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.

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.

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.

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 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.

Experiments were conducted to test the linear theory of internal gravity waves produced in a stably stratified liquid by the steady vertical motion of a sphere. The measurements of the phase configuration in a medium whose density increased linearly with depth were made by means of a Toepler-schlieren system. The agreement between observation and prediction was found to be good.

This paper deals with a detailed application of the Whitham theory of finite amplitude wave propagation to open-channel steady flow of water of infinite depth past a slowly modulated wavy wall. A numerical procedure first used by Garabedian & Lieberstein (1958) for the solution of ‘stably-posed’ elliptic Cauchy problems is employed to obtain a map of the wave pattern on the free surface of the water. The appearance of a ‘shock’ in this solution is discussed in terms of previous analytical and experimental results.

The velocities and the associated pressure gradients of infinitely long liquid-borne cylinders flowing freely in pipes are related analytically to their radial positions. These velocities and pressure gradients are compared with those of liquids in cylinder-free pipes and expressed as ratios. A digital computer was used to evaluate the resultant equations for values of the cylinder/pipe diameter ratio between 0·25 and 0·97, with radial positions varying from the fully eccentric to the fully concentric position. As the clearance between the pipe and the bottom of the cylinder increases, the pressure ratio (RP) decreases and the velocity ratio (RV) increases. The relationship between RP and RV is independent of liquid viscosity and density, capsule density and pipe diameter, and is shown to be nearly linear for the larger diameter ratios. The R, R relationships are compared with data from three experimental capsule pipelines with pipe diameters from ½ to 4 in., involving a variety of diameter ratios, cylinder lengths and densities, and oil viscosities. The experimental results for single capsules of finite length are shown to be in close agreement with the predictions for infinitely long cylinders.

The relevance of the analysis to capsule pipelining is indicated by relating experimental values of capsule velocity over a wide range of densities to the theoretical clearance of the capsules. A general relationship of this type would permit the optimization of the power requirements of any particular throughput of a given commodity.

The initial value problem related to axisymmetric forced oscillations of a rigidly rotating inviscid fluid enclosed in a finite circular cylinder is examined in linear approximation with the aid of the Laplace transform technique. An impulsive starting motion is considered. The solution consists of a ‘periodic’ motion which oscillates with the forcing frequency, together with a doubly infinite set of inertial modes whose presence is determined by the initial conditions and whose frequencies form a dense set in the range (−2ω, 2ω), where ω is the angular velocity. The nature of the periodic or ‘steady-state’ part of the solution is strongly dependent on the precise value of the forcing frequency α (α > 0) when α ≤ 2ω. In particular the system will resonate if α equals any one value of the dense set of resonant frequencies. It is shown that no internal sets of discontinuities in velocity or velocity gradient are present in the inviscid flow for finite times. Effects of viscosity on the inviscid solution are also discussed, and it is argued that when the inertial modes decay the steady-state flow will contain pseudo-random patterns of internal shear layers for some values of α < 3ω. It seems possible that these shear layers may be interpreted as owing their existence indirectly to viscosity.

The equations of mean motion indicate that two-dimensional turbulent wakes, when subjected to appropriately tailored adverse pressure gradients, can be self-preserving. An experimental examination of two nearly self-preserving wakes is reported here. Mean velocity, longitudinal and lateral turbulence intensity, inter-mittency and shear stress distributions have been measured and are compared with Townsend's data from the small-deficit undistorted wake. In comparison with the undistorted case, the present wakes have slightly lower turbulent intensities and significantly lower shear stresses, all quantities being non-dimensionalized by a local velocity scale taken as the maximum mean velocity deficit. A consideration of the reasons for the shear stress reduction leads to an expression from which the shear stresses in any symmetrical free equilibrium shear flow can be found. This relationship is used to calculate the rate of growth in the measured wakes, with reasonable success.

The boundary layer on a horizontal cylinder caused by free convection, when the temperature of the cylinder is oscillating harmonically, is considered in a way which is analogous to the problem of a cylinder performing high-frequency oscillations along its diameter. It is found that, outside the thin boundary layer on the cylinder, a steady flow is induced. An outer steady boundary layer, matched with the inner boundary layer, is required in order to be able to satisfy the conditions which apply at large distances from the cylinder.

In this paper we consider two-dimensional steady cellular motion in a fluid heated from below at large Rayleigh number and Prandtl number of order unity. This is a boundary-layer problem and has been considered by Weinbaum (1964) for the case of rigid boundaries and circular cross-section. Here we consider cells of rectangular cross-section with three sets of velocity boundary conditions: all boundaries free, rigid horizontal boundaries and free vertical boundaries (referred to here as periodic rigid boundary conditions), and all boundaries rigid; the vertical boundaries of the cells are insulated. It is shown that the geometry of the cell cross-section is important, such steady motion being not possible in the case of free boundaries and circular cross-section; also that the dependence of the variables of the problem on the Rayleigh number is determined by the balances in the vertical boundary layers.

We assume only those boundary layers necessary to satisfy the boundary conditions and obtain a Nusselt number dependence $N \sim R^{\frac{1}{3}}$ for free vertical boundaries. For the periodic rigid case, Pillow (1952) has assumed that the buoyancy torque is balanced by the shear stress on the horizontal boundaries; this is equivalent to assuming velocity boundary layers beside the vertical boundaries (rather than the vorticity boundary layers demanded by the boundary conditions) and leads to a Nusselt number dependence N ∼ R¼. If it is assumed that the flow will adjust itself to give the maximum heat flux possible the two models are found to be appropriate for different ranges of the Rayleigh number and there is good agreement with experiment.

An error in the application of Rayleigh's method in this paper is noted and the correct method for carrying the boundary-layer solutions round the corners is given. Estimates of the Nusselt numbers for the various boundary conditions are obtained, and these are compared with the computed results of Fromm (1965). The relevance of the present work to the theory of turbulent convection is discussed and it is suggested that neglect of the momentum convection term, as in the mean field equations, leads to a decrease in the heat flux at very high Rayleigh numbers. A physical argument is given to derive Gill's model for convection in a vertical slot from the Batchelor model, which is appropriate in the present work.

An apparatus is described whereby the distribution of particles in a neutrally buoyant suspension of spheres flowing through a tube may be measured. The discovery of a narrow region of high concentration near to the tube wall, and the behaviour of this ‘wall peak’ at various concentrations and positions along the length of the tube are described. An explanation for this behaviour is put forward in terms of the formation of the peak by interaction of spheres with the tube entrance, and of the formation within this peak of necklace-like strings of particles.