This paper presents a broad investigation into the properties of steady gravity currents, in so far as they can be represented by perfect-fluid theory and simple extensions of it (like the classical theory of hydraulic jumps) that give a rudimentary account of dissipation. As usually understood, a gravity current consists of a wedge of heavy fluid (e.g. salt water, cold air) intruding into an expanse of lighter fluid (fresh water, warm air); but it is pointed out in § 1 that, if the effects of viscosity and mixing of the fluids at the interface are ignored, the hydrodynamical problem is formally the same as that for an empty cavity advancing along the upper boundary of a liquid. Being simplest in detail, the latter problem is treated as a prototype for the class of physical problems under study: most of the analysis is related to it specifically, but the results thus obtained are immediately applicable to gravity currents by scaling the gravitational constant according to a simple rule.

In § 2 the possible states of steady flow in the present category between fixed horizontal boundaries are examined on the assumption that the interface becomes horizontal far downstream. A certain range of flows appears to be possible when energy is dissipated; but in the absence of dissipation only one flow is possible, in which the asymptotic level of the interface is midway between the plane boundaries. The corresponding flow in a tube of circular cross-section is found in § 3, and the theory is shown to be in excellent agreement with the results of recent experiments by Zukoski. A discussion of the effects of surface tension is included in § 3. The two-dimensional energy-conserving flow is investigated further in § 4, and finally a close approximation to the shape of the interface is obtained. In § 5 the discussion turns to the question whether flows characterized by periodic wavetrains are realizable, and it appears that none is possible without a large loss of energy occurring. In § 6 the case of infinite total depth is considered, relating to deeply submerged gravity currents. It is shown that the flow must always feature a breaking ‘head wave’, and various properties of the resulting wake are demonstrated. Reasonable agreement is established with experimental results obtained by Keulegan and others.

The oscillations of a curved interface are considered, neglecting the effects of gravity. The system under consideration consists of a right, circular, cylindrical tank of finite length, partially filled with an inviscid, incompressible, wetting liquid. When the container spins about its axis of revolution, the large-scale vapour cavity takes an elongated spheroid-like shape, symmetric about the axis of rotation. The fluid—vapour interface will oscillate about the equilibrium configuration if disturbing forces are present. The case where the vapour cavity touches the walls of the tank is not included in this investigation.

The equations of motion are linearized. However, the resulting eigenvalue problem is non-linear. Surface tension and rotation are taken into account only to the extent allowed by a linearized stability theory.

The self-sustained oscillations are governed by a partial differential equation of elliptic type, the field equation of the perturbation pressure. According to the results obtained from theory, all eigenfrequencies for this case are greater than twice the angular speed of the tank. The first two eigenfrequencies can be computed with high accuracy. The relation between the bubble shape and the eigen-frequency is shown in a graph for a specific example.

The governing differential equation is hyperbolic for forced oscillations induced by a small force field of constant magnitude and direction in an inertial frame of reference. A solution for this problem exists only in case of a cylindrical tank of infinite length. Discontinuities in the velocity components occur in the flow field. A numerical example has been carried out.

Flow properties for the non-equilibrium two-phase flow of a gas-particle mixture are formulated from the theoretical standpoint. A quasi-one-dimensional flow containing an arbitrary volume of particles is considered, and mass transfer between the phases is allowed. It is shown that meaningful definitions of the flow properties of each phase can be constructed as area-averages of (time-averaged local flow-field properties). Special definitions of averages overcome the difficulties introduced by the fact that one phase does not occupy the entire region at all times. Conservation equations for the newly defined properties are given and criteria for their validity determined. The results give fresh interpretation to several aspects of two-phase flow: the particle-phase pressure is associated with the internal particle pressure, whereas Reynolds-stress terms are introduced by fluctuations in particle velocity. Reynolds stresses for both phases are important in laminar as well as turbulent flow and provide a significant particlephase viscous effect. The interphase momentum transfer because of condensation or vaporization is shown to be characterized by the particle-phase velocity irrespective of the direction of the mass transfer.

The rate of self-propulsion of a doubly-infinite flexible sheet due to transverse waving oscillations in a viscous fluid is shown to decrease with increasing frequency, at a fixed (small) wave amplitude. This result differs from that of Reynolds (1965) who included local inertia, and thereby predicted that the swimming speed increases above the limiting value given by Taylor (1951) at zero frequency. The error in Reynolds’ work is due to his neglect of the simultaneous effect of convection, which induces a non-uniform mean second-order flow, with direction such as to oppose propulsion. Some other results concerning swimming sheets are presented.

A disk (i.e. a body whose maximum thickness is small compared with its lateral dimensions) floats with its central plane of symmetry upright. Its hydrostatic oscillations are lightly damped by the reaction of the gravity waves generated. A damping coefficient is obtained. It is shown that superimposed upon these oscillations is a small displacement which decays with the time t like t−4 or t−5.

The unidirectional flow of an incompressible, electrically conducting, viscous fluid along cylindrical pipes is considered. An external magnetic field, B0, which lies in the plane transverse to the flow is applied. It is shown that the governing equations, written in the co-ordinate system traced out by B0, are mathematically very similar to those for a uniform field.

The paper deals mainly with ducts whose walls are insulators. Though exact solutions (valid for all values of the Hartmann number) are derived, the limit of high Hartmann number is taken for detailed discussion. Transition layers (or, loosely, ‘wakes’) can arise which are centred on curved field lines. In some cases, reversed flow occurs in part of the core (‘radial-type’ fields). Situations also arise where the magnitude (and sign) of the velocity remains the same as for B0 = 0, whatever the strength of the applied, transverse (azimuthal) magnetic field.

A penetrative convection problem which depends on the density maximum of water near 4 °C is solved numerically. When a layer of water has its lower boundary maintained at 0 °C and its upper boundary at some temperature above 4 °C, the layer will be divided into a lower convectively unstable region and an upper convectively stable region. Steady-state finite-amplitude solutions to this problem are obtained using the mean field approximation and free boundaries.

Convective mixing alters the temperature structure of the layer so that the temperature of a large fraction of the layer is slightly below the temperature of maximum density, in agreement with laboratory measurements. The largest motions are found in a principal convective cell which extends from the lower boundary to a temperature of 7° or 8 °C. Above the principal cell one or more counter cells may form, depending on the temperature of the upper boundary. The velocities in the counter cells are substantially less than those in the principal cell and fall off rapidly going upward. When the temperature of the upper boundary is 7 °C or higher, convection first takes place at a finite amplitude and at a Rayleigh number less than that predicted by the linear theory. When the temperature of the upper boundary is 10 °C or higher, the upper boundary is no longer important dynamically to the system.

As the Rayleigh number is increased above critical stability the velocities in the principal cell, heat transport and the distortion of the temperature field all increase. In addition, the principal cell becomes more slender and fills a greater fraction of the layer. Also, as the Rayleigh number increases the counter cells become more flattened, fill a smaller fraction of the layer, and the velocities in the counter cells decrease relative to those in the principal cell.

The most important penetration of convective motions takes place in the form of nearly horizontal motions in the lowest part of the stable region, corresponding to the upper part of the principal cell. The velocities of these motions are a large fraction of the largest vertical velocities in the unstable region.

This study was begun as an attempt to either confirm or disprove conflicting results of previous research upon a classical problem in potential theory. An integral equation resulting from a surface distribution of vorticity is used to solve numerically for the flow through a circular orifice. Free-surface profiles and contraction coefficients are determined for four different ratios of orifice area to pipe area, and a comparison is made between the numerical and experimental results of both this study and previous studies.

Riegels (1938) investigated the breakdown of Hele-Shaw flow in a Hele-Shaw cell with unusually large separation distance 2h* between the walls. A theoretical outer expansion for the velocity was constructed in the case where the obstacle is a circular cylinder, using an intuitive inner boundary condition that seems to be correct in the limit h* → 0, but without explicit matching with the inner expansion.

An inner expansion has now been found, and it shows that the solution in the inner layer forces terms into the outer expansion that are larger than those found by Riegels whenever h* is finite and not zero.

The plane and axisymmetric hypersonic flow past blunted bodies is investigated as an inverse problem (shock shape given). The fluid may behave as a real gas in local thermodynamic equilibrium. Viscosity and heat conduction are neglected. An analytical solution uniformly valid in the whole flow field (from the stagnation region up to large distances from the body nose) is given. The solution is based on two main assumptions: (i) the density ratio ε across the shock is very small, (ii) the pressure at a point P of the disturbed flow field is not very small compared with the pressure immediately behind the shock in the intersection point of the shock surface with its normal through P. Terms O(ε) are neglected in comparison with 1, but it is not necessary for the shock layer to be thin. The change of velocity along streamlines is taken into account. In order to calculate the flow quantities one has to evaluate only two integrals (equations (49) and (53) together with the boundary values (5) and (10)). The application of the solution is illustrated and the accuracy is tested in some examples.