A non-linear analysis is presented for the stability of a liquid film adjacent to a compressible gas and under the influence of a body force directed either outward from or toward the liquid. The effects of the liquid's surface tension are taken into account. The non-linear Rayleigh–Taylor instability is included as a special case. The analysis considers the case of an inviscid liquid adjacent to a subsonic flow and the case of a very viscous liquid adjacent to a subsonic or a supersonic flow. For a subsonic external flow, it is found that the cut-off wave-number is amplitude dependent in the inviscid case whereas it is amplitude independent in the viscous case. It is found that the non-linear motion of the gas may be stabilizing or destabilizing, whereas the non-linear motion of the liquid is found to be stabilizing in the viscous case. For a supersonic external flow and a viscous liquid, the cut-off wave-number is amplitude dependent. Moreover, unstable disturbances with wave-numbers near the cut-off wave-number do not grow indefinitely with time but achieve a steady-state amplitude.

The problem of heat transfer at high Péclet number Pe from a sphere freely rotating in a simple shear field is considered theoretically for the case of small shear Reynolds numbers. It is shown that the present problem is in many respects similar to that of heat transfer past a freely rotating cylinder which was recently solved by Frankel & Acrivos (1968). By taking advantage of the close analogy between these two problems, an approximate method of solution is developed according to which the asymptotic Nusselt number for Pe → ∞ is 9, i.e. 4½ times its value for pure conduction. As in the corresponding case of the cylinder, the fact that the asymptotic Nusselt number is independent of Pe results from the presence of a region of closed streamlines which completely surrounds the rotating sphere.

The surface-wave response of a harbour to a prescribed, incident wave is calculated on the hypotheses of shallow-water theory, an ideal fluid, and a narrow mouth, M. An equivalent electrical circuit is constructed, in which the incidentwave displacement in M appears as the input voltage and the flow through M appears as the input current. This circuit contains a radiation impedance, ZM, which comprises resistive and inductive terms, and a harbour impedance, ZH, which comprises an infinite sequence of parallel combinations of inductance and capacitance that bear a one-to-one correspondence with the natural modes of the closed harbour, together with a single capacitor, which corresponds to the degenerate mode of uniform displacement and dominates the response of the harbour as a Helmholtz resonator. Variational approximations to ZH and ZM are developed. The equivalent circuit exhibits parallel resonance at the resonant frequencies of the closed harbour, ωn, and series resonance at a second set of frequencies, $\tilde{\omega}_n $, where $\tilde{\omega}_n\downarrow \omega_n > 0 $ and $\tilde{\omega}_0\downarrow 0 $ as M → 0; $\tilde{\omega}_0 $ corresponds to the Helmholtz mode. A narrow canal between the coastline and the harbour is represented by a four-terminal network between ZM and ZH. It is shown that narrowing the harbour mouth and/or increasing the length of the canal does not affect the mean response of the harbour to a broad-band, random input except in the Helmholtz mode, but that it does increase significantly the response in that mode, which may dominate tsunami response. The general results are applied to circular and rectangular harbours. The numerical calculation of ZH for an arbitrarily shaped harbour is discussed.

A recent theory for the instability of liquid jets predicted that because of nonlinear interactions, a swelling should occur between the crests of the primary disturbance waves. Experiments are described in which such swellings have been observed. The shape of the swellings is in qualitative agreement with the predictions.

When internal and/or inertial waves reflect from a smooth surface which is not plane, there is in general some energy flux which is ‘back-reflected’ in the opposite direction to that of the incident energy flux (in addition to that ‘transmitted’ in the direction of the reflected rays), provided only that the incident wavelength is sufficiently large in comparison with the length scales of the reflecting surface. The reflected wave motion due to an incident plane wave is governed by a Fredholm integral equation whose kernel depends on the form of the reflecting surface. Some specific examples are discussed, and the special case of the ‘linearized boundary’ is considered in detail. For an incoming plane wave incident on a sinusoidally varying surface of sufficiently small amplitude, in addition to the main reflected wave two new waves are generated whose wave-numbers are the sum and difference respectively of those of the surface perturbations and the incident wave. If the incident wave-number is the smaller, the difference wave is back-reflected.

The reattachment of a supersonic jet with a turbulent separating boundary layer abruptly expanding into a two-dimensional parallel diffuser has been experimentally investigated using a surface-flow technique. The reattachment criterion proposed by Mukerjee & Martin (1969) for axisymmetric confined and unconfined flows is found to correlate equally well similar two-dimensional flow measurements in terms of the free-stream Mach number after separation.

Earlier papers have described a technique whereby a stratified shear flow may be produced under controlled conditions in the laboratory. A comparison between the experiments and theory is made here for small-amplitude unstable disturbances in an accelerating stratified free shear layer at the diffuse interface between layers of brine and water. In the early stages of the observed growth of the instability, which takes the form of growing waves, three measurable quantities can be compared with predicted values: the wavelength of the small amplitude waves, the time, which determines the flow conditions, and the growth rates of the waves. Some observations of the development of the disturbances to finite amplitude, the transition to turbulence and the resulting turbulence are reported.

The influence of surface roughness on the cross-flow around a circular cylinder is the subject of the present experimental work. The investigations were carried out in a high-pressure wind tunnel, thus high Reynolds numbers up to Re = 3 × 106 could be obtained. Local pressure and skin friction distributions were measured. These quantities were evaluated to determine the total drag coefficient and the percentage of friction as functions of Reynolds number and roughness parameter. In addition the local skin friction distribution yields the angular position of boundary-layer transition from laminar to turbulent flow and the location of boundary-layer separation.

The most effective movements of swimming aquatic animals of almost all sizes appear to have the form of a transverse wave progressing along the body from head to tail. The main features of this undulatory mode of propulsion are discussed for the case of large Reynolds number, based on the principle of energy conservation. The general problem of a two-dimensional flexible plate, swimming at arbitrary, unsteady forward speeds, is solved by applying the linearized in viscid flow theory. The large-time asymptotic behaviour of an initial-value harmonic motion shows the decay of the transient terms. For a flexible plate starting with a constant acceleration from at rest, the small-time solution is evaluated and the initial optimum shape is determined for the maximum thrust under conditions of fixed power and negligible body recoil.

The waves generated by the steady motion of an obstacle along the axis of a uniformly rotating, electrically conducting homogeneous fluid have been studied by Lighthill's technique. The wave-number surface consists of a sphere and four coincident planes. The waves corresponding to points on the sphere travel ahead or trail behind the obstacle according as a1 the Alfvén velocity, is greater or less than U, the velocity of the obstacle. By drawing the appropriate normals to the four planes, it is seen that the formation of a Taylor column ahead of the obstacle is possible even at large Rossby numbers when U < a1 in contrast with the non-magnetic case and the case with U > a1.

The airways of the lung form a rapidly diverging system of branched tubes, and any discussion of their mechanics requires an understanding of the effects of the bifurcations on the flow downstream of them. Experiments have been carried out in models containing up to two generations of symmetrical junctions with fixed branching angle and diameter ratio, typical of the human lung. Flow visualization studies and velocity measurements in the daughter tubes of the first junction verified that secondary motions are set up, with peak axial velocities just outside the boundary layer on the inner wall of the junction, and that they decay slowly downstream. Axial velocity profiles were measured downstream of all junctions at a range of Reynolds numbers for which the flow was laminar.

In each case these velocity profiles were used to estimate the viscous dissipation in the daughter tubes, so that the mean pressure drop associated with each junction and its daughter tubes could be inferred. The dependence of the dissipation on the dimensional variables is expected to be the same as in the early part of a simple entrance region, because most of the dissipation will occur in the boundary layers. This is supported by the experimental results, and the ratio Z of the dissipation in a tube downstream of a bifurcation to the dissipation which would exist in the same tube if Poiseuille flow were present is given by \[ Z = (C/4\surd{2})(Re\,d/L)^{\frac{1}{2}}, \] where L and d are the length and diameter of the tube, Re is the Reynolds number in it, and the constant C (equal to one for simple entry flow) is equal to 1·85 (the average value from our experiments). In general, C is expected to depend on the branching angles and diameter ratios of the junctions used. No experiments were performed in which the flow was turbulent, but it is argued that turbulence will not greatly affect the above results at Reynolds numbers less than and of the order of 10000. Many more experiments are required to consolidate this approach, but predictions based upon it agree well with the limited number of physiological experiments available.

In certain accelerated flows the entrainment in the boundary layer, as normally defined, may be either zero or negative; on the other hand, there is no reason to suppose, on physical grounds, that the spread of mean or fluctuating vorticity should cease or become negative in such flows. This paradox is resolved in the present paper. It is also shown that in the equilibrium turbulent sink-flow boundary layer, where the entrainment as normally defined is zero, the reduced advection along streamlines in the outer part of the layer comes about mainly through increased dissipation: there is no reason to assume any radical change in the turbulence structure.

This paper is concerned with the effect on the bulk motion of applying torque by external means to the particles of a suspension. Our investigation is based upon theoretical consideration of the motion of a dilute suspension of permanently magnetized spheroids, in the presence of a uniform magnetic field, for cases in which the motion of a Newtonian fluid would be uni-directional. For convenience, we have divided the work into two parts: in the first, the particles are assumed to be perfectly spherical and the magnetic field strength to be arbitrary; in the second, the particles are taken as arbitrary spheroids (with their magnetic dipole axis coincident with the axis of revolution), but the magnetic field is assumed to be strong enough to ensure that the magnetic dipole axis of each particle is effectively aligned with the field vector H. We concentrate attention on the development of general criteria which allow an a priori determination as to whether the bulk motion remains uni-directional in the presence of particle couples, and (when the motion is uni-directional) whether the resultant velocity profile has the same form as for a Newtonian fluid. In addition, we evaluate the effective viscosity of the suspension for several representative cases in which the velocity field is Newtonian in form. Finally, as an example of the general situation in which the bulk velocity field does not remain uni-directional, we obtain the solution for the motion through a circular tube when the magnetic particles are spherical and the magnetic field is applied at right angles to the tube axis.