The flow set up in an oblate cavity of a precessing rigid body is examined under the assumptions that the ellipticity of the spheroidal boundary of the fluid is large compared with Ω/ω and that the boundary-layer thickness is small compared with the deviations of the boundary from sphericity (ω is the angular velocity of the rigid body about the axis of symmetry, Ω is the angular velocity with which this axis precesses).

The motion of the fluid is found by considering an initial-value problem in which the axis of rotation of the spheroid is impulsively moved at a time t = 0; before that time this axis is supposed to be fixed in space, the fluid and envelope turning about it as a solid body. The solution is divided into a steady motion and transients, and, by evaluating the effects of the viscous boundary layer, the transients are shown to decay with time. The steady motion which remains consists of a primary rigid-body rotation with the envelope, superimposed on which is a circulation with constant vorticity in planes perpendicular to ω × (ω × Ω), the streamlines being similarly situated ellipses.

The possible effects of the luni-solar precession on the fluid motions in the Earth's core are discussed.

The analysis given previously for predicting the average pressure and length of the region of recirculating flow enclosed by a low-speed turbulent jet, issuing parallel to a flat plate, has been modified to take into account the different rates of entrainment by the two edges of the curved jet, the initial mixing region and the pressure forces near reattachment. There is improved correlation between theory and experiment. The analysis has been applied to the flow due to a jet emerging at an angle to a flat plate, and gives good prediction of the length and average pressure of the recirculation region for a particular value of an entrainment-ratio parameter.

Curvature has a considerable effect on the rates of entrainment, but a first-order mixing-length theory indicates that this need not necessarily be accompanied by a marked deviation in jet velocity profile from that of a plane jet.

This paper deals with the initial motion of a gas bubble starting from rest in a liquid in the form of a sphere. Part 1 (Walters & Davidson 1962) was concerned with the similar problem of the initial motion of a two-dimensional bubble starting from rest in the form of a cylinder.

Theory and experiments like those of Part 1 are given for the present problem, and yield qualitatively similar results, the three-dimensional bubble having an initial acceleration equal to twice that of gravity, and distorting into the form of a mushroom. This distortion ultimately causes break-up, but whereas the two-dimensional bubble always detaches two small bubbles at its rear, the three-dimensional bubble breaks up into a small spherical-cap bubble with a large toroid below. A discussion of the toroidal bubble is given, and its relation to the distorted sphere from which it is formed.

The initial-motion theory is extended to deal with the problem of the growing, accelerating bubble, and leads to an expression for the volume of bubbles formed continuously at an orifice, and to a criterion for the gas flow-rate at which coalescence occurs between successive bubbles. These theoretical results are compared with experimental data from the literature and from the authors’ experiments at high gas flow-rates.

A calculation taking into account the interaction of tube wall and ends with a falling sphere is presented, with a view to assessing the change in drag on the sphere due to the end proximity in practical viscometry. It is shown that when the sphere is more than one fall tube radius from the closed end, the extra drag resulting is less than 4·5 × 10−3 of the Faxèn drag, and is therefore usually negligible.

For smaller end-sphere distances the end-effect drag increases rapidly, reaching about 1·5 of the Faxèn drag when the sphere centre is 0·25 tube radius from the end. The drag curve lies substantially below that given by Ladenburg (1907). Satisfactory agreement with experiment is found. It is concluded that with the commonly used types of fall tubes, end effects due to a closed end and an open surface will not be detectable.

The detailed structure of the relaxation region in shock waves in oxygen was investigated using Blackman's experimental results. Oxygen was found to display a behaviour similar in many ways to that found previously for carbon dioxide with the relaxation frequency, as defined by the simple relaxation equation, depending on the departure from equilibrium as well as on temperature. The previous results for carbon dioxide were further analysed by means of a separate relaxation equation for each mode.

This paper is concerned with the stability of a parallel shear flow in an inviscid homogeneous unbounded rotating fluid. A sufficient condition for stability is obtained in terms of the dimensionless parameter N = (cosϕ)/S, where ϕ is the angle between the wave-number K of the disturbance and the axis of rotation, and S is the Rossby number based on the thickness of the shear layer and the change in velocity across the layer. The condition is that infinitesimal disturbances are stable if either $N \ge \frac{1}{2}(1-sin\; \theta)\; \; or\; \; N \le -\frac {1}{2}(1+sin\; \theta)$ Where θ is the angle between k and the direction of streaming. For a shear layer profile of the type U = tanh z, the neutral curves are calculated for various Rossby numbers. These are compared to the stability of a shear layer in a stratified non-rotating fluid. The stability criterion for the large wave-numbers in a cylindrical shear layer is inferred from these results.

The flow of an incompressible second-order fluid due to torsional oscillations of an infinite plate when the fluid is infinite in extent as well as the case when it is bounded by another stationary parallel plate has been considered by expanding the velocity components and the pressure in powers of the amplitude of oscillation of the plate. In both cases the first-order solution consists of a transverse velocity and the second-order solution gives a radial-axial flow composed of a steady part and a fluctuating part. In the case of the unbounded plate the steady part of the radial flow does not vanish outside the boundary-layer region. Hence the equations are solved by another approximate method for the steady part of the flow. The effects of the non-Newtonian terms are to increase the non-dimensional boundary thickness and the shearing stress on the plate. In the case of two plates the velocity components and the shearing stresses on the plates have been expressed in powers of Reynolds number R for its small values. Their asymtotic behaviour for large R has also been studied. The asymtotic expansion of the fluctuating part of the radial-axial flow shows that the boundary layer is developed at both the plates.

An account is given of photographic and pressure observations made on the oblique shock waves occurring in the wake of self-sustaining detonation waves in hydrogen-oxygen mixtures initially at atmospheric pressure. Four explosion tubes were employed, of which three are of circular cross-section with internal diameters of 10, 5 and 1·6 cm and the fourth is a square-section tube of side 1·5 in.

On the assumption that the oblique shocks are sufficiently weak to be regarded as Mach waves, the flow Mach number relative to the detonation front is determined; these are found to be substantially higher than the values predicted by deal one-dimensional theory. The measured flow Mach numbers in the rarefaction are then used to calculate the pressure distribution in this region on the basis of the supersonic nozzle model due to Fay (1959, 1962). The predictions of this model are found to disagree with with the observed static pressure profiles. Moreover, the pressure following the initial peak persists at a higher value than the theoretical for distances of the order of 5–10 cm behind the front. This phenomenon implies that the wall boundary-layer pressure remains higher than the C-J value and it is suggested that the pressure difference across the boundary layer can account for the formation of the oblique waves.

The supersonic features of the flow can be accounted for by the turbulent-structure hypothesis of White (1961). Some validation of this hypothesis is provided here by the observation of the absence of the oblique shocks in overdriven detonation waves caused by the diminished effects of turbulence. This observation is consistent with the view that the oblique shocks are generated by the pressure difference across the boundary layer near the front as this difference would also be diminished in an over-driven wave.

The simplest case of turbulent motion in a conductive fluid is studied. The turbulence is assumed incompressible, isotropic, homogeneous, charge invariant and free of fourth-order cumulants. The emphasis is placed on certain integrals of the correlation functions such as kinetic and magnetic energy, voticity, and current. A system of non-linear ordinary differential equations is derived which governs these integral quantities Several cases are solved numerically, illustrating the decay of ordinary turbulence, the buildup of magnetic energy by a linear or a non-linear process, the buildup of kinetic energy, as well as the destruction of vorticity by Lorentz forces.

In order to handle certain dissipative effects, a special hypothesis is introduced which seems to promote mathematical simplicity. In particular, it leads to a simple decay law very similar to the decay law of ordinary turbulence.

The electrical conductivity of shock-ionized argon produced in an electromagnetic shock tube of low attenuation has been measured at shock speeds of March 10–33, with initial pressures of 0·01–2·0 mm Hg. These measurements extend considerably the range of previous measurements performed with pressure-driven shock tubes. With the higher initial pressures or at the highest Mach numbers the measured conductivity is in good agreement with the previous measurements and with the Spitzer–Harm (1953) formula for the conductivity of a fully ionized gas. With the lower initial pressures (which have not previously been investigated) and at the lower March numbers the conductivity falls to less than half of the Spitzer-Harm value. Order-of-magnitude calculations show that diffusion of atoms, and heat conduction by the plasma atoms from the plasma to the shock-tube walls, can cause appreciable plasma cooling (and hence a reduction of the electrical conductivity) with the lowest initial pressures. This mechanism in conjunction with non-attainment of equilibrium ionization appears to explain the observed diminution in conductivity at the lowest pressures, but not the reduced conductivity at the medium pressures.

Induced e.m.f. flow-velocity measurements indicate steady-flow conditions in the shock tube while photomultiplier measurements of the plasma radiation indicate that the column of shock-heated gas is 10–20 cm long; this latter figure is supported by the conductivity measurements. The fact that the length of the shock-heated gas column is not drastically shortened at low initial pressures in constrast to the work of Duff (1959), Roshko (1960) and Hooker (1961) is attributed to the fact that in this experiment both driver and driven gases are at high temperature.

The stability of thermally stratified plane Couette flow of visco-elastic liquids, with respect to disturbances of small amplitude, is considered. It is found that an initial state of finite elastic stress is necessary for elasticity to effect stability. The critical Rayleigh number of the flow is shown to be decreased for all non-zero rates-of-strain. This greater instability is due solely to the variations of the apparent viscosity with shear rate. In this way, the presence of elasticity can be said to have a destabilizing effect on the flow.

The calculation by shock-expansion theory of supersonic aerofoil flow fields with varying entropy is simplified by assuming the flow behind the curved attached shock to be a small perturbation of a simple wave. A characteristic perturbation method is used.

The stability of Couette flow between concentric, co-rotating cylinders in an axial magnetic field is examined for fluids of arbitrary magnetic Prandtl number Pm = ν/η, where ν is the kinematic and η the magnetic viscosity of the fluid. It is assumed that the gap spacing d between the cylinders is small compared to the mean radius and that no magnetic disturbances penetrate into the cylinder walls. The critical Taylor number at which non-oscillatory disturbances are marginally stable is determined as a function of the magnetic Prandtl number and the dimensionless parameter S = (Vad/v)2, where Va is the Alfvén velocity. Asymptotic formulas relating the critical Taylor number to the magnitude of the magnetic field are derived for the limiting conditions of very small and very large magnetic Prandtl number.

Smooth- and rough-wall boundary layers and fully developed pipe and duct flow investigations are reviewed. It is shown that the effect of roughness on the flow away from the wall can be accounted for by using an equivalent viscosity νe. This viscosity is thought to depend only on the variables at the wall, such as shear stress τ0, fluid density ρ, viscosity μ and the roughness size and geometry and that the relationship between these variables is the same for both boundary layers and duct flow. However, experiments to date have been confined to the ‘rough régime’ and to boundary layers with a zero pressure gradient.

Experiments were performed and the results show that the above finding can be extended to boundary layers with adverse pressure gradients in the rough régime.

A general method for measuring the local boundary-layer characteristics, with roughness and pressure gradients present, is developed.

In this paper methods are given for establishing qualitative and quantitative measures of the effects of rotation in supercavitating flows past slender bodies. A linearized theory is developed for steady, two-dimensional flow under the assumption that the flow has a constant rotation throughout. The stream function of the rotational flow satisfies Poisson's equation. By using a particular solution of this equation, the rotational problem is reduced to a problem involving Laplace's equation and harmonic perturbation velocities. The resulting boundary-value problem is solved by use of conformal mapping and singularities from thinairfoil theory. The theory is then applied to asymmetric shear flow past wedges and hydrofoils and to symmetric shear flow past wedges. The presence of rotation is shown to create significant changes in the forces acting on the slender bodies and in the shape and size of the trailing cavities.

The diffusion of small floating objects and of fluorescent dyehas been studied at Douglas Point, Lake Huron. The dispersal of floating objects was complicated by surface confluences, slicks and windrows, which under certain circumstances could completely reverse the diffusion process. In the absence of such disturbing effects, however, the dispersal of floating objects exhibited an increase in rate of growth with the size of the diffusing cloud, characteristic of relative turbulent diffusion. A similar conclusion holds with regard to the diffusion of dye perpendicular to the mean current. Along the direction of the current the diffusion of dye was accelerated by the changes of current velocity with depth, much as in a pipe or channel. In the vertical, diffusion was very slow, the effective diffusivity being barely 10 times the molecular constant. Even in the horizontal the effective diffusivities describing relative diffusion were much less than values typical of the lower atmosphere or of the ocean (only of order 103 times the molecular diffusivity). A study of the meandering of the dye plume showed that this was a more important agency than relative diffusion in dispersing the dye over a large area.

We consider here the manner in which the state of rigid rotation of a contained viscous fluid is established. It is found that the motion consists of three distinct phases, namely, the development of the Ekman layer, the inviscid fluid spin-up, and the viscous decay of residual oscillations. The Ekman layer plays the significant role in the transient process by inducing a small circulatory secondary flow. Low angular momentum fluid entering the layer from the geostrophic interior is replaced by fluid with greater angular momentum convected inward to conserve mass. The rotational velocity in the interior increases as a consequence of conservation of angular momentum, and the vorticity is increased through the stretching of vortex lines. The spin-up time is $T = (L^2|v \Omega)^{\frac {1}{2}}.$ Boundary-layer theory is used to study the phenomenon in the case of general axially symmetric container configuration and explicit formulas are deduced which exhibit the effect of geometry in spin-up. The special case of cylindrical side walls is also investigated by this method. The results of very simple experiments confirm the theoretical predictions.

The results of an experimental investigation are presented to show that the stability of the steady laminar wake behind a circular cylinder is strongly influenced by the proximity of the walls of the confining experimental equipment.

A general theorem for drag is given according to which the boundary-layer drag of a body is equal to the inviscid drag of the displacement surface together with a term which is given as an integral involving the ‘streamwise’ momentum and displacement thicknesses taken round the trailing edge. A less accurate result for thin slender wings is that the boundary-layer drag is equal to the line integral $\rho _\infty U^2_\infty \int \Theta d \sigma$ taken round the trailing edge, where Θ is the streamwise momentum thickness. This result leads to the possibility of finding boundary-layer drag by means of a traverse round the trailing edge. The extension of the results to wings with swept trailing edges is also given.

The process of mixing at the top of a turbulent cloud layer contains a new feature which has not been considered in previous studies of mixing; evaporation of liquid water can cause density changes which may affect the dynamics. A model experiment has been devised to study this problem, using liquids whose density behaviour is non-linear to simulate evaporation.

The existence of a moist, stable, turbulent region above cloudtop can be explained using the laboratory results, which suggest that this region can be regarded dynamically as part of the cloud. Comparison of the rates of mixing in the model experiments with and without ‘evaporation’ suggests that evaporation could cause a small decrease in the mixing rate for a given density difference, but the change would be negligible in practice. This result also sheds some light on the mechanism of mixing, in both the linear and non-linear cases.