This paper studies the two-dimensional convective motion in a rectangular cavity, the two vertical sides of which are maintained at different temperatures. This system is studied for the special case in which the temperature difference ΔT between the two vertical walls is so large that the transfer of heat from one vertical wall to the other is achieved almost entirely by convection. Heat transfer by conduction is assumed to be of importance only in thin boundary layers adjoining the walls. For a cavity of height H, the boundary layers on the two vertical walls are found to have thickness proportional to [ell ], where [ell ]4 = κνH/γgΔT, and the condition for the boundary-layer regime to be established is that [ell ] be small compared with the width of the cavity. An approximate solution of the problem is obtained for the case of large values of the Prandtl number ν/κ, and found to be in satisfactory agreement with experimental results obtained by Elder (1965).

This paper elaborates on the assertion that energy methods provide an always mathematically rigorous and a sometimes physically precise theory of sub-critical convective instability. The general theory, without explicit solutions, is used to deduce that the critical Rayleigh number is a monotonically increasing function of the Nusselt number, that this increase is very slow if the Nusselt number is large, and that a fluid layer heated from below and internally is definitely stable when $RA < \widetilde{RA}(N_s) > 1708/(N_s + 1)$ where Ns is a heat source parameter and $\widetilde{RA}$ is a critical Rayleigh number. This last problem is also solved numerically and the result compared with linear theory. The critical Rayleigh numbers given by energy theory are slightly less than those given by linear theory, this difference increasing from zero with the magnitude of the heat-source intensity. To previous results proving the non-existence of subcritical instabilities in the absence of heat sources is appended this result giving a narrow band of Rayleigh numbers as possibilities for subcritical instabilities.

The maximum amplitude of the solitary wave of constant form is determined to be 0·83b, where b is the depth far from the crest. In the analysis it is assumed that the crest is pointed and the motion is two-dimensional and irrotational. The complex velocity potential is expressed in terms of known singularities and an infinite power series with unknown coefficients. Approximate solutions are obtained by truncating the power series after N terms, where N = 1, 3, 5, 7, and 9. The amplitude, a measure of the error, and several other pertinent quantities are computed for each value of N.

The problem solved concerns the disturbance to a stream of shallow water due to an immersed slender body, with special application to the steady motion of ships in shallow water. Formulae valid to first order in slenderness are given for the wave resistance and vertical forces at both sub- and supercritical speeds. The vertical forces are used to predict sinkage and trim of ships and satisfactory comparisons with model experiments are made.

In this paper we consider the effect of internal heat generation and a spatial variation of the gravity field on the onset of thermal convection in spherical shells. If the temperature gradient and gravity fields have the same spatial variation, then initially quiet fluids are subcritically stable. For these flows the effect of inertially non-linear disturbances is not destabilizing if the Rayleigh number is below the critical value set by linear theory plus ‘exchange of stabilities’. For subcritically-stable flows a principle of exchange of stabilities is not necessary; a stronger statement of stability for the same stability limit can be made. For the many cases calculated here in which subcritical instabilities can exist, the difference between the linear and energy limits is small and can be contracted only toward the energy limit by an improved linear theory.

The incompressible viscous flow along a right-angle corner, formed by the intersection of two semi-infinite flat plates, is considered. The effect of the three-dimensional geometry on the second-order ‘boundary layer’ flow away from the corner is determined and an interesting secondary flow is deduced. It is observed that this cross-flow prescribes the necessary asymptotic boundary conditions for the equations governing the flow inside the ‘corner layer’. A systematic matching scheme is specified and the corner flow problem is reformulated in terms of the ‘corner layer-boundary layer’ matching conditions.

The magnetohydrodynamic lubrication flow in an externally pressurized thrust bearing is investigated both theoretically and experimentally. The ordinary magnetohydrodynamic lubrication theory for this bearing is extended to include fluid inertia effects. Very good agreement is obtained between theory and experiment.

The structure and mode of propagation of spinning detonation waves in stoichiometric oxyhydrogen, at initial pressures of 20–30 mm, have been investigated. The waves were generated in a square-section tube and observations have been made by the smoked-film technique, spark schlieren photography and pressure gauges. At the front of the self-sustaining detonation waves obtained at these pressures, two Mach interactions exist, the trajectories of which are derived from the imprints made on the smoked foil. As the triple point traverses the tube section, its direction of motion is found to vary between 50° and 70° with the tube diameter. An analysis of a Mach triple point for these conditions predicts the absence of chemical reaction behind the Mach stem in the immediate neighbourhood of the triple point. Experimentally determined pressures and triple shock angles confirm, to within experimental error, the postulated theoretical configuration.

The motion of turbulent jets of heavy salt solution injected upwards into a tank of fresh water has been compared with that of plumes which are initially buoyant but become heavy as they mix with the environment. The reversal of buoyancy in the latter case is produced by using fluids having a non-linear density change on mixing, a laboratory analogue of the density changes occurring at the top of a cumulus cloud due to evaporation. The behaviour in the two cases is quite different; salt jets reach a steady height about which only small fluctuations occur, while the plumes with reversing buoyancy exhibit violent regular oscillations. This phenomenon, which is clearly a property of the ‘evaporation’ and not just of the geometry, is suggested as a likely explanation of the observed oscillation of the tops of cumulus towers. Dimensional arguments have been used to relate the experimental results to the volume, momentum and buoyancy fluxes at the source. An application of one of the deduced relations to the atmosphere gives realistic periods for the cloud-top oscillations.

A theory of finite-amplitude secondary flow between concentric rotating cylinders has been published by Davey (1962). A necessary feature of the theory is the generation of harmonics of the spatial periodicity in the axial direction of the velocity field. A method has been devised to measure the amplitude of each harmonic separately and experimental results for the fundamental and first three harmonics are presented here for Taylor numbers up to 100 times the critical value. The agreement with Davey's theory is excellent, and the agreement extends far beyond the range where the theory is expected to be valid. It is shown that all the harmonics are in phase with the fundamental. This result requires that jets and shock-like structure must be present in the velocity field.

A series of experiments is described in which a jet issues from an orifice at the nose of a body in supersonic flow to oppose the mainstream. An analytical model of the flow is developed which suggests that the aerodynamic features of a steady flow depend primarily on a jet flow-force coefficient, and the Mach number of the jet in its exit plane. A sufficient condition for steady flow is developed. The experiments are found to agree well with predictions based on the flow model. A short account is presented of some previous investigations, and some of their conclusions are re-examined in the light of the present study.

A series of flat-plate laminar boundary layer experiments carried out in an arc-heated supersonic wind tunnel is described. Measurements of the static pressure distribution, the total pressure profiles through the boundary layer, and the heat-transfer rates to catalytic and non-catalytic surfaces at moderately high enthalpies are given and compared with various theoretical predictions. The observations indicate that the dissociation fraction affects the boundary-layer profiles through its influence on the local transport properties. It is confirmed that the atomic recombination at the surface is retarded by using a non-catalytic oxide coating, and hence the heat transfer to the wall is reduced.

In the presence of an air stream, a uniform liquid film on a horizontal flat plate may be unstable to small disturbances, and waves may arise. In this paper the hydrodynamic stability of thin liquid films is examined both experimentally and theoretically.

The experiments concern water films thinner than those which have been examined in the past. It is found that, when the film thickness is sufficiently small, a previously unknown type of instability occurs. The theoretical analysis explains this surprising phenomenon.

Due to interaction of the mean airflow and small disturbances of the liquid-air interface, normal and tangential stress perturbations are produced at the liquid surface. It is shown that small wave-like disturbances become unstable when the joint influence of the component of normal stress in phase with the wave elevation and the component of tangential stress in phase with the wave slope is sufficient to overcome the ‘stiffness’ of the liquid surface due to gravity and surface tension. It is found that the destabilizing role of the tangential stress component is dominant for very thin films, and that instability may occur whatever the velocity of the air stream, provided the film is made sufficiently thin.

The dynamical properties of a fluid, occupying the space between two concentric rotating spheres, are considered, attention being focused on the case where the angular velocities of the spheres are only slightly different and the Reynolds number R of the flow is large. It is found that the flow properties differ inside and outside a cylinder [Cscr ], circumscribing the inner sphere and having its generators parallel to the axis of rotation. Outside [Cscr ] the fluid rotates as if rigid with the angular velocity of the outer sphere. Inside [Cscr ] the fluid rotates with an angular velocity intermediate to the angular velocities of the two spheres and determined by the condition that the flux of fluid into the boundary layer of the faster-rotating sphere is equal to the flux out of the boundary layer of the slower-rotating sphere at the same distance from the axis. The return of fluid is effected by a shear layer near [Cscr ] and we show that it has a complicated structure for it can be divided into three separate layers, two outer ones, of thickness $\sim R^{-\frac{2}{7}}$ and ∼R−¼, and an inner layer of thickness $\sim R^{-\frac{1}{3}}$.

The approach to equilibrium in a non-equilibrium-dissociating boundary-layer flow along a catalytic or non-catalytic surface is treated from the standpoint of a singular perturbation problem, using the method of matched asymptotic expansions. Based on a linearized reaction rate model for a diatomic gas which facilitates closed-form analysis, a uniformly valid solution for the near equilibrium behaviour is obtained as the composite of appropriate outer and inner solutions. It is shown that, under near equilibrium conditions, the primary non-equilibrium effects are buried in a thin sublayer near the body surface that is described by the inner solution. Applications of the theory are made to the calculation of heat transfer and atom concentrations for blunt body stagnation point and high-speed flat-plate flows; the results are in qualitative agreement with the near equilibrium behaviour predicted by numerical solutions.

The separation points which occur in the Newtonian theory of hypersonic flow are treated by locally modifying the shock-layer equations. This approach leads to direct verification of the free-layer theory for separation on certain bodies with discontinuous curvature. Separation points on bodies with continuous curvature have already been treated by matching the upstream shock-layer solution to the downstream free-layer solution. The agreement that is found between these results and those of the present approach provides further confirmation of the free-layer theory.

The characteristics of the laminar boundary layer on a continuous moving surface are described and an experiment is performed to demonstrate that such a flow is physically realizable. The hydrodynamic stability of the flow is analysed within the framework of small-perturbation stability theory. A complete stability diagram is mapped out. The critical Reynolds number is found to be substantially higher than that for the Blasius flow and, correspondingly, the critical layer lies closer to the wall. The disturbance amplitude function and its derivative are numerically evaluated, from which are derived the vector flow field of the disturbance, the resultant flow field (main flow plus disturbances), the root-mean-square distributions of the disturbance velocity components, and the distributions of the kinetic energy and the Reynolds stress. The energy criterion for stability is also investigated and is found to be consistent with the solutions of the eigenvalue problem.

A technique for the quantitative measurement of fluid velocities in the range 0–5 cm/sec is described. The technique uses a pH indicator, is applicable in aqueous solutions and permits visualization and measurement of three-dimensional flow fields.

A symposium on ‘Rotating fluid systems’ was held at La Jolla, California, from 28 March to 1 April 1966. The meeting was organized under the auspices of the International Union of Theoretical and Applied Mechanics, and took place at the Institute of Geophysics and Planetary Physics of the University of California. There were 75 participants, and attendance was by invitation. The subjects discussed included steady flows, both thermally driven and non-thermal; transient motions and instabilities; planetary waves and gravity waves affected by rotation; inertial oscillations; and hydromagnetic flows relating to the Earth's interior. The following is a brief account of the proceedings as seen by the authors. No formal volume of the papers presented is to be published, but references to published or unpublished work are given at the end of this article.

The effect of uniform rotation on surface-tension-driven convection in an evaporating fluid layer is considered both theoretically and experimentally. The theoretical analysis follows the usual small-disturbance approach of perturbation theory and leads, at the neutral state, to a functional relation between the Marangoni and Taylor numbers which is then computed numerically. In addition, it is shown analytically that, in the limit of rapid rotation, the velocity and temperature fluctuations are confined to a thin Ekman layer near the surface, and that Mc = 4·42T½ and ac = 0·5T¼, where Mc and ac are, respectively, the critical Marangoni number and the critical wave number for neutral stability, and T is the Taylor number.

The experimental part deals primarily with the flow pattern of a 50% solution of ethyl ether in n-heptane evaporating into still air. In this case, the convective flow is surface-tension-driven and its structure was observed using schlieren optics. In the absence of rotation, the flow shows a remarkable cellular pattern when the layer is shallow, but when the depth of the layer is increased the pattern quickly becomes highly irregular. In contrast, for T > 103, a cellular structure is always observed even for deep layers, a result which is attributable to the stabilizing effect of the Coriolis force. A further increase in T leaves the flow pattern unchanged except that the size of the cells is found to decrease as T−¼ which is in agreement with the results of the linear stability analysis.