The process of nonlinear geostrophic adjustment in the presence of a boundary (i.e. in a half-plane bounded by a rigid wall) is examined in the framework of a rotating shallow water model, using an asymptotic multiple-time-scale theory based on the assumed smallness of the Rossby number ε. The spatial scale is of the order of the Rossby scale. Different initial states are considered: periodic, ‘step’-like, and localized. In all cases the initial perturbation is split in a unique way into slow and fast components evolving with characteristic time scales f−1 and (εf)−1, respectively. The slow component is not influenced by the fast one, at least for times t [les ] (fε)−1, and remains close to geostrophic balance. The fast component consists mainly of linear inertia–gravity waves rapidly propagating outward from the initial disturbance and Kelvin waves confined near the boundary.

The theory provides simple formulae allowing us to construct the initial profile of the Kelvin wave, given arbitrary initial conditions. With increasing time, the Kelvin wave profile gradually distorts due to nonlinear-wave self-interaction, the distortion being described by the equation of a simple wave. The presence of Kelvin waves does not prevent the fast–slow splitting, in spite of the fact that the frequency gap between the Kelvin waves and slow motion is absent. The possibility of such splitting is explained by the special structure of the Kelvin waves in each case considered.

The slow motion on time scales t [les ] (εf)−1 is governed by the well-known quasigeostrophic potential vorticity equation for the elevation. The theory provides an algorithm to determine initial slow and fast fields, and the boundary conditions to any order in ε. For the periodic and step-like initial conditions, the slow component behaves in the usual way, conserving mass, energy and enstrophy. In the case of a localized initial disturbance the total mass of the lowest-order slow component is not conserved, and conservation of the total mass is provided by the first-order slow correction and the Kelvin wave.

On longer time scales t [les ] (ε2f)−1 the slow motion obeys the so-called modified quasi-geostrophic potential vorticity (QGPV) equation. The theory provides initial and boundary conditions for this equation. This modified equation coincides exactly with the ‘improved’ QGPV equation, derived by Reznik, Zeitlin & Ben Jelloul (2001), in the step-like and localized cases. In the periodic case this equation contains an additional term due to the Kelvin-wave self-interaction, this term depending on the initial Kelvin wave profile.

The large negative Soret coefficient of 1N-LiI gives rise in a Rayleigh-Bénard experiment to a density distribution which is observed to stabilize the fluid layer for values of the Rayleigh number as large as 196 times the value of 1708 for the onset of convection in a pure fluid. The Soret transport also affects the convective heat flux. A power law relating heat flux and temperature difference is found with the same exponent as is found in pure fluids but with a lower value of the multiplicative constant. The Rayleigh number at the onset of power-law behaviour depends on the Soret coefficient. Three types of oscillation are seen: transient oscillations at onset, low frequency fluctuations at low Rayleigh number, and higher frequency oscillations similar to those observed in pure water. The intermediate state found after onset in NaCl solutions is not found in LiI.

The asymptotic structure of the discrete spectrum of a compressible inviscid swirling flow with arbitrary radial distributions of density, pressure and velocity is described for disturbances with large wavenumbers. It is shown that discrete eigenmodes are unstable when a criterion derived by Eckhoff & Storesletten (1978) is satisfied. In general, these modes are characterized by a length scale of order $|m|^{-3/4}$ where $|m|\,{\gg}\,1$ is the azimuthal wavenumber of the disturbance. They have a spatial structure similar to the incompressible modes obtained by Leibovich & Stewartson (1983). In the particular case of solid-body rotation with a positive gradient of entropy, the unstable discrete spectrum contains modes which scale with $|m|^{-1/2}$. If the modes are localized near a solid boundary, they scale with $|m|^{-2/3}$.

The unsteadiness of a shock-wave/turbulent-boundary-layer interaction induced by an axisymmetric step (cylinder/$90^{\circ }$-disk) is investigated experimentally at Mach 3.9. A large-scale separation of the order of previously reported incoming turbulent superstructures is induced ahead of the step ${\sim}30\unicode[STIX]{x1D6FF}_{o}$ and followed by a downstream separation of ${\sim}10\unicode[STIX]{x1D6FF}_{o}$ behind it, where $\unicode[STIX]{x1D6FF}_{o}$ is the incoming boundary-layer thickness. Narrowband high-frequency instabilities shift gradually to more moderate frequencies along the upstream separation region exhibiting a strong predominance of shear-induced disturbance levels – arising between the outer high-speed flow and the subsonic bubble. Through spectral/time-resolved analysis of this high Reynolds number and large-scale separation, results offer new insights into the shear layer’s inception and evolution (convection, growth and instability) and its influence on interaction unsteadiness.

We examine the dynamics of a thin film of viscous fluid on the inside surface of a cylinder with horizontal axis, rotating about this axis. Using the so-called lubrication approximation, we derive an asymptotic equation for three-dimensional motion of the film and use this equation to examine its linear stability. It is demonstrated that: (i) there are infinitely many normal modes (harmonic in the axial variable and time), which are all neutrally stable and their eigenfunctions form a complete set; (ii) but the film is nonetheless unstable with respect to non-harmonic disturbances, which develop singularities in a finite time.

The diffraction problem of a plane shock wave at the apex of an obtuse wedge treated by Lighthill (1950) is extended by assuming that the shock wave strikes the walls of the obtuse wedge at some finite oblique angle of incidence (not exceeding the critical angle). Transformations similar to that performed in the above-mentioned paper lead to a non-symmetrical boundary-value problem for an analytic function of a complex variable having a non-homogeneity in the form of a delta-function. It was found possible to extend, for the case considered, the method developed by Lighthill and construct the solution in almost as simple a form as given in the above-mentioned paper. The case of three-dimensional stationary flow is considered when the line of reflexion makes a finite angle with the edge of the wedge.

The evolution of the shape of a slender inviscid drop in an axisymmetric straining motion is studied at low Reynolds numbers. It is found that the shape equation can be solved by polynommals with time-dependent coefficients. A global stability result can be used to show simply that only one possible equilibrium is stable. It is further shown that if the slender drop starts with a long-wavelength waist then it cannot go to this stable equilibrium and must either extend indefinitely or burst. In the class of trinomial shapes, it is shown that the drop either bursts or goes to the stable equilibrium, depending on whether or not the initial shape has a waist.

A method for generating controllable two-dimensional velocity fluctuations using two pitching foils was derived theoretically in a previous companion paper. The present work describes the experimental implementation of the method. The experiments are carried out in a re-circulating water channel optimised to provide low turbulence intensity in the incoming flow. Velocities are measured using an acoustic Doppler velocimeter (ADV). The pitching motions of the foils are position-controlled using a closed-loop control system. Two velocity fluctuation patterns are investigated. They consist of a combination of sinusoidal components. Theoretical predictions and experimental measurements are compared in the time and frequency domain. Although some discrepancies are observed, the agreement is generally good and therefore validates the theoretical method for the conditions investigated.

In this paper, unsteady thin aerofoil theory is extended to the case of two interacting aerofoils that oscillate harmonically perpendicular to the direction of the incident flow. The two aerofoils represent the headsail and mainsail of a yacht that sails upwind in waves. The developed theory is validated against theoretical data from the literature and results from wind tunnel tests with rigid, high-aspect ratio sail models oscillating at reduced frequencies from k = 0 to k = 0.68. Good agreement is found between the predicted and measured chordwise pressure distributions. An application of the theory to the case of an International America's Cup Class yacht reveals that the time-varying components of the aerodynamic forces are small and that the thrust gain is minimal, i.e. only very little energy can be extracted from the unsteady flow about the sails. No attempt is made to investigate the influence of the flexibility of the sails, three-dimensional effects or phenomena related to dynamic stall.

The propagation of axisymmetric solitary waves on the surface of an otherwise cylindrical ferrofluid jet subjected to a magnetic field is investigated. An azimuthal magnetic field is generated by an electric current flowing along a stationary metal rod which is mounted along the axis of the moving jet. A numerical method is used to compute fully nonlinear travelling solitary waves, and the predictions of elevation waves and depression waves made by Rannacher and Engel (New J. Phys., vol. 8, 2006, pp. 108–123) using a weakly nonlinear theory are confirmed in the appropriate ranges of the magnetic Bond number. New nonlinear branches of solitary wave solutions are identified. As the Bond number is varied, the solitary wave profiles may approach a limiting configuration with a trapped toroidal-shaped bubble, or they may approach a static wave (i.e. one with zero phase speed). For a sufficiently large axial rod, the limiting profile may exhibit a cusp.

A rotating magnetic field (RMF) is used in crystal growth applications during the solidification process in order to improve the crystal quality. Its influence on the convective flows in molten metals and on their stability is studied here in the case of a horizontal infinite cylindrical channel subjected to a longitudinal temperature gradient. The steady convective flows, which correspond to the usual longitudinal counterflow structure, with four vortices in the cross-section for non-zero Prandtl number, Pr, are modified by the RMF (parametrized by the magnetic Taylor number Tam). For zero Prandtl number, the flow in the cross-section corresponds to circular streamlines and the longitudinal flow structure is moved in the direction of the magnetic field rotation, with a decrease in its intensity and an asymptotic variation as 1/Tam. For non-zero Prandtl numbers, depending on the respective values of Tam on one side and Prandtl and Grashof numbers on the other side, different structures ranging from the circular streamlines with transport by rotation of the longitudinal velocity and the temperature field, to the more usual counterflow structure almost insensitive to the RMF with four cross-section vortices, can be obtained. The decrease in the flow intensity with increasing Tam is also delayed for non-zero Pr, but the same asymptotic limit is eventually reached. The stability analysis of these convective flows for Tam = 0 shows a steep increase of the thresholds around Pr = Prt,0 ≈ 3 × 10−4, corresponding to the transition between the usual counterflow shear mode and a new sidewall shear mode. This transition is still present with an RMF, but it occurs for smaller Pr values as Tam is increased. Strong stabilizing effects of the rotating magnetic field are found for Pr < Prt,0, particularly for Pr = 0 where an exponential increase of the threshold with Tam is found. For Pr > Prt,0 (i.e. in the domain where the sidewall instability is dominant), in contrast, the stabilization by the RMF is weak.

Linear mode conversion is the partial transfer of wave energy from one wave type (a) to another (b) in a weakly non-uniform background state. For propagation in one dimension (x), the local wavenumber kjx of each wave (j = a, b) varies with x; if these are equal at some xR, the waves are locally in phase, and resonant energy transfer can occur. We model wave propagation in the Gulf of Guinea, where wave a is an equatorially trapped Rossby–gravity (Yanai) wave, and wave b is a coastal Kelvin wave along the (zonal) north coast of the Gulf, both propagating in zonal coordinate x. The coupling of the waves is due to the overlap of their eigenfunctions (normal modes in y, the meridional coordinate). We derive coupled mode equations from a variational principle, and obtain an analytic expression for the wave-energy conversion coefficient, in terms of the wave frequency and the scale length of the thermocline depth.

A class of similarity solutions for two-dimensional unsteady flow in the neighbourhood of a front or rear stagnation point on a plane boundary is considered, and a wide range of possible behaviour is revealed, depending on whether the flow in the far field is accelerating or decelerating. The solutions, when they exist, are exact solutions of the Navier–Stokes equations, having a boundary-layer character analogous to that of the classical steady front stagnation point flow. The velocity profiles are obtained by numerical integration of a nonlinear ordinary differential equation. For the front-flow situation, the solution is unique for the accelerating case, but bifurcates for modest deceleration, while for sufficient rapid deceleration there exists a one-parameter family of solutions. For the rear-flow situation, a unique solution exists (remarkably!) for sufficiently strong acceleration, and a one-parameter family again exists for sufficient strong deceleration. Analytic results, which are consistent with the numerical results, are obtained in the limits of strong acceleration or deceleration, and for the asymptotic behaviour far from the boundary.

Low-Reynolds-number polymer solutions exhibit a chaotic behaviour known as ‘elastic turbulence’ when the Weissenberg number exceeds a critical value. The two-dimensional Oldroyd-B model is the simplest constitutive model that reproduces this phenomenon. To make a practical estimate of the resolution scale of the dynamics, one requires the assumption that an attractor of the Oldroyd-B model exists; numerical simulations show that the quantities on which this assumption is based are bounded. We estimate the Lyapunov dimension of this assumed attractor as a function of the Weissenberg number by combining a mathematical analysis of the model with direct numerical simulations.