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.

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.

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.

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.

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.

The basic method described by Bradshaw, Ferriss & Atwell (1967) is extended to compressible flow in two-dimensional boundary layers in arbitrary pressure gradient (excluding shock waves and expansion fans) by invoking Morkovin's hypothesis (Favre 1964) that the turbulence structure is unaffected by compressibility. Using the same empirical functions as in incompressible flow, skin friction in zero pressure gradient is predicted to within 3% of Spalding & Chi's (1964) correlation for free-stream Mach numbers less than 5. Comparisons with experiments in pressure gradient are restricted by the lack of data, but, since Morkovin's hypothesis does not depend on pressure gradient, methods which use it (of which the present method seems to be the first) can be checked fairly adequately by comparisons with data in zero pressure gradient.

No ‘compressibility transformations’ are needed, although the Crocco relation is used, provisionally, for the temperature: since the calculations take only about 20% longer than in incompressible flow, Morkovin's hypothesis does as much as any transformation could do. It is pointed out that, in supersonic flow, surface curvature which is large enough to induce a significant longitudinal pressure gradient is also large enough to have a very significant effect on the turbulence structure.

Sound waves of finite but small amplitude propagating into a quasi-steady, supersonic flow in a non-uniform duct are analyzed by means of a perturbation method. General properties of the flow and of the wave propagation are studied using a one-dimensional approximation. A shock propagation law in the unsteady flow is obtained. As an example, the formation and development of shock waves are discussed for a duct with a conical convergence. Comparisons of the theory with an experiment are also made; fairly good agreement is found.

The scattering of surface gravity waves by a circular dock is considered in order to determine the horizontal and vertical forces and torque on the dock. An incident plane wave is expanded in Bessel functions, and for each mode the problem is formulated in terms of the potential on the cylindrical surface containing the dock and extending to the bottom. The solution is shown to have phase independent of depth and so may be obtained from an infinite set of real equations, which are solved numerically by Galerkin's method. The convergence of the solution is discussed, and some numerical results are presented.

This problem has been investigated previously by Miles & Gilbert (1968) by a different method, but their work contained errors.

Complementary variational formulations are developed for the scattering of a gravity wave by a circular dock. These formulations, which are based on assumed distributions of the radial velocity and the potential, respectively, on the projection of the cylindrical boundary, yield lower and upper bounds to an impedance parameter that determines the difference between the scattered wave for the dock and the corresponding wave for a circular cylinder. Numerical results, using trial functions based on the incident wave, are compared with the results implied by a Galerkin solution (Garrett 1971). The maximum errors in the variational approximations to the total scattering cross-section are found to be of the order of 2% for a typical depth/radius ratio, draft/depth ratios of 0, ½ and 1, and all wavelengths. The axisymmetric component of the scattering cross-section is found to be very close to the value for scattering by a circular cylinder (dock extending to bottom). The intensity of the scattered wave on the forward axis for long wavelengths and a certain range of the geometric parameters is significantly less than that for a circular cylinder, and may vanish for critical combinations of these parameters.

Schwinger's variational formulation is applied to the radiation of surface waves due to small oscillation of bodies. By means of Haskind's theorem the wave forces on a stationary body due to a plane incident wave are found using only far-field properties. Results for horizontal rectangular and vertical circular cylinders are presented.

The time-mean skin friction of the laminar boundary layer on a flat plate which is fixed at zero incidence in a fluctuating stream is investigated analytically. Flow oscillation amplitude outside the boundary layer is assumed constant along the surface. First, the small velocity-amplitude case is treated, and approximate formulae are obtained in the extreme cases when the frequency is low and high. Next, the finite velocity-amplitude case is treated under the condition of high frequency, and it is found that the formula obtained for the small-amplitude and high-frequency case is also valid. These results show that the increase of the mean skin friction reduces with frequency and is ultimately inversely proportional to the square of frequency.

The corresponding energy equation is also studied simultaneously under the condition of zero heat transfer between the fluid and the surface. It is confirmed that the time-mean surface temperature increases with frequency and tends to be proportional to the square root of frequency. Moreover, it is shown that the timemean recovery factor can be several times as large as that without flow oscillation.

Measurements on square and circular plates in turbulent flow show the mean base pressure to be considerably lower than that measured in smooth flow. Power spectral density measurements of the fluctuating component of the drag on square plates in both smooth and turbulent flow are presented. The measurements in turbulent flow show the importance of the ratio of turbulence scale to plate size. There is shown to be a strong correlation between the fluctuating drag force and the velocity fluctuations in the approaching flow. The distortion of the turbulence structure approaching a plate is also discussed.

In this paper, an attempt has been made to model the dynamics of ciliary propulsion through the concept of an ‘envelope’ covering the ends of the numerous cilia of the microscopic organism. This approximation may be made in the case when the cilia are close together, as can occur in the case of the symplectic metachronal wave (i.e. the wave travels in the same direction as the effective beat). For simplicity, a spherical model has been chosen, and the analysis which follows is a correction to Lighthill's (1952) paper on squirming motions of a nearly spherical organism. The velocity and efficiency compared to the work done in pushing an inert organism are obtained, and compared to that of a ciliated organism.