The ordinary nonlinear Schrödinger equation for deep-water waves (found by a perturbation analysis to O(ε3) in the wave steepness ε) compares unfavourably with the exact calculations of Longuet-Higgins (1978) for ε > 0·10. Dysthe (1979) showed that a significant improvement is found by taking the perturbation analysis one step further to O(ε4). One of the dominant new effects is the wave-induced mean flow. We elaborate the Dysthe approach by investigating the effect of the wave-induced flow on the long-time behaviour of the Benjamin–Feir instability. The occurrence of a wave-induced flow may give rise to a Doppler shift in the frequency of the carrier wave and therefore could explain the observed down-shift in experiment (Lake et al. 1977). However, we present arguments why this is not a proper explanation. Finally, we apply the Dysthe equations to a homogeneous random field of gravity waves and obtain the nonlinear energy-transfer function recently found by Dungey & Hui (1979).

The thermal convective instability of a nematic layer aligned perpendicular to horizontal plates displays original characteristics, due to the coupling of the nematic distortion with the temperature gradient; in particular the adverse temperature gradient threshold ΔTc can be modified by the application of a vertical (stabilizing) or horizontal (destabilizing) magnetic field. In addition, the application of a magnetic field H controls both the threshold of this instability and the geometric form of the instabilities above ΔTc.

A linear and irrotational model is constructed to represent the formation of water waves by ground motions of a sloping bed. A survey of the constant depth case, given first, helps in understanding the mechanism of formation, and, in this oversimplified case, wave propagation away from a source, which is usually very asymmetric. The importance of asymmetry, which may produce trapped waves, is illustrated by an estimate of the propagation in a three-dimensional case. The formation of waves by a ground motion on a slope is then studied in detail. The problem is reduced to linear integral equations of the first kind. Using an inversion technique one constructs a source–response pair in which the source is ‘δ-like’ and the response is close to that which would be found if the depth was constant around the source. A general approximate solution is then derived, in both the two-dimensional and three-dimensional cases. Results for the sloping-bottom case are given for small times. They give initial values of surface displacement. They also enable one to determine the important physical parameters in the ground motion and to evaluate the efficiency of wave production.

The existence of a rarefaction shock wave or negative shock wave in a substance whose unperturbed state is close to the thermodynamic critical liquid–vapour point has been demonstrated experimentally. Its evolution and propagation velocity in a shock tube with Freon-13 as the test substance are described. It is shown that the steepness of the wave front does not diminish as the wave evolves. An equation is derived that describes the evolution of long-wave perturbations near the critical point.

Three-dimensional and time-dependent numerical simulations of thermal convection are carried out for rotating layers in which the rotation vector is tilted from the vertical to represent various latitudes. The vertical component of the rotation vector produces narrow convection cells and a reduced heat flux. As this vertical component of the rotation vector diminishes in the lower latitudes, the vertical heat flux increases. The horizontal component of the rotation vector produces striking changes in the convective motions. It elongates the convection cells in a north–south direction. It also tends to turn upward motions to the west and downward motions to the east in a manner that produces a large-scale circulation. This circulation is directed to the west and towards the poles in the upper half of the layer and to the east and towards the equator in the bottom half. Since the layer is warmer on the bottom this circulation also carries an equatorward flux of heat. When the rotation vector is tilted from the vertical, angular momentum is always transported downwards and toward the equator. For rapidly rotating layers, the pressure field changes in a manner that tends to balance the Coriolis force on vertical motions. This results in an increase in the vertical heat flux as the rotation rate increases through a limited range of rotation rates.

In this article we present numerical solutions to multicellular natural convection in a vertical enclosure. The calculated streamlines faithfully represent what has been seen in the laboratory by smoke traces in air and particle traces in oils. The calculated isotherms for air correspond to reported interferometric patterns. Solutions exhibiting travelling waves for water were calculated near conditions where they should occur according to linear stability theory. Heat-transfer results for air are given and their dependence on the aspect ratio of the enclosure exhibited.

When a laser beam is used as the energy source for welding two pieces of metal together, a hole is formed perpendicular to the plane of the workpiece. The latter is moved relative to the laser and metal is transferred from the front to the rear by fluid flow round the hole. The equations governing the process are set out and the conditions at the two boundaries in the problem (one between the hole and the molten metal, and the other between the liquid and the solid states of the metal) are considered.

Approximate solutions of the problem for low welding speeds are obtained for four different models. The first is one in which the viscosity is taken to be constant. In the second, the viscosity is allowed to depend linearly on temperature. The third model divides the liquid into a region in which the cooler part is taken to be viscous and the hotter part inviscid; the fourth model is then constructed as a limit, with the liquid motion considered as wholly inviscid. It is found that the motion is not irrotational in this last model. The models all display a downstream displacement of the boundary between the solid and liquid states, in agreement with observations. An expression for the minimum power of the laser is calculated.

Measurements of the vortex-shedding frequency behind a vertical rectangular prism and a vertical circular cylinder attached to a plane wall are correlated with the characteristics of the smooth-wall turbulent boundary layer in which they are immersed. Experimental data were collected to investigate the effects of (i) the aspect ratio of these bodies and (ii) the boundary-layer characteristics on the vortex-shedding frequency. The Strouhal number for the rectangular prism and the circular cylinder, defined by S = fcw/U0 and fcd/U0 respectively, was found to be expressed by a power function of the aspect ratio h/w (or h/d). Here fc is the vortex-shedding frequency, U0 is the free-stream velocity, h is the height, w is the width and d is the diameter. As the aspect ratio is reduced, the type of vortex shedding behind each of the two bodies was found to change from the Karman-type vortex to the arch-type vortex at the aspect ratio of 2·0 for the rectangular prism and 2·5 for the circular cylinder.

This paper reconsiders formal expansion methods for the analysis of the nonlinear properties of a modal disturbance. A survey is given of the various types of expansions based on different assumptions, and their range and shortcomings are discussed. By introducing a well-defined amplitude, Watson's expansion in a time-dependent amplitude is developed into a rational method for uniquely determining Landau constants of arbitrary order. Complementary to the common orthogonality condition for points at the neutral curve, an alternative definition of the Landau constants is given for points in the unstable domain. The method is not restricted to small amplification rates but is invalid in the stable domain. The method of Reynolds and Potter for a direct attack on equilibrium states is extended into a class of rational methods. The methods in this class agree to within a rearrangement of the infinite expansion series but differ in their respective range of validity.

The impingement of concentrations of vorticity upon a corner is studied using simultaneous flow visualization and correlation of pressure fluctuations, revealing the relations between the instantaneous pressure fields along the top and front faces of the corner and the instantaneous distortion of the incident vortex.

By varying the transverse offset of the corner relative to the incident vortex, several distinct and consistent patterns of incident vortex–corner interaction are evident. The corresponding form and phasing of the instantaneous pressure fields are strongly dependent upon the nature of the vortex–corner interaction, though the maximum amplitude on the top and front faces is always of the same order. For certain interactions, there is separation of flow from the edge of the corner, as well as from the front face of the corner, leading to secondary-vortex formation and corresponding peaks in the local pressure fields.

By integrating the instantaneous pressure fields along the top and front faces of the corner, the phasing between the respective forces is shown to exhibit varying degrees of dipole-like behaviour, depending upon the character of the incident vortex–corner interaction. Amplitudes of the corresponding moments of the pressure fields are remarkably insensitive to this interaction.

A cloud of solute injected into a pipe or channel is known to spread out by a dispersion process based on cross-sectional diffusion across a velocity shear. The original description of the process is due to Taylor (1953, 1954), and an important subsequent contribution was by Aris (1956), who framed and partially solved equations for the integral moments of the cloud of contaminant. The present work resolves some technical difficulties that occur when Aris’ solution method (separation of variables) is pursued in depth. In particular, it is shown that Aris’ technique has to be modified to give the moments at short and moderate times after the injection of solute into the flow. The paper is concerned with dispersion in those parallel flows for which an associated eigenvalue problem has a discrete spectrum of eigenvalues; fortunately, this case appears to be the rule rather than the exception. Expressions are obtained for the second and third moment about the mean, and the theory is applied to three cases of interest.

The effect of depth variations near the mouth of harbour-like geometries is investigated. It is found that some modes of oscillation that, for uniform depth, would decay rapidly can be trapped very effectively when moderate increases in depth occur at the seaward end of the harbour.

In this investigation the creeping flow due to the motion of a liquid drop or a bubble in another immiscible fluid is examined when the interface is partially covered by a stagnant layer of surfactant. The associated boundary-value problem involves mixed boundary conditions at the interface, which lead to a set of dual series equations. An inversion of these equations yields the exact solution to the stagnant cap problem.

Several useful results are obtained in closed form. Among these are the expressions for the drag force, the difference between the maximum and the minimum interfacial tensions, and the amount of adsorbed surfactant. A shifting of the centre of the internal vortex is observed.

Three velocity components of water particles in a plunging breaker over a horizontal step on the bed of a two-dimensional laboratory wave tank have been determined simultaneously by means of an elaborate flowmeter that measures the flow drag on three ‘tension threads’, with each recording a separate flow component.

It is found that all three of the r.m.s. values in the plunging breaker become maximum at x/L ≈ 0·7, where x is the distance from the breaking point to the shore and L is the wavelength. It is found that both the velocity and r.m.s. values of the transverse flow component generated by the shoaling and wave breaking become comparable to those of the other two flow components.

On the basis of spectral analyses it is found that major wave frequencies in both the longitudinal and vertical flow components of the original two-dimensional wave survive even after experiencing relatively strong shoaling and wave breaking, and part of the original wave energy is transferred to the transverse flow component and is located at these major frequencies. It is found that the majority of the higher-harmonic-frequency components (or turbulent fluctuations) are generated in the shoaling process and that the wave breaking provides a relatively minor contribution to the generation. Finally, it is found that, through the shoaling and wave breaking, the original wave energy is transported to a frequency range lower than the primary wave frequency (negative cascade), as well as to the higher frequency range (positive cascade) in each flow component.

A liquid drop held captive between parallel disks that are differentially rotated is a model for the swirling flows induced by crystal rotation in the floating-zone process for growing semiconductor materials. An asymptotic analysis for a cylindrical drop is presented that elucidates the structure of the axisymmetric cellular motions caused by disk rotation at low Reynolds number. Variations of meniscus shape induced by these flows are described in the limit of small capillary number. Most cellular flow fields break the bifurcation point that corresponds to the Plateau–Rayleigh limit for the length of a static drop into two disjoint shape families and lower the maximum stable drop length. This effect is studied by a singular bifurcation analysis.

Convection in a rapidly rotating spherical layer with constant-temperature boundary conditions is studied in a laboratory experiment. The asymptotic theory of Busse (1970) is extended to permit a comparison with the observations of the onset of convection and its properties. It is found that the prediction of the power-law dependences of the critical buoyancy number and the critical wavenumber on the rotation rate are borne out, although discrepancies in the actual values of these quantities do exist. Calculations on the basis of equations proposed by Roberts (1968) show that a thermal wind that is present in the basic state of the model has a stabilizing influence on the onset of convection. Stewartson layers not taken into account in the asymptotic analysis for vanishing Ekman number E appear to be responsible for the remaining disagreement between theoretical predictions and observations at finite values of E.

An exact solution of the free convective flow of a viscous incompressible fluid from a heated disk, rotating in a vertical plane, is obtained. The non-axisymmetric fluid motion consists of two parts; the primary von Kármán axisymmetric flow and the secondary buoyancy-induced cross-flow. A highly accurate solution of the energy equation is also derived for its subsequent use in the analysis of the cross-flow.

Experiments were performed to investigate some aspects of turbulence in rotating and non-rotating fluid systems where the turbulence was induced by a horizontal grid oscillating vertically. An earlier theory by the second author made use of a planar source of energy, which appeared to be similar to the energy source of the grid, in determining the characteristics of the turbulence at points some distance away. The simplicity of the theory was in the parameterization of the grid ‘action’ by a single quantity K, with dimensions and characteristics of eddy viscosity.

The experimental results provide additional confirmation of the theory in the non-rotating case, and indicate the usefulness of the idealized energy source in the rotating case. In the latter, we measured the propagation of the front separating disturbed and undisturbed fluid, moving along the axis of rotation. The thickness d(t) of the disturbed region increases at first as (Kt)½, as in a non-rotating fluid, until the Rossby number K/Ωd2k becomes of order unity.

Beyond this the disturbances are wavelike and rotationally dominated, and the thickness now increases linearly with time, yielding a speed of propagation for the front proportional to the wave speed (KΩ)½. Finally, the disturbances reach the bottom and the vessel is in statistical steady state. Then a region of thickness dk independent of time is found, and it contains motion that resembles ordinary, three-dimensional turbulence. dk ∼ (K/Ω)½ is analogous to the depth of the turbulent Ekman layer H ∼ (K/Ω)½, where K is taken as an eddy viscosity.

McEwan constructed a similar rotating experiment, although with a different energy source, and observed vortices parallel to the axis of rotation, provided that the Rossby number was less than a critical value. Our observations and theory indicate that the disappearance of the vortices corresponds to h < dk, where h is the total depth of the fluid. At that point, the whole tank is filled with three-dimensional turbulence.

The inviscid instability of columnar vortex flows in unbounded domains to three-dimensional perturbations is considered. The undisturbed flows may have axial and swirl velocity components with a general dependence on distance from the swirl axis. The equation governing the disturbance is found to simplify when the azimuthal wavenumber n is large. This permits us to develop the solution in an asymptotic expansion and reveals a class of unstable modes. The asymptotic results are confirmed by comparisons with numerical solutions of the full problem for a specific flow modelling the trailing vortex. It is found that the asymptotic theory predicts the most-unstable wave with reasonable accuracy for values of n as low as 3, and improves rapidly in accuracy as n increases. This study enables us to formulate a sufficient condition for the instability of columnar vortices as follows. Let the vortex have axial velocity W(r), azimuthal velocity V(r), where r is distance from the axis, let Ω be the angular velocity V/r, and let Γ be the circulation rV. Then the flow is unstable if $ V\frac{d\Omega}{dr}\left[ \frac{d\Omega}{dr}\frac{d\Gamma}{dr} + \left(\frac{dW}{dr}\right)^2\right] < 0.$

In this paper we investigate the manner in which finite-amplitude disturbances are set up in viscous fluid flows that are changing slowly in time. It is shown that, when the appropriate Reynolds or Rayleigh number is slowly increased, then, no matter how slowly this change takes place, there is always a short time interval where a quasi-steady approach breaks down. In this time interval a finite-amplitude solution is set up which ultimately approaches that predicted by a quasi-steady theory. In order to demonstrate our ideas we discuss the Taylor-vortex problem in a situation in which the speed of the inner cylinder changes slowly in time. In particular we discuss the case when the speed of the inner cylinder is modulated slowly in time and it is found that at low frequencies the disturbances of most physical relevance are not periodic solutions of the equations of motion.