The Cauchy problem for the β-plane form of the tidal equations is solved for both oscillatory and delta function initial data. The radius of deformation is assumed to be much less than the radius of the earth, and in accord with this assumption a ray approximation is employed.

It is shown that, owing to the rapid rate of propagation of inertio-gravity waves, the motion in its initial development tends towards geostrophic balance. However, the solution given by the ray approximation is singular on certain surfaces in space and time, the envelopes of the rays. A local boundary-layer theory is employed to correct this deficiency. The existence of these caustics implies that the process of geostrophic adjustment is more complicated than hitherto imagined.

The hydrodynamic stability of the developing laminar flow in the entrance region of a parallel-plate channel is investigated using the theory of small disturbances. The stability of the fully developed flow is also re-examined. A wide range of analytical (i.e. asymptotic) and numerical methods are employed in the stability investigation. Among the asymptotic methods, each of three viscous solutions (singular, regular and composite) is used along with the inviscid solution to provide critical Reynolds numbers and complete neutral stability curves. Two numerical methods, finite differences and stepwise integration, are applied to calculate critical Reynolds numbers. The basic flow in the development region is treated from two stand-points: as a channel velocity profile and as a boundary-layer velocity profile. Extensive comparisons among the various methods and flow models disclose their various strengths and ranges of applicability. As a general result, it is found that the critical Reynolds number decreases monotonically with increasing distance from the channel entrance, approaching the fully developed value as a limit.

The stability of cellular convection flow in a layer heated from below is discussed for Rayleigh number R close to the critical value Rc. It is shown that in this region the stable stationary solution is determined by a minimum of the integral \[ \int_0^{H_0}R(H)\,dH, \] where R(H) is a functional of arbitrary convective velocity fields which satisfy the boundary conditions. For the stationary solutions R(H) is equal to the Rayleigh number. H0 is a given value of the convective heat transport. In a second part of the paper explicit results are derived for the convection problem with deviations from the Boussinesq approximation owing to the temperature dependence of the material properties.

This paper considers the mixing of a turbulent radial wall jet with a secondary fluid introduced into the impingement area of the wall jet so as to form a steady axisymmetric state. Similarity of the concentration profiles perpendicular to the wall has been assumed and, by solving the momentum and mass flow equations, the concentration distribution through the layer and a similarity exponent giving the variation of concentration along the wall have been determined.

The interval of Rayleigh numbers in Bénard convection corresponding to cellular motion is determined in the case of free-free boundaries, rigid-free boundaries and rigid-rigid boundaries, taking into account the variation of the kinematic viscosity with temperature. Neglecting the effect of surface tension, it is shown that this interval is largest for the rigid-rigid case. The most important feature from the obtained formula (6.1) is, however, that the interval is extremely dependent on the depth of the fluid layer. To obtain a cellular pattern it is therefore necessary to have very small fluid depths. For example, with Silicone oil and a fluid depth of 7 mm, cellular motion will, according to the theory, be observed for Rayleigh numbers larger than the critical value and less than 1·07 times the critical value. For a fluid depth of 5 mm, however, the formula (6.1) gives that cellular motion will be observed for Rayleigh numbers up to 1·54 times the critical value.

This paper presents the results of an experimental investigation of the behaviour of artificial disturbances produced by an oscillating ribbon in the free convection boundary layer over a vertical uniform flux plate. The boundary layer was observed using a Mach Zehnder interferometer. The observations resulted in the approximate location of a neutral curve which is in good agreement with calculations from the linear stability theory.

In a rotating system, the vertical transport of angular momentum by internal gravity waves is independent of height, except at critical levels where the Doppler-shifted wave frequency is equal to plus or minus the Coriolis frequency. If slow rotation is ignored in studying the propagation of internal gravity waves through shear flows, the resulting solutions are in error only at levels where the Doppler-shifted and Coriolis frequencies are comparable.

A shallow layer of silicone oil on a plane, circular copper plate was uniformly heated from below. The air on its surface was kept at constant temperature by a uniformly cooled glass plate close to the oil, which at the same time inhibited air motions. Motions began with concentric circular rolls which, after the centre ring had formed, broke down into a hexagonal pattern. A rather accurate determination of the wavelength of the motions was possible. The wavelength was found to be variable with the depth of the fluid layer in qualitative accordance with the theory of Nield. Supercritical motions are briefly discussed.

The evolution of disturbances after a laminar, slightly supercritical flow between parallel planes is disturbed is considered as an initial-value problem. An asymptotic solution of the disturbances for large time possesses the same characteristic features as the turbulent spots observed by Emmons (1951).

Most suspensions exhibit a rheological behaviour which cannot be represented by either Bingham's or Ostwald–De Waele's law. In studying such cases a very simple expression with only three parameters may be used. Starting with an intermediate law of this sort, this paper gives velocity profiles and head losses in laminar flow, which have been computed and plotted on diagrams in non-dimensional co-ordinates.

It has been found that transition flow rates in circular tubes for data taken from the literature and from experiments conducted on drilling muds at the Institut Français du Pétrole, are efficiently predicted by an empirical criterion (Ryan & Johnson 1959) which establishes a relation between a generalized Reynolds number and a generalized Hedström number.

These studies constitute an extension of the work of Doyle, Moffett & Vonnegut (1964).

Experiments showed that the evaporation of a charged droplet of water, aniline or toluene supported in a vertical electric field is accompanied by no discernible loss of charge and a consequent increase in electrical pressure in the drop surface owing to the decrease in radius. When the relationship between drop charge Q and radius R becomes consistent with the Rayleigh criterion the drop disintegrates at its uppermost point, where the electrical pressure is a maximum, to eject about 25% of its mass in the form of highly charged droplets. The measured relationship between the quantity of ejected charge, ΔQ, the mass loss, ΔM, and R was close to that derived theoretically from Rayleigh's equation, indicating that the value of ΔQ for a particular ΔM and R is approximately the theoretical minimum and that an individual drop may undergo a sequence of disintegrations, as was observed. The range of droplet radii studied by means of this technique was 30–200 μ.

The absence of disintegrations during the evaporation of much larger charged drops suspended from insulating fibres was attributed to corona discharge.

Townsend's (1961) hypothesis that the turbulent motion in the inner region of a boundary layer consists of (i) an ‘active’ part which produces the shear stress τ and whose statistical properties are universal functions of τ and y, and (ii) an ‘inactive’ and effectively irrotational part determined by the turbulence in the outer layer, is supported in the present paper by measurements of frequency spectra in a strongly retarded boundary layer, in which the ‘inactive’ motion is particularly intense. The only noticeable effect of the inactive motion is an increased dissipation of kinetic energy into heat in the viscous sublayer, supplied by turbulent energy diffusion from the outer layer towards the surface. The required diffusion is of the right order of magnitude to explain the non-universal values of the triple products measured near the surface, which can therefore be reconciled with universality of the ‘active’ motion.

Dimensional analysis shows that the contribution of the ‘active’ inner layer motion to the one-dimensional wave-number spectrum of the surface pressure fluctuations varies as τ2w/k1 up to a wave-number inversely proportional to the thickness of the viscous sublayer. This result is strongly supported by the recent measurements of Hodgson (1967), made with a much smaller ratio of microphone diameter to boundary-layer thickness than has been achieved previously. The disagreement of the result with most other measurements is attributed to inadequate transducer resolution in the other experiments.

A largely qualitative experimental investigation has been made of cellular convection patterns produced by the instability of a horizontal layer of fluid heated uniformly throughout its body and cooled from above. The fluid was aqueous zinc sulphate solution, and the heating was produced by an electrolytic current. The flow was visualized and photographed using polystyrene beads which, because of a differential expansion, came out of suspension in regions where the fluid was hotter or colder than average.

The development of the cellular patterns as the Rayleigh number (a modified form) was varied was in many respects similar to that for Bénard convection. There were, however, two striking differences. First, the fluid descended in the centres of the cells and ascended at the peripheries; this was probably associated with the asymmetry between the heating and cooling. Secondly, and more surprisingly, the horizontal scale of the convection patterns was unusually large (except at the lowest Rayleigh numbers); the distance between rising and falling currents was observed to reach typically five times the depth of the layer.

This last observation may be relevant to theories of convection within the Earth's mantle, to mesoscale convection in the atmosphere, and to patterns formed in ice.

The linear stability of a quiescent, three-dimensional rectangular box of fluid heated from below is considered. It is found that finite rolls (cells with two non-zero velocity components dependent on all three spatial variables) with axes parallel to the shorter side are predicted. When the depth is the shortest dimension, the cross-sections of these finite rolls are near-square, but otherwise (in wafer-shaped boxes) narrower cells appear. The value of the critical Rayleigh number and preferred wave-number (number of finite rolls) for a given size box is determined for boxes with horizontal dimensions h, ¼ ≤ h/d ≤ 6, where d is the depth.

The light-scatter technique facilitates study of the concentration field when one of the streams entering a mixing region is marked with colloidal particles. The present paper provides basic information for the effective application of the technique. The system response is analysed theoretically. The problems of noise and spatial resolution are examined and experimental data are presented from which the response parameters are evaluated numerically. The capabilities of the technique for characterizing turbulent concentration fluctuations are described. The technique is excellent in the convection region of the turbulence spectrum but fails where molecular diffusion is important.

A normal magnetic field has a destabilizing influence on a flat interface between a magnetizable and a non-magnetic fluid. Stabilizing influences are provided by interfacial tension and gravity if the lighter fluid is uppermost. The critical level of magnetization for onset of the instability is derived for a fluid having a non-linear relation between magnetization and magnetic induction. Experiments using a magnetizable fluid, which contains a colloidal suspension of ferromagnetic particles, at interfaces with air and water are made and cover a wide range of density differences. Measurements confirm the prediction for critical magnetization, and it was found that, after onset, the interface took a new form in which the elevation had a regular hexagonal pattern. The pattern was highly stable, and the measured spacing of peaks agreed reasonably with that derived from the critical wave-number for the instability of a flat interface.

The light-scatter technique has been used to study the nozzle-fluid concentration field in an isothermal, turbulent, axisymmetric air/air free jet with the nozzle air marked by an oil smoke. The data on the mean concentration field appear to be the most accurate yet obtained, due to the peculiar advantages of the technique. The turbulent concentration fluctuations have been characterized as to intensity, spectral distribution, and two-point correlation. The intermittency factor has been measured and the properties of the turbulent fluid computed. Comparison with the results of other investigators who used heat to mark the nozzle fluid indicates a close similarity between the concentration and temperature fields.

The acoustic streaming of an idealized elastico-viscous fluid near a cylindrical obstacle is considered using the boundary-layer type of approximation. It is shown that this streaming phenomena can be greatly influenced by the presence of elasticity in the fluid.

A theoretical study has been made of an experiment by Tritton & Zarraga (1967) in which eonvective motions were generated in a horizontal layer of water (cooled from above) by the application of uniform heating. The marginal stability problem for such a layer is solved, and a critical Rayleigh number of 2772 is obtained, at which patterns of wave-number 2·63 times the reciprocal depth of the layer are marginally stable.

The remainder of the paper is devoted to the finite amplitude convection which ensues when the Rayleigh number, R, exceeds 2772. The theory is approximate, the basic simplification being that, to an adequate approximation, Fourier decompositions of the convective motions in the horizontal (x, y) directions can be represented by their dominant (planform) terms alone. A discussion is given of this hypothesis, with illustrations drawn from the (better studied) Bénard situation of convection in a layer heated below, cooled from above, and containing no heat sources. The hypothesis is then used to obtain ‘mean-field equations’ for the convection. These admit solutions of at least three distinct forms: rolls, hexagons with upward flow at their centres, and hexagons with downward flow at their centres. Using the hypothesis again, the stability of these three solutions is examined. It is shown that, for all R, a (neutrally) stable form of convection exists in the form of rolls. The wave-number of this pattern increases gradually with R. This solution is, in all respects, independent of Prandtl number. It is found, numerically, that the hexagons with upward motions in their centres are unstable, but that the hexagons with downward motions at their centres are completely stable, provided R exceeds a critical value (which depends on Prandtl number, P, and which for water is about 3Rc), and provided the wave-number of the pattern lies within certain limits dependent on R and P.

The structure of normal shock waves in a gas composed of rigid sphere molecules is investigated by numerical experiments with a simulated gas on a digital computer. The non-equilibrium between the temperatures based on the longitudinal and lateral velocity components is studied and the results compared with the theory of Yen (1966). Details of the velocity distribution function are presented for a shock of Mach number 10. The distribution functions for both the longitudinal and lateral velocity components are plotted for a number of locations in the shock profile and are compared with the equilibrium distribution.