A solution is presented for the flow field in and around a single spherical droplet or bubble moving axially at an arbitrary radial location, within a long circular tube. In the tube there is viscous fluid flowing with a constant Poiseuillian velocity distribution far from the droplet.

The settling velocity of the droplet or bubble is \begin{eqnarray*} U = \frac{2(\rho_i-\rho_e)ga^2}{9\mu_e}\frac{1+\alpha}{\frac{2}{3}+\alpha}\left[1-\frac{2+3\alpha}{3(1+\alpha)}\left(\frac{a}{R_0}\right)f\left(\frac{b}{R_0}\right)\right]+U_0\left[1-\left(\frac{b}{R_0}\right)^2\right.\\ \left. - \frac{2\alpha}{2+3\alpha}\left(\frac{a}{R_0}\right)^2\right] + O\left(\frac{a}{R_0}\right)^3. \end{eqnarray*} This is a general equation and it contains as special cases the familiar solutions of Stokes, Hadamard-Rybczynski, Brenner & Happel, Greenstein & Happel and Haberman & Sayre.

The function describing the deviation of the interface from sphericity is solved and an iterative procedure is suggested for obtaining higher order solutions.

The phenomenon of salt fingers has been investigated optically to determine the geometry of the cells as seen from above. When the fingers are short, the flow appears to be highly turbulent, though a dominant scale is evident. When the fingers are longer, a cellular structure is clear. This structure changes only slowly, apparently in response to disturbances in the convecting layers which bound the fingers above and below, and becomes more nearly stationary as the fingers grow. Cell boundaries show a strong tendency to intersect at right angles, which favours the emergence of cells with a square horizontal section. As the fingers get longer the cell width increases, but more slowly than the length.

Results for three interrelated problems are obtained by making use of solutions of boundary-value problems obtained in a different context. The first one concerns a thin rigid circular disk rotating in a slow stream of viscous fluid, both when the fluid is conducting and when it is non-conducting. For the case of a conducting fluid formulae are given for both small and large Hartmann numbers. The second problem concerns a disk performing simple harmonic rotary oscillations about its axis of symmetry in a non-conducting viscous fluid which is at rest at infinity. The last problem is that of an arbitrary axisymmetric solid oscillating about its axis of symmetry in a bounded viscous fluid, and the solution is illustrated by the case of an oscillating disk.

In rotating flow moving axially through a straight tube, a helical vortex will be generated if the angular momentum flux is sufficiently large relative to the flux of linear momentum. This paper describes an experimental study of the occurrence, frequency and peak-to-peak amplitude of the wall pressure generated by this vortex. The experimental results are displayed in dimensionless form in terms of a Reynolds number, a momentum parameter and tube geometry.

A three-dimensional boundary layer developing along a semi-infinite swept stagnation line from a starting edge and evolving into that associated with such a line of infinite extent is calculated. A series solution useful for assessing the counteracting effects of cross-flow and mass transfer near the starting edge and for providing initial data for a subsequent streamwise, numerical solution is developed. The asymptotic behaviour far from the starting edge is examined and shown to involve only eigenfunction contributions associated with the far upstream flow. However, it is not presently possible to determine the relevant eigenvalues and eigenfunctions. Numerical solutions based on a difference-differential analysis yield the entire development of the boundary layer and indicate the streamwise length required for the case of the boundary layer at an infinite stagnation line to be obtained.

Conditions for two gas bubbles in a liquid to rise steadily in a vertical line are derived theoretically with these assumptions: large Reynolds number, no surface contamination, spherical shape, negligible gas density and viscosity. Drag coefficients are found, and are lower than for single bubbles. The bubbles have equilibrium distances apart, which are calculated to a first approximation. The equilibrium is shown to be stable to small vertical disturbances but unstable to horizontal ones. Similar results exist for lines of more than two bubbles, but are not calculated in detail.

The flow of an inviscid incompressible fluid, with imbedded identical spherical particles, around an arbitrary corner is treated by the method of small perturbations. Application is made to convex flows only and the approximate effects of separation are also considered. The assumption of arbitrary initial particle density k0 leads to a complex system of equations, which appears to have no simple solution in general, and is not considered. For small k0 the particle density distribution is governed by a first-order partial differential equation which, when solved by the method of characteristics, yields ordinary differential equations, whose solutions are simple and analytic for unseparated flow and numerical only when separation is considered. In the former flow, spiral type curves partially encircling the plate tip, and then trailing downstream, delineate the particle-free zones, and it is found that the particle density increases monotonically in the downstream direction on all particle streamlines. In the separated flow, the most noteworthy effect is the disappearance of the infinite velocity at the origin and the consequent considerable reduction in the magnitude of the perturbation.

The method of matched asymptotic expansions is applied to the problem of a wing of finite span in very close proximity to the ground. The general lifting surface problem is shown to be a direct problem, represented by a source-sink distribution on the upper surface of the wing and wake, with concentrated sources around the leading and side edges plus a separate confined channel flow region under the wing and wake. The two-dimensional flat plate airfoil is examined in detail and results for upper and lower surface pressure distribution and lift coefficient are compared with a numerical solution. A simple analytic solution is obtained for a flat wing with a straight trailing edge which has minimum induced drag. To lowest order, this optimally loaded wing has an elliptical planform and a lift distribution which is linear along the chord, resulting in a parabolic spanwise lift distribution. An expression for the lift coefficient at small clearance and angle of attack, valid for moderate aspect ratio, is derived. The analytic results show reasonable agreement when compared with numerical results from lifting surface theory.

Experiments have been performed to explore the qualitative and quantitative characteristics of thermals which ascend through the fluid environment above a heated horizontal surface. With water as the participating fluid, an electrochemical technique was employed which made the flow field visible and facilitated the direct observation of thermals. Measurements were also made of the fluid temperature above an active site of thermal generation.

As seen in flow field photographs, a thermal has a mushroom-like appearance, with a blunted nearly hemispherical cap. At a given heating rate, thermals are generated at fixed sites which are spaced more or less regularly along the span of the heated surface. At these sites, the generation of thermals is periodic in time, thereby validating a prediction of Howard. Both the spatial frequency of the sites and the rate of thermal production increase with increases in heating rate. The break-up Rayleigh number of the conduction layer is shown to be a constant (within the uncertainties of the experiment), which is in accord with Howard's phenomenological model.

Interactions of suitable pairs of gravity waves in a shear flow are found to give rise to aperiodic or weakly periodic secondary motions. These secondary flows resemble the ‘Langmuir vortices’ which are associated with the formation of windrows. It seems likely that such wave interactions will play a substantial part in determining the quasi-steady structure of the flow when wind blows over a water surface.

A simplified example of this type of flow was examined in detail by developing two series, eventually matched, one about the nodal point and the other about the saddle point, and also by finite differences, marching from the nodal point to the saddle point. It was found that the results of marching the two series were in agreement with the finite difference method. The series solution near the saddle point is not unique, but numerical evidence indicates that the correct solution is that which has ‘exponential decay’ at infinity, and that this type of solution, if such exists, automatically emerges when the finite difference method is used.

Cross-waves are standing waves with crests at right angles to a wave-maker. They generally have half the frequency of the wave-maker and reach a steady state at some finite amplitude. A second-order theory of the modes of oscillation of water in a tank with a free surface and wave-makers at each end leads to a form of Mathieu's equation for the amplitude of the cross-waves, which are thus an example of parametric resonance and may be excited at half the wave-maker frequency if this is within a narrow band. The excitation depends on the amplitude of the wave-maker at the surface and the integral over depth of its amplitude. Cross-waves may be excited even if the mean free surface is stationary. The effects of finite amplitude are that the cross-waves approach a steady state such that a given amplitude is achieved at a frequency greater than that for free waves by an amount proportional to the amplitude of the wave-maker. The theory agrees reasonably well with the experimental results of Lin & Howard (1960). The amplification of the cross-waves may be understood in terms of the rate of working of the wave-maker against transverse stresses associated with the cross-waves, one located at the surface and the other equal to Miche's (1944) depth-independent second-order pressure. The theory applies to the situation where the primary motion consists of standing waves and the cross-waves are constant in amplitude away from the wave-maker, but certain generalizations may be made to the situation where the primary waves are progressive and the cross-waves decay away from the wave-maker.

An analysis is made of the free outflow of fluid from those underflow gates known as radial or Tainter gates. Attention is focused on a correct treatment of the effects of gate curvature and of gravity on the flow. The rapidly convergent, iterative solution is based on the combined use of conformal mapping and the Riemann-Hilbert solution to a mixed boundary-value problem. A limited comparison with some experimental results shows agreement to be good.

A fluid contained in a rotating cylinder has an inertial mode which is excited by forced precession of the container. Wood's (1965, 1966) early work specifically excluded resonance phenomena. Recently McEwan (1970) has discussed resonance phenomena for strong amplitude of excitation, corresponding to rapid precession in this work.

In this paper the magnitude of the resonant response for small precession rate is precisely calculated by matching the Ekman layer suction to the precessional forces. The procedure is to find the resonant mode v1, compute its boundary layers, $\tilde{{\bf v}}_1$, and the associated Ekman layer suction. The second-order problem has a solvability condition which is satisfied by matching the Ekman layer suction to the precession.

A WKB-perturbation technique is applied to study the slow modulation of a Stokes wave train on the surface of water. It is found that new terms directly representing modulation rates must be included to extend the scope of validity of Whitham's theory based on an averaged Lagrangian. Two examples are discussed. In the first, a monochromatic wave normally incident on a mild beach is studied and the local rate of depth variation is found to affect the wave phase. In the second, the ‘side-band instability’ problem of Benjamin & Feir is discussed from both linear and non-linear points of view.