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 effect of the presence of a catheter upon the pressure distribution inside the ureter is considered. Under the assumption of Stokes flow and long wavelengths it is shown that during the contraction a thin lubrication-type layer is formed between the catheter and the ureteral wall, capable of sustaining high pressures. Furthermore, it is found that the insertion of a catheter does not change the pressure distribution inside the ureter appreciably, leading to the conclusion that a urometrogram obtained with a catheter gives a good representation of the pressure inside an undisturbed ureter.

The two-dimensional prediction method of Bradshaw, Ferriss & Atwell (1967), which was based on the empirical conversion of the turbulent energy equation into a ‘transport’ equation for shear stress, is extended to three-dimensional flows satisfying the boundary-layer approximation (which excludes flows near bluff obstacles or streamwise corners). Predictions, using exactly the same empirical data as in two-dimensional flow, agree to within the likely experimental error with a variety of experiments on ‘infinite’ swept wings.

The initial-value problem for the linear, inviscid motion of a contained, rotating stratified fluid is considered in the limit of weak stratification, that is, for small values of the stratification parameter S = N2/Ω2, where N is the Brunt–Väisälä frequency and Ω is the rotational frequency. The limiting flow is of interest because, although the initial-value problem has been studied, both for the case of a homogeneous, rotating fluid and for the case of a stratified, rotating fluid, the exact relationship of the two flows, in the limit of vanishing stratification, is not straightforward. For example, the method of determining, from the initial conditions, the steady geostrophic component of the flow of a rotating, stratified fluid does not in general give a motion that reduces, in the limit S → 0, to the steady component of the flow of a homogeneous fluid. By including a consideration of slow unsteady motions that vary on a time scale dependent on the stratification parameter, the relationship of the limiting flow to the flow of a homogeneous fluid is established.

The development of optical methods for the quantitative study of the fluctuating properties of turbulent flows can provide a supplement to conventional hot-body anemometry techniques. In particular, the study of high-speed flows by hot-wire or hot-film anemometry is often difficult owing to the presence of temperature and velocity fluctuations in the flow, thereby complicating the correct interpretation of measured signals. In addition, restrictions are placed on the application of such anemometers by their physical strength, frequency-response characteristics and the introduction of disturbances by the measuring probes into the flow.

The operation of an optical detection system depends primarily on the mechanism by which the detected radiation intensity is modulated by the flow. Methods which have been used successfully include scattering or absorption of incident light by tracer constituents or particles, the absorption or emission of infra-red radiation by the flow and quantitative adaptations of the schlieren and interferometer systems which are sensitive to the flow density structure. All these systems detect a summation of signals from different parts of the flow and in consequence it is necessary to consider in detail the relation of the integrated signal to the local properties of the flow. This paper deals in particular with the application of the schlieren principle to an axisymmetric turbulent jet.

The pattern and propagation of waves excited by forcing effects that may oscillate with a frequency σo and are travelling with a uniform velocity U along the axis of rotation in a rotating stratified liquid are studied by employing the technique of Lighthill. The fluid is assumed to be an unbounded, in viscid, incompressible, non-diffusive, rotating stratified liquid, rotating with a constant angular velocity Ω about a vertical axis and with a density decreasing vertically upwards. It is found that the effect of rotation on stratification or vice versa is to reduce the region of disturbance and to split the wave crests for Ω0 ≠ 0. The periodic nature of the forcing effect excites various systems of waves which are otherwise coincident.

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.

In this paper we consider the flow field induced in an incompressible viscous conducting fluid, occupying the interior of a right circular cone, by an electriccurrent source situated at the vertex of the cone. We assume that the velocity field is small and its effect on the electromagnetic field is negligible. A similarity solution is obtained for the non-linear problem. This solution is an adaptation of Slezkin's solution for the momentum transfer through a viscous jet and, apart from the numerical solution of a Riccati type of equation, is exact. In particular, we investigate the case when the half angle of the cone is ½π and the fluid occupies the whole space on one side of an infinite plane. We also consider the corresponding inviscid flow problem that was recently investigated by another author and suggest that the solution obtained is not physically realistic.

A solution is developed for the deflexion of an inviscid, incompressible, twodimensional jet by a series of segments of arbitrary number, lengths, and angles. The Schwarz–Christoffel transformation and free-streamline theory are used. Results are calculated for a number of configurations, using an IBM 360 computer. Excellent comparison is found with several previous calculations for special cases.

In this paper the general analysis, developed in part 1, of three-dimensional duct flows subject to a strong transverse magnetic field is used to examine the flow in diverging ducts of rectangular cross-section. It is found that, with the magnetic field parallel to one pair of the sides, the essential problem is the analysis of the boundary layers on these (side) walls. Assuming that they are highly conducting and that those perpendicular to the magnetic field are non-conducting, the flow is found to have some interesting properties: if the top and bottom walls diverge, the side walls remaining parallel, then an O(1) velocity overshoot occurs in the side-wall boundary layers; but if the top and bottom walls remain parallel, the side walls diverging, these boundary layers have conventional velocity profiles. The most interesting flows occur when both pairs of walls diverge, when it is found that large, 0(M½), velocities occur in the side-wall boundary layers, either in the direction of the mean flow or in the reverse direction, depending on the geometry of the duct and the external electric circuit!

The mathematical analysis involves the solution of a formidable integral equation which, however, does have analytic solutions for some special types of duct.

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.

The flow of a two-phase system past an arbitrary corner has been studied by Healy (1970a), using the method of small perturbations, and some results were presented for convex-type flows. This paper, which is an extension of the abovementioned one to concave-type flows, also compares the particle streamlines found from the perturbation with those obtained by numerically integrating the unperturbed equations. The agreement is found to be quite good for ξ0.2 and excellent when ξ = 0·1 or less, where ξ = ατ is the particle parameter, α is proportional to the speed of the fluid and τ is the particle relaxation time. The nature of concave corner flows abruptly changes when the angle β through which the flow is deflected is 90°. For β < 90°, all particles collide directly except those approaching on streamlines near the stagnation line. When β = 90° the critical value of ξ is ξc = 0·25 and, for 180° > β > 90°, only particles approaching on streamlines near the stagnation line all collide. No particle-free zones exist in concave-type flows and the particle density increases monotonically in the downstream direction along all particle streamlines. The approximate effects of viscosity are also discussed.

Measurements of the noise field from a 25 mm diameter subsonic air jet are presented. These results are analysed in some detail by determining both the jet velocity dependence and the directivity of the intensity of the radiation in 1/3-octave bands at particular values of the frequency parameter, \[ (fD/V_J)(1-M_c\cos\theta). \] This procedure should ensure that a particular source in a geometrically similar position in the jet is always observed, whatever the jet velocity, diameter and emission angle.

These results are compared with the predictions of Lighthil's (1952) theory of convected quadrupoles. It is shown that the theory predicts the variation of the intensity with jet velocity and emission angle provided that the observed frequency is below a certain critical value, which depends on jet diameter and emission angle and is independent of jet velocity. Above this critical frequency, the predicted variations overestimate the measurements and it appears that the convective amplification predicted by the theory is much reduced. The variation of this critical frequency is explained by assuming that substantial interaction occurs between the radiated sound and the jet flow when the wavelength of the sound becomes shorter than the sound path length in the jet flow.

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.

Axisymmetric particles in zero Reynolds number shear flow execute closed orbits. In this paper we consider the role of small Brownian couples in establishing a steady-state probability distribution for a particle being on any particular orbit. After presenting the basic equations, we derive an expression for the equilibrium distribution. This result is then used to calculate some bulk properties for a suspension of such particles, and these predicted properties are compared with available experimental observation.

Experiments were made on a pulsating water flow at a mean flow Reynolds number of 3770 in a cylindrical tube of diameter 3·81 cm. Pulsations were produced by a piston oscillating in simple harmonic motion with a period of 12 s. Turbulence was made visible by means of a sheet of dye produced by electrolysis from a fine wire stretched across a diameter. The sheet of dye is contorted by the turbulent eddies, and ciné-photography was used to find the velocity of convection which was shown to be the flow speed except in certain circumstances which are discussed. By subtracting the mean flow velocity profile the profile of the component of the motion oscillating at the imposed frequency was determined.

The Reynolds number of these experiments lies in the turbulent transition range, so that large effects of laminarization are observed. In the turbulent phase, the velocity profile was found to possess a central plateau as does the laminar oscillating profile. The level and radial extent of this were little different from the laminar ones. Near to the wall, the turbulent oscillating profile is well represented by the mean velocity power law relationship, u/U ∝ (y/a)1/n. In the laminarized phase, the turbulent intensity is considerably reduced at this Reynolds number. The velocity profile for the whole flow (mean plus oscillating) relaxes towards the laminar profile. Laminarization contributes appreciably to the oscillating component.

Extrapolation of the results to higher Reynolds numbers and different frequencies of oscillation is suggested.

A uniformly valid asymptotic solution has been constructed for three-dimensional jet-flapped wings by the method of matched asymptotic expansions for high aspect ratios. The analysis assumes that the flow is inviscid and incompressible and is formulated on the thin airfoil theory in accordance with the well-established Spence (1961) theory in two dimensions.

A simple method emerges in treating the bound vortices along the jet sheet which forms behind the wing with the aid of the following physical picture. Three distinct flow regions—namely inner, outer and Trefitz—exist in the problem. Close to the wing the flow approximates to that in two dimensions. Therefore, Spence's solution in two dimensions applies. In the outer region a wing shrinks to a line of singularities from which the main disturbances of flow in this region arise. In particular, we find that the shape of the jet sheet, hence the strength of vortices, is now predetermined by the strength of the singularities there. Hence a complete flow field in the outer region can now be determined first by evaluating the flow due to various degrees of singularities along this line and then adding the effect of the jet bound vortices which is now known. Far removed from the wing, the well-known Trefftz region exists in which calculations of aerodynamic forces can be most easily done.

The result has been applied to various wing planforms such as cusped, elliptic and rectangular wings. The present result breaks down for rectangular wings. However, we can apply Stewartson's (1960) solution for lifting-line theory for semi-infinite rectangular wings, because, to this second-order approximation it is established that the jet sheet in the outer region makes no contribution to lift, with the direct contribution of the deflected jet at the exit being cancelled by the reduced circulation in the jet vortices. This result for the rectangular wing gives excellent agreement with the experiments made on a rectangular wing, while the result for elliptic wings underestimates them considerably.

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.

A Green's function for a semi-infinite rotating ocean of uniform depth is obtained, and the resulting near and far fields are estimated asymptotically.

Given a tide of uniform height at the mouth of a narrow channel on a semiinfinite ocean, the Green's function is used to calculate the diffracted Kelvin and Poincaré waves propagating up the channel and into the ocean.

We present a linear stability analysis, to second order in initial amplitude, of Bénard convection of a Boussinesq fluid in a thin rotating annulus for modest Taylor numbers T ([les ] 104). The work is motivated in part by the desire to study further a mechanism for maintaining, through horizontal Reynolds stresses induced in the convection, the sun's ‘equatorial acceleration’, which has been demonstrated for a rotating convecting spherical shell by Busse & Durney. The annulus is assumed to have stress free, perfectly conducting top and bottom (which allows separation of the equations) and non-conducting non-slip sides. A laboratory experiment which fulfills these conditions (except perhaps the free bottom) is being developed with H. Snyder.

We study primarily annuli with gap-width to depth ratios a of order unity. The close, non-slip side-walls produce a number of effects not present in the infinite plane case, including overstability at high Prandtl numbers P, and multiple minima in Rayleigh number R on the stability boundary. The latter may give rise to vacillation. The eigenfunctions for stationary convection for a = 2, T [lsim ] 2000 clearly show momentum of the same sense as the rotation is transported from the inner to the outer half of the annulus, corresponding to transport toward equatorial latitudes on the sphere. The complete second-order solutions for the induced circulations indeed give faster rotation in the outer half, except for large P (> 102), in which case thermal stresses dominate. At all P, this differential rotation is qualitatively a thermal wind. Overstable convective cells, and stationary cells at higher T, induce more complicated differential rotations.