The asymptotic behaviour for large Rayleigh number and Prandtl number of O(1) of two-dimensional convection cells in a fluid between horizontal plates heated from below has been discussed by Pillow (1952) and more recently by Robinson (1967). The flow models derived by Pillow and Robinson differ from each other. The purpose of the present paper is to point out the likelihood of an inconsistency in Robinson's flow model, and to discuss qualitatively a few hitherto unnoticed features of the flow.

In gas-liquid flows a certain sudden change of the flow structure may occur, which can be described as a transition from ‘jet flow’ to ‘froth flow’ accompanied by energy dissipation and pressure build-up. Upstream of this phenomenon the gas is the continuous phase; downstream the liquid is the continuous phase. The phenomenon, which has been called ‘mixing shock’, shows some similarity and also some differences with the plane shock wave in gasdynamics. In the first part of this paper the mixing shock is treated as a one-dimensional macroscopic process. With the aid of the laws of conservation of mass, momentum and energy, expressions are obtained for the pressure and entropy change across the mixing process. In addition the stability of the mixing shock in a cylindrical flow channel is treated. Next, a theory that explains the gas entrainment mechanism in the mixing shock is proposed. As an experimental tool a water-air ejector with the water as a driving medium was used. The experiments confirm the macroscopic and the microscopic theory. In the last section of this paper theoretical and experimental evidence is combined to construct a model of the processes that play a role in the shock.

The paper describes an investigation of the internal waves that are produced in a stratified fluid having constant Brunt—Väisälä frequency by a cylinder which executes small vibrations at a lower frequency. Explicit solutions are found for slender cylinders having arbitrary cross-sections. When the cross-sectional area of the cylinder varies with time it is found necessary in calculating the surface pressures and power output to take account of terms in the governing equations that are significant only at distances from the cylinder comparable to or larger than the scale height of the density variations. For this case a simple expression for the power output is obtained in terms of the rate of change of the cross-sectional area of the cylinder.

When the vibrating cylinder is rigid its cross-sectional area is independent of time and then the expression for the power output is very similar to von Kármán's expression for the drag of a body of revolution in supersonic flow.

In both the above cases it is found that one quarter of the power is radiated in each of the four directions that are inclined at a particular angle to the horizontal.

Many of the known analytic solutions of the equation for neutral disturbances to a stably stratified, inviscid, parallel shear flow are shown to belong to a wider family of solutions when a transformation to the hypergeometric differential equation is possible. Two particular cases in which the transformation can be made are examined in some detail and the solutions are expressed in a simple analytical form. A number of novel solutions are presented as examples.

The stability of plane Couette flow is examined for liquids in which the viscosity varies with depth. Under suitable conditions, the flow may be stable or unstable at small Reynolds numbers for disturbances with wavelengths long compared with the liquid depth.

A mechanism which may be stabilizing or destabilizing is found to derive from the role of diffusion in the neighbourhood of a ‘critical layer’, where the liquid velocity equals the phase velocity of a wave-like disturbance. This mechanism, which requires a viscosity gradient, is quite distinct from that found by Yih (1967) for the case of a viscosity discontinuity. An example is considered, for which these two mechanisms may be of comparable importance.

A theoretical and experimental investigation has been made of the initial ionization processes in a strong shock wave in hydrogen. The relaxation length for ionization, which is principally determined by the rate of excitation, was measured and from a comparison with the theory an estimate was obtained for the cross-section for atom-atom excitation collisions.

Detailed theoretical calculations showed that the electron temperature approaches to within 1 % of the atomic temperature in a distance that is small compared with the total relaxation length for ionization. This enabled considerable simplification, for it indicated that a single-temperature model could be used in calculating the theoretical relaxation profile over the experimental range of operating conditions. An electromagnetic shock tube, with a Philippov pinch to create the driver plasma, was employed to produce shock waves of the required velocity. The ionization behind the shock front was studied by means of a double-frequency Mach-Zehnder interferometer, with a ruby laser and a K.D.P. crystal as the light source. A close agreement between the theoretical and experimental electron density profiles, behind the shock front, was obtained for small relaxation lengths, when the cross-section for the atom-atom excitation collisions was assumed to be about 7 × 10−2 times that of the corresponding cross-section for electron-atom excitation collisions.

Rigorous estimates of amplification rates, wave speeds and sufficient conditions for linear stability are derived for the manifold of solutions of the Orr—Sommerfeld problem governing parallel motion in the boundary layer and in round pipes. The estimates for channel flow (part I) are improved and compared with numerical results for the neutral stability of Jeffery—Hamel flow.

A lifting-surface theory is presented for a cascade in subsonic shear flow by applying Fourier integral methods to the expressions of the perturbed flow field. The pressure distribution on the blade surface is determined by means of the socalled singularity method. Some numerical examples are presented and discussed in comparison with the results according to the lifting-line theory.

A significant difference is found in the effect of compressibility between a shear flow and a uniform flow. In shear flows with the maximum Mach number close to one, no such great local lift force is found near the sonic station as would be predicted by the linearized subsonic uniform flow theory. The correlation between the local lift and the local effective angle of attack at high Mach number span-stations shows a great deviation from that according to the uniform flow theory.

A generalized linear theory is developed for a compressible shear flow by extending the Fourier integral representations for the pressure field disturbed by pressure dipoles, and it is applied to a transonic shear flow past a thin aerofoil.

A method of determining the pressure distribution on the aerofoil surface is shown. Some numerical examples are presented and discussed.

It is found that the three-dimensional effect due to non-uniformity of the Mach number appears in the greatest degree at lower supersonic span-stations. Even within the scope of the linearized theory not such a great lift force arises near the sonic station as would be expected from the linearized uniform flow theory.

A train of internal gravity waves in a stratified liquid exerts a stress on the liquid and induces changes in the mean motion of second order in the wave amplitude. In those circumstances in which the concept of a slowly varying quasi-sinusoidal wave train is consistent, the mean velocity is almost horizontal and is determined to a first approximation irrespective of the vertical forces exerted by the waves. The sum of the mean flow kinetic energy and the wave energy is then conserved. The circulation around a horizontal circuit moving with the mean velocity is increased in the presence of waves according to a simple formula. The flow pattern is obtained around two- and three-dimensional wave packets propagating into a liquid at rest and the results are generalized for any basic state of motion in which the internal Froude number is small. Momentum can be associated with a wave packet equal to the horizontal wave-number times the wave energy divided by the intrinsic frequency.

The paper presents the method of solving a class of two-dimensional problems of the similarity flow of an incompressible fluid with a free surface. The fluid is assumed to be non-viscous and weightless. We consider two-dimensional irrotational similarity flows with dimensionless hydrodynamic characteristics depending only on the ratios x/v0t, y/v0t, where x, y are Cartesian co-ordinates, t is time and v0 is a constant of the velocity dimension.

The proposed method is based upon using the function introduced by Wagner (1932) and can be applied to the problems where the flow region is bounded by free surfaces and uniformly moving (or fixed) rectilinear impermeable boundaries. Introduction of Wagner's function makes it possible to reduce each of the problems under consideration to a non-linear singular integral equation for the real function.

The method is illustrated by solving the classical problem of the uniform symmetrical entry of a wedge into a half-plane of a fluid.