In this paper a two-dimensional free boundary arising from the steady thermo-capillary flow in a viscous incompressible fluid is studied numerically. The problem is considered in the context of the open-boat crystal-growth technique. The motion of the fluid is governed by the Navier-Stokes equations coupled with the heat equation. The problem is solved numerically by a finite-element-method discretization. Three iterative methods are introduced for the computation of the free boundary. The non-dimensional form of the problem gives rise to the following characteristic parameters: Reynolds, Grashof, Prandtl, Marangoni, Bond, Ohnesorge, Biot numbers. The influence of these parameters on the flow field, the temperature distribution and the shape of the free boundary is studied.

Numerical results are presented for the approach of a rigid sphere normal to a deformable fluid-fluid interface in the velocity range for which inertial effects may be neglected. Both the case of a sphere moving with constant velocity, and that of a sphere moving under the action of a constant non-hydrodynamic body force are considered for several values of the viscosity ratio, density difference and interfacial tension between the two fluids. Two distinct modes of interface deformation are demonstrated: a film drainage mode in which fluid drains away in front of the sphere leaving an ever-thinning film, and a tailing mode where the sphere passes several radii beyond the plane of the initially undeformed interface, while remaining encapsulated by the original surrounding fluid which is connected with its main body by a thin thread-like tail behind the sphere. We consider the influence of the viscosity ratio, density difference, interfacial tension and starting position of the sphere in deter-mining which of these two modes of deformation will occur.

The unstable transition behaviour of a bounded, current-carrying, two-dimensional magnetofluid is explored, using the hydrodynamic theory developed for parallel shear flows as a guide. The time development of a perturbed driven magnetohydrodynamic sheet pinch is simulated numerically. The nonlinear partial differential equations of two-dimensional, incompressible, viscoresistive magnetohydrodynamics are advanced in time numerically by means of a semi-implicit, mixed Fourier collocation-finite difference algorithm. Nonlinear excitation of the higher wavenumbers results in the development of electric current sheets of finite extent, as well as the formation of ‘attraction currents’ centred at the magnetic O-points. A secondary instability mechanism, the dynamic tearing of the electric current sheet, is also observed. This dynamic tearing leads to sawtooth-like temporal oscillations in certain global quantities. The long time state of the system resembles a nonlinearly saturated state with significant excitation of many wavenumbers. Some features of this state can be understood by means of a Landau nonlinear stability theory based on certain assumptions about the perturbation energy balance.

Several experiments have suggested that streamwise vortices, with their accompany-ing low-momentum streaks in a turbulent boundary layer have a characteristic spanwise wavelength of approximately λz+ = 100. Here a mechanism is proposed which selects a comparable spanwise wavelength and produces counter-rotating streamwise vortices in a turbulent boundary layer. Examining the equations which describe the deviation of the velocity field from its time-average, it is found that a resonance (Benney & Gustavsson 1981), is associated with the mean-velocity profile. As an integral part of this resonance, there is a mean secondary flow which has a spanwise wavelength λz+ = 90 and whose velocities exhibit a streamwise vortex structure similar to those observed.

Despite its central importance, the pressure field at a leading edge has remained uncharacterized for the classical jet-edge interaction at a single predominant frequency. This investigation shows that the force, due to the integrated instantaneous pressure field on the edge, is located a distance downstream of the tip of the edge as much as one-quarter of a wavelength (λ) of the incident instability; this distance also corresponds to about one-quarter of the geometric length (L) between the nozzle and tip of the edge. Consequently, the traditional assumption that the phase-locking criterion for self-sustained oscillations can be expressed as a ratio of L/λ is inappropriate for low-speed jet flows, which have been of primary interest over the past two decades.

The edge pressure field is made up of two regions bounded by the maximum amplitude at the onset of separation from the surface of the edge: a near-tip region (0 ≤ x/λ [lsim ] 0.1) where the amplitude drops to a minimum as the tip is approached; and a downstream region (x/λ [gsim ] 0.1) where the amplitude varies as x−a. Since the drop in pressure in the near-tip region does not occur over a streamwise length commensurate with the length of the edge, imposition of a Kutta condition is inappropriate in simulations of the edge region. Moreover, in the near-tip region (0 [lsim ] x/λ [lsim ] 0.2), the pressure field is non-propagating; a wave-type representation is appropriate only downstream of this region.

At the tip of the edge, occurrence of the pressure minimum is due to the minimum in fluctuating angle of attack a of the approaching shear layer, deducible from the velocity eigenfunctions of linear theory; correspondingly, flow separation occurs downstream of, not at, the tip of the edge. When the tip is displaced off centreline, there is a rise in a, giving a rise in tip pressure amplitude; nevertheless, the overall x−a amplitude distribution persists.

This overall x−a (a ∼ ½) variation of the pressure amplitude commences downstream of the tip of the edge near the onset of flow separation, which leads to secondary-vortex formation; in turn, it is driven by development of the primary vortex in the unstable jet shear layer, having initially distributed vorticity. The role of this flow separation and subsequent secondary-vortex formation is, therefore, not to relieve a singularity at the tip of the edge; it is simply a consequence of growth of the primary vortex along the edge.

In Part 1 of this investigation we addressed the instantaneous pressure field at a leading-edge due to single frequency jet-edge interactions; here we consider edge pressure fields and associated vortex interaction patterns at the edge having a number of (at least six) well-defined spectral components. Each of the spectral components is present along the entire extent of the approach shear layer upstream of the edge; the relative amplitudes of these components are preserved in the conversion process from the free (approach) shear layer to the surface pressure field, the key link being complex, but ordered patterns of vortex interaction at the edge. Moreover, the predominant spectral components can be reasoned on the basis of these visualized vortex interactions by considering the vortex array in the incident shear layer.

The spectral character of the surface pressure field changes dramatically with edge location in the incident shear layer, or vortex array. If the edge is symmetrically located within the vortex array of the incident jet, a low-frequency component prevails due to large-scale vortex formation; however, with an asymmetrically disposed edge, the most unstable frequency of the jet dominates due to direct vortex impingement upon the edge, and the mean-square pressure amplitude (encompassing all spectral components) is double that of the symmetrical interaction.

Irrespective of the type of interaction and the manner in which energy is partitioned between spectral components, the mean-square pressure due to the sum of all spectral components decays approximately as x−c immediately downstream of the tip of the edge. However, each spectral component tends either to a maximum or minimum amplitude as the tip of the edge is approached, depending upon the class of vortex array-edge interaction.

Steady-state two-dimensional results obtained from numerical solutions to the transient Navier-Stokes equations are given for laminar convective motion of a gas in an enclosed vertical slot with large horizontal temperature differences. We present results for air using the ideal-gas law and Sutherland-law transport properties, although the results are also valid for hydrogen. Wide ranges of aspect-ratio, Rayleigh-number and temperature-difference parameters are examined. The results are compared in detail with the exact solution in the conduction and fully developed merged boundary-layer limits for arbitrary temperature difference, and to the well-established Boussinesq limit for small temperature difference. It is found that the static pressure, and temperature and velocity distributions are very sensitive to property variations, even though the average heat flux is not. In addition we observe a net vertical heat flux to be the same as that obtained from the Boussinesq equations. We concentrate on the boundary-layer regime, but we present a rather complete picture of different flow regimes in Rayleigh-number, aspect-ratio and temperature-difference parameter space. We observe that, with increasing temperature difference, lower critical Rayleigh numbers for stationary and oscillatory instabilities are obtained. In addition we observe that in some cases the physical nature of the instability changes with increasing temperature difference.

Numerical solutions of the three-dimensional equations for buoyancy-driven flows in cylinders with differentially heated endwalls have been obtained by a finite-difference method. Special attention has been devoted to the complex three-dimensional flow structures arising in horizontal cylinders for configurations relevant for crystal growth by vapour transport. The characterization of the transition between the core-driven regime and the boundary-layer-driven regime is considered with the properties of the main flow and also the transverse flow.

Transition to turbulence in a circular Couette system is followed numerically up to the state beyond the second bifurcation in the limit of the narrow gap between coaxial cylinders which rotate with almost equal speeds in the same direction. Taylor-vortex flow, which emerges after the Taylor number T exceeds its critical value Tc = 1708, becomes unstable to two different types of non-axisymmetric disturbances, depending on [Rscr ] (the Reynolds number) and Ω, which measure the velocity difference between the cylinders and the mean angular velocity respectively. Finite-amplitude calculations show that one part of the bifurcating flow is characterized by the vortices winding out of phase in the axial direction but still keeping the boundaries between vortices unaffected. The other is distinguished by the vortex dislocation. The inflow as well as the outflow boundaries between vortices are wavy. Both types of solutions are stationary in the frame of reference rotating with the angular velocity Ω.

Several points of interpretation are reviewed bearing on the celebrated discovery by Darwin (1953) that the added mass for a body translating uniformly in an infinite expanse of perfect fluid equals the drift-volume times the density of the fluid. The discussion focuses on the delicate qualifications needed to secure this equality as a mathematical proposition. In § 2 a different approach to the matter is presented, leading to a new fact about added mass. In § 3 a model of infinity in the fluid is proposed which clarifies an aspect of Darwin's original analysis.

The question is addressed of how much energy is reflected when a Kelvin wave propagating along a straight channel hits a bend. The solution is expressed as a truncated series of Kelvin waves and several evanescent cross-channel Poincaré modes. The bend acts as a diffraction grating – for bends of certain angles there is complete transmission and between these angles there are lobes of reflection. The width of the lobes of the diffraction pattern is directly proportional to the wavelength of the incident Kelvin wave, as in optics, electromagnetism, etc. The effect of changing the inside radius of the bend is also examined. The reflection of energy is generally small unless the Poincaré modes are nearly propagating.

The flow created by an impulsively started pressure distribution travelling at a constant velocity in a shallow channel is investigated. The restricted Green-Naghdi theory of fluid sheets is used to perform the three-dimensional calculations. The results show remarkable similarity to model tests. In particular, these calculations predict the periodic generation of two-dimensional solitons in front of and travelling faster than the disturbance if the disturbance is large enough. Behind the disturbance a complicated, doubly corrugated set of waves is formed. The computations also predict that periodic creation of solitons is accompanied by a correspondingly periodic oscillation of the wave drag, as well as a dramatic increase in the mean wave drag.

The effects of viscosity and heat conduction on the propagation of internal gravity waves are examined. These waves propagate in a stably stratified, parallel shear flow with one critical level. The Boussinesq approximation is adopted. For large Reynolds number the governing sixth-order differential equation is solved by analytical methods. In the limit of large Reynolds number it is found that the reflection and transmission coefficients for a wave incident in a viscous fluid are the same as in the inviscid case. Hence over-reflection can also occur in a viscous fluid. For the perturbed velocity components at the critical level, asymptotic expressions are derived. The results we obtain are valid for smooth, but otherwise arbitrary, shear-flow and density profiles.

The effects of circumferential curvature and strong density variation on E¼ Stewartson layers are investigated. The solution of the third-order ordinary differential equation found to govern the flow is obtained by numerical integration for layers extending in the positive radial direction and in terms of a Frobenius-series solution for layers extending in the opposite direction. Due to the variation of the basic density field, the E¼ layer is compressed in the positive radial direction. E1/4 layers extending in the negative radial direction are likely to extend fully to the axis of symmetry because of the density variation and, consequently, a distinction in terms of a geostrophic flow and an E¼ layer flow cannot be made. Curvature effects are found to play a significant role in this case. A simple case of driving by a differential rotation of part of the horizontal boundaries is examined.

We examine weakly dissipative, weakly nonlinear waves in which the fundamental derivative $\overline{\Gamma}$ changes sign. The undisturbed state is taken to be at rest, uniform and in the vicinity of the $\overline{\Gamma} = 0$ locus. The cubic Burgers equation governing these waves is solved numerically; the resultant solutions are compared and contrasted to those of the inviscid theory. Further results include the presentation of a natural scaling law and inviscid solutions not reported elsewhere.

A phase-plane method is used to study the existence of similarity solutions of the two-dimensional and axisymmetric shallow-water equations representing gravity currents with volumes proportional to tα, where α ≥ 0 and t is the time after the flow is initiated. Only currents for which there is a balance between the inertia of the current and the driving buoyancy force are considered. It is found that similarity solutions exist for the two-dimensional problem for all α ≥ 0, with some restrictions on the condition at the current front. It is shown that no similarity solutions satisfying the boundary conditions on the axis of symmetry exist for the axisymmetric problem except when α = 0.

The nonlinear growth of Tollmien-Schlichting disturbances in a boundary layer is considered as an initial-value problem, for the unsteady two-dimensional triple deck, and computational and analytical solutions are presented. On the analytical side, the nonlinear properties of relatively high-frequency/high-speed disturbances are discussed. The disturbances travel at the group velocity and their amplitude is controlled by a generalized cubic Schrödinger equation, during a first stage of the nonlinear development. The equation, which has been studied in other contexts also, is integrated numerically here, and the resulting large-time/far-downstream behaviour is then deduced analytically. This behaviour comprises an exponentially fast growth and spreading of the disturbance, the spreading being governed only by an integral property of the initial disturbance. Secondary sideband instability does not occur, and there is no conclusive sign of a chaotic response, during this stage, although the three-dimensional counterpart could well yield both phenomena. In the subsequent (and more nonlinear) second stage further downstream, however, where the amplitude is larger, spiked behaviour and spectrum broadening can occur because of vorticity bursts from the viscous sublayer. Computationally, two forms of numerical solution of the triple-deck problem, one spectral, the other finite-difference, are given. The results from each form tend to support the conclusions of the high-frequency analysis for initial-value problems, and recent calculations of the two-dimensional unsteady Navier-Stokes equations also provide some backing. One implication is that the unsteady planar interacting-boundary-layer equations, or a composite version, can capture much of the physics involved in the beginnings of boundary-layer transition although, again, three-dimensionality is undoubtedly an important element which will need to be incorporated eventually.

We examine the problem of a steady, inviscid, reduced-gravity rotating flow in a wedge around a sharp corner. Solutions to nonlinear equations are obtained via a power-series expansion in a Rossby number, diffraction theory and Green's function method. The wedge of an angle $\frac{3}{2}\pi $ is used, as an example, to show details of the solution. The results exhibit the relative importance of the pressure gradient, centrifugal and Coriolis forces. For re-entrant corners, a centrifugal upwelling of the interface occurs very close to the apex and, hence, is not important if coastal radii of curvature are comparable to, or larger than, the Rossby radius; the flow is also supercritical within an arc, whose size depends upon the Rossby number and the angle of the wedge. Using two or more corner solutions, plausible flow streamlines can be generated in more complicated domains, as long as no two corners are closer than the Rossby radius of deformation. This procedure is illustrated with two examples: (i) circulation in a channel mouth; and (ii) flow around a square bump in a coastline.

Two-dimensional periodic waves beneath an elastic sheet resting on the surface of an infinitely deep fluid are investigated using a high-order series-expansion technique. The solution is found to have certain features in common with capillary-gravity waves; specifically, there is a countably infinite set of values of the flexural rigidity of the sheet at which the series solution fails, and these values are conjectured to be bifurcation points of the solution. Limiting waves of maximum height are found at each value of the flexural rigidity investigated. These are characterized by a cusp singularity in the elastic bending moment at the wave crest, and infinite fluid pressure there. For at least one value of the flexural rigidity, the series solution shows that the wave of maximum height also travels with infinite speed.

The flow of a stratified fluid over topography is considered in the long-wavelength weakly nonlinear limit for the case when the flow is near resonance; that is, the basic flow speed is close to a linear long-wave phase speed for one of the long-wave modes. It is shown that the amplitude of this mode is governed by a forced Korteweg-de Vries equation. This equation is discussed both analytically and numerically for a variety of different cases, covering subcritical and supercritical flow, resonant or non-resonant, and for localized forcing that has either the same, or opposite, polarity to the solitary waves that would exist in the absence of forcing. In many cases a significant upstream disturbance is generated which consists of a train of solitary waves. The usefulness of internal hydraulic theory in interpreting the results is also demonstrated.