which is situated upstream of the disk. This paper analyses and discusses the effects caused by the density decrease when the velocity of flow through the injector-plate has a fractional non-uniformity ue, when there is a curvature θe, of the injector-plate, and when the density decrease has a fractional non-uniformity δ2. It is shown that the non-uniformity of the flow far downstream of the disk caused by the disturbances ue and θe is diminished by the density decrease; while that due to the disturbances δ2 is increased, unless the density decrease occurs close to the injector-plate. The distance separating the injector-plate and actuator-disk is found to be an important parameter. When this distance is small, compared to the radius of the tube, and the density decrease is severe, a small disturbance ue, θe or δ2 may set up a pressure variation at the injector-plate of many times the inlet dynamic pressure of the flow. It is conjectured that this pressure variation could affect the uniformity of velocity and composition of the flow through the injector-plate, and thus cause the disturbances ue and δ2 to grow or diminish.

Far from a shoreline, the spreading of a contaminant plume in a shallow-water flow can be predicted easily and accurately by a ray-tracing method. Unfortunately, the concentration predictions become singular at a beach, where the ray paths have a caustic. In this paper a uniform approximation is derived which remains valid at a beach. It is shown how the singular ray solutions corresponding to rays incident to and transmitted from the beach can be combined to construct the uniform approximation.