We examine the effects of compressiblity on the structure of a single row of hollowcore, constant-pressure vortices. The problem is formulated and solved in the hodograph plane. The transformation from the physical plane to the hodograph plane results in a linear problem that is solved numerically. The numerical solution is checked via a Rayleigh-Janzen expansion. It is observed that for an appropriate choice of the parameters M∞ = q∞/c∞, and the speed ratio, a = q∞/qv, where qv is the speed on the vortex boundary, transonic shock-free flow exists. Also, for a given fixed speed ratio, a, the vortices shrink in size and get closer as the Mach number at infinity, M∞, is increased. In the limit of an evacuated vortex core, we find that all such solutions exhibit cuspidal behaviour corresponding to the onset of limit lines.

The reflection of shock waves over straight reflecting surfaces in steady flows was investigated experimentally using the supersonic wind tunnel of Laboratoire d'Aerothermique du CNRS, Meudon, France. The results for a flow Mach number M0 = 4.96 contradict the state of the art regarding the regular [harr ] Mach reflection transition in steady flows. Not only was a hysteresis found to exist in this transition, but, unlike previous reports, regular reflection configurations were found to be stable in the dual-solution domain in which theoretically both regular and Mach reflection are possible.

The reflection of shock waves over straight reflecting surfaces in steady flows was investigated numerically with the aid of the LCPFCT algorithm. The findings completely supported the experimental results which were reported in Part 1 of this paper (Chpoun et al. 1995). In addition, the dependence of the resulting shock wave configuration on the distance between the trailing edge of the reflecting wedge and the bottom surface, inside the dual-solution domain, was studied. As a result of this study, as well as the one reported in Part 1, the state of the art of shock wave reflections in steady flows was reconsidered.

Front tracking simulations of the Richtmyer-Meshkov instability produce significantly better agreement with experimentally measured growth rates than obtained in nontracking computations. Careful analysis of the early stages of the shock acceleration process show that nonlinearity and compressibility play a critical role in the behaviour of the shocked interface and are responsible for the deviations from the linear theories. The late-time behaviour of the interface growth rate is compared to a nonlinear potential flow model of Hecht et al.

The Magellan images of Venus have revealed a number of intriguing volcanic features, including the steep-sided or ‘pancake’ domes. These volcanic domes or flows have morphologies that suggest formation by a single continuous emplacement of lava with a higher viscosity than that of the surrounding basaltic plains. Numerous investigators have suggested that such high viscosity is due to high silica content, leading to the conclusion that the domes are evidence of evolved magmatic products on Venus. However, viscosity depends on crystallinity as well as on silica content: high viscosity could therefore also be due to a cooler (and therefore higher crystal content) lava. Models of dome emplacement which include both cooling and composition factors are thus necessary in order to determine the ranges of crystallinity and silica content which might lead to the observed gross dome morphologies. Accordingly, in this study domes are modelled as radial viscous gravity currents with an assumed cooling-induced viscosity increase to include both effects. Analytical and numerical results indicate that pancake dome formation is feasible with compositions ranging from basaltic to rhyolitic. Therefore, observations of gross dome morphology alone are insufficient for determining composition and the domes do not necessarily represent strong evidence for evolved magmatism on Venus.

As in a previous theory (Longuet-Higgins 1963) parasitic capillary waves are considered as a perturbation due to the local action of surface tension forces on an otherwise pure progressive gravity wave. Here the theory is improved by: (i) making use of our more accurate knowledge of the profile of a steep Stokes wave; (ii) taking account of the influence of gravity on the capillary waves themselves, through the effective gravitational acceleration g* for short waves riding on longer waves.

Nonlinearity in the capillary waves themselves is not included, and certain other approximations are made. Nevertheless, the theory is shown to be in essential agreement with experiments by Cox (1958), Ebuchi, Kawamura & Toba (1987) and Perlin, Lin & Ting (1993).

A principal result is that for gravity waves of a given length L > 5 cm there is a critical steepness parameter (AK)c at which the surface velocity (in a frame of reference moving with the phase-speed) equals the minimum (local) speed of capillary-gravity waves. On subcritical gravity waves, with steepness AK < (AK)c, capillary waves may be generated at all points of the wave surface. On supercritical waves, with AK > (AK)c, capillary waves can only be generated in the wave troughs; they are trapped between two caustics near the crests. Generally, the amplitude of the parasitic capillaries is greatest on gravity waves of near critical (but not maximum) steepness.

A viscous or inviscid cylindrical jet with surface tension in a surrounding medium of negligible density tends to pinch owing to the mechanism of capillary instability. We construct similarity solutions which describe this phenomenon as a critical time is encountered, for three distinct cases: (i) inviscid jets governed by the Euler equations, (ii) highly viscous jets governed by the Stokes equations, and (iii) viscous jets governed by the Navier-Stokes equations. We look for singular solutions of the governing equations directly rather than by analysis of simplified models arising from slender-jet theories. For Stokes jets implicitly defined closed-form solutions are constructed which allow the scaling exponents to be fixed. Navier-Stokes pinching solutions follow rationally from the Stokes ones by bringing unsteady and nonlinear terms into the momentum equations to leading order. This balance fixes a set of universal scaling functions for the phenomenon. Finally we show how the pinching solutions can be used to provide an analytical description of the dynamics beyond breakup.

We report direct measurements of slip flow during and after plane-Couette shearing of poly(styrene) solutions between glass surfaces. Slip is studied by visualizing the microscopic motions of micron-sized silica spheres suspended in the solutions. Slip velocities as large as 4 μm s−1 are observed during shear, and recoil displacements of as much as 80 μm are observed at the glass surfaces after the driving surface stops moving. Significant slip is observed when the Weissenberg number $\lambda\dot{\gamma}$ exceeds 0.5 or so, where λ is the relaxation time and $\dot{\gamma}$ the shear rate. The magnitude of the slip correlates strongly with the fluid's birefringence, indicating that slip is induced by polymer stress or orientation. Away from the centre of the sample, toward the free edges, a secondary flow is observed, which appears to be driven by a normal-stress imbalance at these edges.

We introduce a possibly new system of orthogonal curvilinear coordinates, whose coordinate curves are logarithmic spirals in the plane, supplemented by a cylindrical coordinate for three dimensions. It is shown that plane spiral coordinates form a oneparameter family, with equal scale factors along the two orthogonal coordinate curves, and constant Christoffel symbols. The equations of magnetohydrodynamics, which include those of fluid mechanics, are written in spiral coordinates and used to find a state of magnetohydrostatic equilibrium under a radial gravity field and spiral magnetic field, and to solve the equation of non-dissipative Alfvén waves in a spiral magnetic field in terms of Bessel functions. This exact solution specifies the evolution of wave perturbations (velocity and magnetic field) and energy variables (kinetic and magnetic energy densities and energy flux) with distance, for waves of arbitrary frequency. Both the frequency and the spiral angle are varied in plots of the waveforms, which show the effect on Alfvén wave propagation of three simultaneous effects: change in the mass density of the medium and in the strength and direction of the external magnetic field.

We study the hydrodynamic stability of a thin layer of liquid that is sheared by a gas. First, the interface conditions for the free surface approximation of the problem are discussed. We then study the stability of the flow to disturbances with phase speeds smaller than the maximum velocity in the liquid film, i.e. the internal mode, extending previous results and resolving some apparent contradictions.

The dynamic effect of the gas is studied by dropping the free surface approximation and solving the Orr-Sommerfeld equation for the gas together with that for the liquid. The effect on the stability of the liquid film is very large, which is explained by the fact that the imaginary part of the wave speed (which determines the stability of the film) is very small. Consequently the free surface approximation is, in general, not correct.

We then study the dependence of the critical Reynolds number on the Weber number, on the curvature of the liquid velocity profile and on the properties of the gas. With the gas included, a second mode of instability is found which has a phase velocity that is, in general, larger than the maximum liquid velocity and corresponds to capillary-gravity waves. We compare results with experiments from the literature; good agreement is found. Finally, a suggestion on the relevance of this study to the generation of ‘roll waves’, which are important from a practical point of view, is given.

The flow field behind a constricted channel is studied numerically. A pulsating incoming flow with a non-vanishing mean is imposed at the entrance and the flow field is investigated for a wide range of Reynolds and Strouhal numbers (1500 > Re > 45, 12 > St > 0.01). In most cases (except at the two ends of the Strouhal number regime or for Re < 90), propagating vortices are found downstream of the constriction with a wavy core flow between them. The size and number of coexisting vortices depend on St but less on Re. The strength and structure of the vortical regions depend on both Re and St. The formation of the vortices is discussed for the various St regimes and the characteristics of the vortical flow are described.

The two-dimensional inviscid transonic flow about a circular cylinder is investigated. To do this, the Euler equations are integrated numerically with a time-dependent technique. The integration is based on an high-resolution finite volume upwind method.

Time scales are introduced and the flow at very short, short and large times is studied. Attention is focused on the behaviour of the numerical solution at large times, after the breakdown of symmetry and the onset of an oscillating solution have occurred. This solution is known to be periodic at Mach number between 0.5 and 0.6.

At higher speed, however, a richer behaviour is observed. As the Mach number is increased from 0.6 to 0.98 the numerical solution undergoes two transitions. Through a first one the periodical, regular flow enters a chaotic, turbulent regime. Through the second transition the chaotic flow comes back to an almost stationary state. The flow in the chaotic and in the almost stationary regimes is investigated. A numerical conjecture for the behaviour of the solution at large times is advanced.

The possible excitation of synchronous edge waves by a monocromatic wave normally approaching a plane beach is studied. Use is made of the full three-dimensional water wave theory but only beach angles β = π/(2N), where N is an integer, are considered. A weakly nonlinear stability analysis is used to investigate the interaction of subharmonic, synchronous and 3/2-frequency edge waves with the incoming wave field. It is shown that values of β exist for which an energy transfer from the incoming wave to the synchronous edge waves takes place through the intermediary of the subharmonic components.

The three-dimensional evolution of near-inertial internal gravity waves is investigated for the case of a laterally unbounded fluid layer of constant finite depth. A general Green's function formulation is derived which can be used to solve initial value problems or study the effect of forcing. The Green's function is expanded in vertical normal modes, and is very singular. Convolutions with finite-sized initial conditions lead however to well-behaved solutions. Expansions in similarity solutions of the diffusion equation are shown to be an alternative for finding exact solutions to initial value problems, with respect to one normal mode. For the case of constant buoyancy frequency normal modes expansions are shown to be equivalent to expansions in an alternative series of which the first term is the response on the infinite domain, all the others being corrections to account for the no-flux boundary condition on the upper and lower boundaries.

A thin rigid disk translates edgewise perpendicular to the rotation axis of an unbounded fluid undergoing solid-body rotation with angular velocity Ω. The disk face, with radius a, is perpendicular to the rotation axis. For arbitrary values of the Taylor number, [Tscr ] = Ωa2/ν, and in the limit of zero Reynolds number [Rscr ]e, the linearized viscous equations reduce to a complex-valued set of dual integral equations. The solution of these dual equations yields an exact representation for the velocity and pressure fields generated by the translating disk.

For large rotation rates [Tscr ] [Gt ] 1, the O(1) disturbance velocity field is confined to a thin O([Tscr ]−1/2) boundary layer adjacent to the disk. Within this boundary layer, the flow field near the disk centre undergoes an Ekman spiral similar to that created by a nearly geostrophic flow adjacent to an infinite rigid plate. Additionally, flow within the boundary layer drives a weak O([Tscr ]−1/2) secondary flow which extends parallel to the rotation axis and into the far field. This flow consists of two counter-rotating columnar eddies, centred over the edge of the disk, which create a net in-plane flow at an angle of 45° to the translation direction of the disk. Fluid is transported axially toward/away from the disk within the core of these eddies. The hydrodynamic force (drag and lift) varies as O([Tscr ]1/2) for [Tscr ] [Gt ] 1; this scaling is consistent with the viscous stresses created in the Ekman boundary layer. Additionally, an approximate expression, suitable for all Taylor numbers, is given for the hydrodynamic force on a disk translating broadside along the rotation axis and edgewise transverse to the rotation axis.

A general class of exact solutions is presented for a time-evolving bubble in a two-dimensional slow viscous flow in the presence of surface tension. These solutions can describe a bubble in a linear shear flow as well as an expanding or contracting bubble in an otherwise quiescent flow. In the case of expanding bubbles, the solutions have a simple behaviour in the sense that for essentially arbitrary initial shapes the bubble its asymptote is expanding circle. Contracting bubbles, on the other hand, can develop narrow structures (‘near-cusps’) on the interface and may undergo ‘breakup’ before all the bubble fluid is completely removed. The mathematical structure underlying the existence of these exact solutions is also investigated.

The effect of rotation about a vertical axis on the linear stability of a salt-stratified fluid enclosed in a vertical slot when subjected to a temperature difference between the walls is investigated. It is found that for large salinity stratifications there are three distinct regimes of instability for different values of the rotation rate. For small rotation rates the convection cells resemble the thin, almost flat, convection cells predicted by non-rotating theory. The effect of the rotation is to marginally destabilize the fluid whilst inducing a small slope in the convection cells along the parallel walls. As the rotation rate is increased there is an abrupt change in the form of the most unstable convection cells as their aspect ratio and slope parallel to the walls both become of order one in magnitude. As the rotation rate is further increased there is a second less abrupt transition when the internal Rossby radius of deformation based on the vertical scale of the cells becomes of the same order of size as the slot width. After this point the slope of the cells increases in proportion to the rotation rate. The asymptotic nature of these three regimes is found.

The effect of rotation on double-diffusive instabilities caused by more general horizontal temperature and salinity gradients in a salt-stratified fluid is also investigated, with particular reference to the case of heating the salinity gradient from a single sidewall. This analysis is restricted to the case where the rotation rate is low.

Convection driven by centrifugal buoyancy in a cylindrical fluid annulus cooled from the inside, heated from the outside, and rotating about its axis is described. While at high values of the dimensionless rotation parameter τ convection rolls aligned with the axis are preferred, three-dimensional patterns of convection are introduced at low values of τ through either the cross-roll instability or a subharmonic varicose instability. The instabilities are studied in terms of simple analytical relationships as well as through numerical methods based on the Galerkin scheme. Analytical expressions for the steady three-dimensional patterns induced by the instabilities are also derived.

Instabilities due to a vertically stratified horizontal magnetic field (magnetic buoyancy instabilities) are believed to play a key role in the escape of the Sun's internal magnetic field and the formation of active regions and sunspots. In a star the magnetic diffusivity is much smaller than the thermal diffusivity and magnetic buoyancy instabilities are double-diffusive in character. We have studied the nonlinear development of these instabilities, in an idealized two-dimensional model, by exploiting a non-trivial transformation between the governing equations of magnetic buoyancy and those of classical thermosolutal convection. Our main result is extremely surprising. We have demonstrated the existence of finite-amplitude steady convection when both the influential gradients (magnetic and convective) are stabilizing. This strange behaviour is caused by the appearance of narrow magnetic boundary layers, which distort the mean pressure gradient so as to produce a convectively unstable stratification.