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.

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 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 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.

Upper bounds for the heat flux through a horizontally infinite layer of fluid heated from below are obtained by maximizing the heat flux subject to (a) two integral constraints, the ‘power integrals’, derived from the equations of motion, and (b) the continuity equations. This variational problem is solved completely, for all values of the Rayleigh number R, when only the constrains (a) are imposed, and it is thus shown that the Nusselt number N for any statistically steady convective motion cannot exceed a certain value N1(R), which for large R is approximately (3R/64)½. When (b) is included as a constraint, the variational problem is solved for large R, under the additional hypothesis that the solution has a single horizontal wave number; the associated upper bound on the Nusselt number is (R–248)⅜. The mean properties of this maximizing ‘flow’, in particular the mean temperature and mean square temperature deviation fields, are found to resemble the mean properties of the real flow observed by Townsend; the results thus tend to support Malkus's hypothesis that turbulent convection maximizes heat flux.

The velocity fields associated with a variety of flows which may be described by perturbations of the Blasius solution are considered. These are flows which, for example, because of localized mass transfer, involve the initial-value problem of boundary-layer theory, or which involve a variable ratio of the viscosity-density product, or finally which involve mass transfer. The perturbation solutions are presented so that in accord with the usual linearization procedures further applications for the determination of first-order effects can be readily made. In addition, each of these perturbations involves a common differential operator whose eigenfunctions form a complete orthogonal set. Accordingly, a procedure for systematically improving each perturbation solution to obtain higher-order effects by quadrature is presented. The results of applications in several cases are given and are compared to more accurate solutions where available.

The nature of the flow of a dissociating diatomic gas in a convergent-divergent supersonic nozzle is investigated by measuring the distribution of flow parameters along the nozzle. Two parameters are measured, namely the degree of dissociation and the static pressure. The degree of dissociation is measured by a spectroscopic method. The pressure is measured by [xcup ]-tube manometers. The results are compared with Bray's (1959) theory and Logan's (1957) prediction.

The statistical density function is derived for a variable (such as the surface elevation in a random sea) that is ‘weakly non-linear’. In the first approximation the distribution is Gaussian, as is well known. In higher approximations it is shown that the distribution is given by successive sums of a Gram–Charlier series; not quite in the form that has sometimes been used as an empirical fit for observed distributions, but in a modified form due to Edgeworth.

It is shown that the cumulants of the distribution are much simpler to calculate than the corresponding moments; and the approximate distributions are in fact derived by inversion of the cumulant-generating function.

The theory is applied to random surface waves on water. The third cumulant and hence the skewness of the distribution of surface elevation is evaluated explicity in terms of the directional energy spectrum. It is shown that the skewness λ3 is generally positive, and positive upper and lower bounds for λ3 are derived. The theoretical results are compared with some measurements made by Kinsman (1960).

It is found that for free, undamped surface waves the skewness of the distribution of surface slopes is of a higher order than the skewness of the surface elevation. Hence the observed skewness of the slopes may be a sensitive indicator of energy transfer and dissipation within the water.