From an analysis of small oscillations of a viscous fluid droplet immersed in another viscous fluid a general dispersion equation is derived by which frequency and rate of damping of oscillations can be calculated for arbitrary values of droplet size, physical properties of the fluids, and interfacial viscosity and elasticity coefficients. The equation is studied for two distinct extremes of interfacial characteristics: (i) a free interface between the two fluids in which only a constant, uniform interfacial tension acts; (ii) an ‘inextensible’ interface between the two fluids, that is, a highly condensed film or membrane which, to first order, cannot be locally expanded or contracted. Results obtained are compared with those previously published for various special cases.

When the viscosities of both fluids are low, the primary contribution to the rate of damping of oscillations is generally the viscous dissipation in a boundary layer near the interface, in both the free and inextensible interface situations. For this reason inviscid velocity profiles, which do not account for the boundarylayer flow, do not lead to good approximations to the damping rate. The two exceptions in which the approximation based on inviscid profiles is adequate occur when the interface is free and either the interior or exterior fluid is a gas of negligible density and viscosity.

Arguments are given which suggest that the mean flow of a turbulent fluid is irrotational far from the boundaries. Mathematical difficulties precluded a detailed theory so a simple experiment was performed. This experiment qualitatively confirmed the suggestion. Other well-known experiments on the mean flow of turbulent fluid may also be interpreted in terms of vorticity expulsion, an idea closely related to Scorer's hypothesis on turbulent mixing.

Theoretical papers on stellar rotation usually assume that convective regions rotate uniformly because of a high turbulent viscosity. However, if vorticity expulsion occurs, convective stellar cores will not rotate. The concept of a turbulent viscosity is criticized.

It has been suggested on theoretical grounds that a vortex could be initiated in a cylindrical region of fluid, originally in solid rotation, by the horizontal mixing of angular momentum produced by external stirring. In this paper various arguments for and against the mechanism are examined and their difficulties exposed. No firm conclusion is reached. A series of mathematical models of the mixing motions has been used, to bring out the differences between mechanical stirring and the agitation of a gas by random molecular motions. They suggest the introduction of a diffusion coefficient for angular momentum, to be determined empirically. These theoretical ideas are then applied to the interpretation of the results of a laboratory experiment which has been designed to test the proposed mechanism directly.

A wide, flat tank of liquid was set up on a rotating table and stirred with a vertically oscillated grid, whose elements were much smaller than the width of the tank. A neutrally buoyant particle was used as a tracer of fluid motions, to measure relative circulation velocities and the properties of the turbulence. The motion observed was dominated by the loss of angular momentum to the walls and the grid, an effect which has not been taken into account in previous theoretical assessments of the effects of mixing of angular momentum. The relative circulation present was not significantly different from zero, and the limits of error of the measurements imply that the rate of diffusion of angular momentum is less than 5% of that for fluid particles, with 95% probability.

A simple frictional wind-driven model of the Antarctic circumpolar current is examined. The geometry includes what are thought to be the main features effecting the current, namely a gap corresponding to Drake Passage and a partial barrier corresponding to the South American peninsula. Solutions obtained by both numerical and analytical methods are presented. The analytic solution, valid for small values of a friction parameter, enables the total rate of transport of water by the current to be calculated as a function of the friction parameter and the wind stress distribution. The width which controls the rate of transport by the current tends to be the narrowest encountered rather than the average width as assumed in earlier zonal models. However, the current spreads out to several times this width due to frictional effects. The values of the eddy viscosity required to give a rate of transport of the observed order are about 103 cm2/s for a bottom-friction model and of the order of 108 cm2/s for a lateral friction model.

Two-dimensional internal gravity waves in a rectangular container are examined theoretically and experimentally in (a) fluids which contain a single density discontinuity and (b) fluids in which the density gradient is everywhere continuous. The fractional density difference between the top and bottom of the fluid is small.

Good agreement is found between the observed and calculated wave profiles in case (a). Unlike surface standing waves, which tend to sharpen at their crests as the wave amplitude increases, and which eventually break at the crests when fluid accelerations become equal to that of gravity, internal wave crests are found to be flat and exhibit no instability. In the case (a) breaking is found to occur at the nodes of the interfacial wave, where the current shear, generated by the wave itself, is greatest. For sufficiently large wave amplitudes, a disturbance with the form of a vortex but with direction of rotation reversing twice every cycle, grows at the wave node and causes mixing. This instability is found to be followed by the generation of cross-waves, of which two different forms are observed.

Several modes of oscillation can be generated and are observed in a fluid with constant density gradient. The wave frequencies and shape are well predicted by theory. The experiments failed to establish any limitation of the possible wave amplitudes.

The structure of eddies embedded in a swirling stream of an incompressible, inviscid fluid is examined. Laminar motion is assumed, as is axial symmetry of the flow field.

It is supposed that the eddies are formed either (i) as wakes behind solid obstacles placed in the stream, or (ii) as the products of vortex breakdown occurrences, in which case the eddies are completely surrounded by fluid of the outer stream. In either event, it is assumed that the presence of an eddy only slightly disturbs the motion of the outer stream, i.e. that the eddies are slender. In particular, this requires that r0Ω∞/W∞ [Lt ]1, where r0 is the maximum radial extent of the eddy, and Ω∞ and W∞ are defined below.

The analysis is carried to completion when the undisturbed outer stream is a wheel flow. Principal results derived are summarized below. (1) Eddy lengths are related directly and simply to the internal vorticity, providing that there is swirl in the outer flow. (2) A corollary is that very slender eddies have a maximum possible length. In the case (ii) above, this length is 2πW∞/Ω∞, where W∞ is the axial velocity and Ω∞ the angular velocity of the undisturbed outer flow. (3) The shapes of the free streamlines bounding the eddy are calculated. A feature is the cusped nature of an eddy free end. (4) The motion inside the eddy and the disturbance to the outer stream are calculated.

Irrotational base flows are exceptional. Results (1) and (2) above cannot be found for a potential outer flow, but (3) and (4) remain.

The lee-wave amplitudes and wave drag for a thin barrier in a two-dimensional stratified flow in which the upstream dynamic pressure and density gradient are constant (Long's model) are determined as functions of barrier height and Froude number for a channel of finite height and for a half-space. Variational approximations to these quantities are obtained and validated by comparison with the earlier results of Drazin & Moore (1967) for the channel and with the results of an exact solution for the half-space, as obtained through separation of variables. An approximate solution of the integral equation for the channel also is obtained through a reduction to a singular integral equation of potential theory. The wave drag tends to increase with decreasing wind speed, but it seems likely that the flow is unstable in the region of high drag. The maximum attainable drag coefficient consistent with stable lee-wave formation appears to be roughly two and almost certainly less than three.

An investigation is made of the properties of inertial boundary currents in a stably stratified, inviscid, non-diffusive ocean. The Boussinesq and β-plane approximations are adopted. The formalism developed by Robinson (1965) is used: the equations are transformed so that density replaces the vertical coordinate as an independent variable, and after a suitable non-dimensionalization of variables, the various fields are expanded as power series in the downstream co-ordinate η. The motion is shown to conserve potential vorticity. The equations and boundary conditions are obtained to order η2. Solutions are obtained in the region of formation of the coastal jet (i.e. the case of no mass flux through the plane η = 0) for several cases in which the potential vorticity function depends on stream function and density in a simple way. For these cases, it is found that in a constant depth ocean, a boundary current can exist only if the geostrophic drift at the boundary-layer edge is westward at all depths. This constraint is relaxed if the depth increases rapidly enough in the downstream (northward) direction. For slopes just in excess of the critical value, a deep onshore counter-current is predicted. Solutions of the first-order problem, using realistic values of the various parameters, have been computed and are found to be in qualitative agreement with observed features of the Florida Current.

In an appendix, it is shown that the constraint of westward geostrophic drift at all levels must hold in a flat-bottomed ocean for arbitrary potential vorticity distributions consistent with stable stratification.

In the presence of a surface-active agent waves generated on the surface of a liquid by a wind may be stabilized. This problem is explored, and in particular the critical wind velocity is calculated for wave inception in the presence of such a film.

The dominant parameter associated with the surface layer is the surface elasticity χ the ‘inextensible film’ of Lamb is obtained as a limit as χ → ∞. Such a value is hypothetical as for real films χ is not large, being in the range 0 to 80 dyne/cm approximately. Nevertheless, we show that damping exceeding that of the inextensible film can be obtained for short wavelengths, and that the critical wind speed for capillary ripples can be increased by a factor of ten in the presence of a film of experimentally attainable characteristics.

The reason for the effectiveness of films of low χ is that the damping in the liquid is related to χ and the wavelength λ in such a way that for small λ, damping is a maximum for small χ.

Using Stuart's shape assumption and a condition of maximum heat transport it is found that the preferred number of finite roll cells present in Bénard convection in a three-dimensional rectangular box tends to decrease with increasing supercritical Rayleigh number in contradiction to the behaviour in an infinite layer but in accordance with experimental observation.

This ‘end effect’ might explain the similar observation of wave-number decrease in the Taylor instability between rotating cylinders.