We study vibrational instabilities of the thermocapillary return flow driven by a constant temperature gradient along the free surface of an infinite layer that vibrates in its normal direction with acceleration of amplitude $g_1 $ and frequency $\omega _1 $. The layer is unstable to hydrothermal waves in the absence of vibrations beyond a critical Marangoni number $M$. Modulated gravitational instabilities with $M\,{=}\,0$ are also possible beyond a critical Rayleigh number $R$ based on $g_1 $. We employ two-time-scale high-frequency asymptotics to derive the equations governing the mean field. The influence of vibrations on the hydrothermal waves is found to be characterized by a dimensionless parameter $G$ that is proportional to $R^2.$ The return flow at $G\,{=}\,0$ is also a mean field basic flow and we study its linear instability at different Prandtl numbers $P$. The hydrothermal waves are stabilized with increasing $G$ and reverse their direction of propagation at particular values of $G$ that decrease with increasing $P$. At finite frequencies, a time-periodic base state exists and we study its linear instability by calculating the Floquet exponents. The stability boundaries in the $(R,M)$-plane are found to be composed of two intersecting branches emanating from the points of pure thermocapillary or buoyant instabilities. Three-dimensional modes are always preferred and the region of stability, while anchored at the point of hydrothermal waves corresponding to $R\,{=}\,0$, is found to grow without bound along the $R$-axis with increasing frequencies. Results from the two approaches are shown to be in asymptotic agreement at large frequencies.

The paper presents a new analytical and experimental study of transonic flow around spheres. The results of the analytical study, which employs the method of orthogonal collocation for simultaneous solution of the momentum equations, the equation of continuity and the energy equation, are compared with hitherto unpublished measurements obtained on spheres of various sizes (1·02, 2·54 and 3·81 cm in diameter) in air, in dry steam and in wet steam with free-stream Mach numbers in the transonic range (0·58 < M∞ < 0·97). The relationship θ h = 91·78 + 8·59 M∞ between the attached-shock angle and the free-stream Mach number was obtained by fitting the theoretical pressure distributions to the experimental ones.

We calculate the drag coefficient of a circular liquid domain in a flat fluid membrane surrounded by three-dimensional fluids on both sides. The coefficient of a rigid disk is well known, while that of a circular liquid domain is also well known when the membrane viscosity inside the domain equals the one outside the domain. As the ratio of the former viscosity to the latter increases to infinity, the drag coefficient of a liquid domain should approach that of the disk of the same size in the same ambient viscosities. This approach has not yet been shown explicitly, however. When the ratio is not unity, the continuity of the stress makes the velocity gradient discontinuous across the domain perimeter in the membrane. On the other hand, the velocity gradient is continuous in the ambient fluids, whose velocity field should agree with that of the membrane as the spatial point approaches the membrane. This means that we need to assume dipole singularity along the domain perimeter in solving the governing equations unless the ratio is unity. In the present study, we take this singularity into account and obtain the drag coefficient of a liquid domain as a power series with respect to a dimensionless parameter, which equals zero when the ratio is unity and approaches unity when the ratio tends to infinity. As the parameter increases to unity, the sum of the series is numerically shown to approach the drag coefficient of the disk.

This paper discusses the long-wave limit of the asymptotic theory of hypersonic boundary-layer stability for a gas with the Prandtl number ½ < σ < 1 and with the viscosity–temperature law being a power function. The investigation is confined to the local-parallel approximation.

In the long-wave limit the vorticity mode starts to interact with the acoustic disturbances in the boundary-layer region. The general solution of the linear problem in the boundary-layer inner region is analysed numerically and analytically. This solution is matched with the long-wave vorticity-mode solution near the transition layer. As a result, the inviscid instability problem for a hypersonic boundary layer is formulated. The analytical solution of this problem is found and analysed. The different limits of the solution are considered and the universal forms of the dependence are obtained. A similarity parameter is found which is a function of the Prandtl number and the power in the viscosity–temperature law. A significant change of the solution behaviour is noticed when this parameter passes a critical value. The asymptotic structure of the amplification rate, as a function of the wavenumber, is described and discussed.

Here we study the spin-up and spin-down of a homogeneous fluid with a free surface on an experimental ‘β-plane’ and describe the important features for both cases over a range of parameters. Quantitative values are found for the velocity fields using a new image processing technique that analyses a video record of particle motion and stores the results digitally. Streamlines, pressure fields and vorticity values are found by interpolation techniques and result in a complete description of the flow characteristics. We discuss the relationship between the results of these experiments and those observed in large-scale homogeneous models of ocean circulation, e.g. Moore (1963). This study extends the work of van Heijst et al. (1990) to the case of spin-up in a rectangular container but of non-uniform depth and we note the differences to and similarities with their observations. It is related, also, to more recent results of Maas et al. (1992), who considered spin-up on a β-plane but in a tank of very different proportions to the one considered here.

The linear field induced by the sudden starting of a wave on an infinite plane surface is found exactly. An acoustic wavefront, on which the pressure remains constant and finite, moves outwards from the surface at the sound speed c. Behind this front the pressure field consists of two distinct components. The first is recognizable as the field due to steady motion over a wavy wall, while the second is an acoustic transient which propagates through the fluid to accelerate it into its steady asymptotic state. We find that whenever the surface phase speed is subsonic, there is an equipartition of the energy between the steady evanescent field attached to the surface and the outward-travelling sound. If the surface has sonic phase speed then the pressure on the surface grows like t½, and becomes unbounded. For subsonic waves the final momentum of the fluid parallel to the surface is shown to be equal to the total energy radiated divided by the surface phase speed. In the asymptotic state the ratio of momentum in the far field to that in the near field is ½m2/(1 − ½m2), where m is the ratio of the surface phase speed to the sound speed. The transient field on the surface can be identified as consisting of two travelling sound waves. One travels in the same direction as the surface wave, and the other in the opposite direction. They have amplitudes which are proportional to (1 − m)−l and (1 + m)−1 respectively, and decay in time as t−½. Their wavelength parallel to the surface is, of course, the same as that of the surface wave that induced them. We show that when the surface wave is very subsonic, the evanescent field is established very slowly, settling down only after the surface wave has travelled about m−1 wavelengths.

Two of the simpler nonlinear wave systems on water of uniform depth are permanent waves and wave groups of permanent envelope. The interaction of each of these wave systems with swell of much smaller amplitude and greater wavelength, propagating in the same direction, is investigated analytically and numerically. A linear-stability analysis of the modulation of these systems by swell shows that they are unstable over short times. Calculations of their evolution over longer times confirms that the initial exponential growth of the modulations is not sustained, and that cyclic recurrence of the modulations occurs in some cases. The modulation of a wave train by swell is found to concentrate the energy of the wave train into single waves in turn, a process which may cause irreversible nonlinear changes such as wave breaking. In contrast, the only observable effect in the modulation of a wave group by swell is a small slow oscillation of the envelope of the group as it propagates. The conclusion is that wave trains on the ocean, generated for example by a wind system of long fetch and duration, disintegrate under the modulation of swell. Wave groups, however, either wind-generated or resulting from the breakdown of wave trains, propagate almost unchanged by the presence of swell.

The flow structures generated and drag experienced by a rigid cylinder moving in an arbitrary direction through a rotating electrically conducting fluid in the presence of an applied magnetic field are investigated, with he aim of understanding better the nature of the small-scale flow in the core of the Earth which may be responsible for maintaining the geomagnetic field through dynamo action. Three cases are considered in the limit of small Rossby and magnetic Reynolds numbers. In the case of very weak rotation, the possible flow structures consist of a thin Hartmann layer and a long wake extending in the direction of the magnetic field, in which Lorentz and viscous forces balance, but only the long wake plays a dynamical role. The dominant drag force is experienced for motion that cuts magnetic lines of force. Motion of the cylinder parallel to its axis induces a much weaker drag, while that in the direction of the magnetic field induces none to dominant order. The cylinder also experiences weak lateral forces due to the Coriolis effect. In the case of weak rotation, the balance in the long wake is now magnetostrophic: between Lorentz and Coriolis forces. The drag is qualitatively identical to that in the first case, but the drag induced by motion parallel to the axis of the cylinder is increased, though still smaller than that for motions cutting magnetic lines of force. In the case of strong rotation, the flow structures consist of a thin Ekman layer and a foreshortened Taylor column extending in the direction of the rotation axis. In this column, the force balance is again magnetostrophic. Again only the large-scale structure plays a dynamical role. Motion of the cylinder perpendicular to its axis induces a larger drag than does motion parallel to its axis. The cylinder also experiences large lateral Coriolis forces.

The study of nonlinear magnetosonic waves in a turbulent plasma is extended to include the effects of the Hall term. The turbulence and Hall effect are characterized by an effective electrical conductivity and an ion gyrofrequency respectively. It is shown that the magnetosonic waves are governed by a nonlinear equation which can be considered as the generalization of a Korteweg & de Vries (1895) equation with dispersion. For a stationary solution two cases are considered in detail: (a) an unperturbed magnetic field is almost parallel to a wave vector, and (b) they are almost perpendicular. In the case (a) it is shown that the presence of the Hall term can lead to an oscillatory solution which decays due to the finite conductivity. In the second case the Hall effect does not affect the monotonous character of a decaying Taylor-shock profile.

The nonlinear development of the Batchelor–Nitsche instability (of a periodically stratified fluid) is considered, utilizing the disparity between vertical and horizontal scales of motion. The resulting evolution equation is used to show that the preferred pattern of convection takes the form of rolls, and that the motion evolves to larger and larger horizontal scales as time increases.

The phenomena of momentum and heat transfer associated with an impulsively started spherical particle in a quiescent argon thermal plasma environment is considered. The changing plasma thermodynamics and transport property effects are studied using a Chebyshev-Legendre spectral method. Steady-state solutions for the case of constant sphere surface temperature are obtained and compared with previously published results. Transient solutions with particle internal heat conduction included are also presented. Results indicate that the magnitude of the drag force increases as the plasma free-stream temperature increases, while the Nusselt number decreases with increasing free-stream temperature. Effects due to different initial particle temperatures on the transient Nusselt number and drag coefficient are demonstrated.