In this paper a theoretical investigation of convection currents in anisotropic porous media is performed. The porous layer is homogeneous and bounded by two infinite perfectly heat-conducting horizontal planes. The criterion for the onset of convection is derived. The supercritical steady two-dimensional motion, the heat transport and the stability of the motion are investigated. Some of the results may be applied in insulation techniques.

Swirling water bells are studied theoretically and experimentally. It is shown theoretically that, if the effects of gravity and the surrounding air are neglected, the shape of a swirling water bell will, under certain circumstances, be periodic along the axis of rotation. Under ideal conditions, a swirling water bell may thus be infinitely long. However, the experiments show that in reality the length of a swirling water bell will be limited owing to Kelvin-Helmholtz instabilities. Theoretically calculated shapes of swirling water bells are found to agree reasonably well with experimental results.

The stability characteristics of some simple steady magnetohydrodynamic flows within an axisymmetric container of arbitrary electrical conductivity are investigated. Attention is focused upon rapidly rotating fluids in which the unperturbed velocity and magnetic field are axially symmetric and purely zonal. Detailed solutions are obtained for the particularly simple basic state representing a rigidly rotating homogeneous fluid with a uniform axial current. The theory of dynamic (dissipationless) instabilities is reviewed and its shortcomings are elucidated. A stability criterion is derived for an inviscid fluid of small electrical conductivity within a perfectly conducting axisymmetric container and it is shown that a certain class of inertial modes is unstable for any non-zero magnetic field strength. When the effects of container conductivity are included it is found that a class of slow modes with westward phase speed may be unstable. These modes are shown to be unstable within a cylinder but appear to be stable within a sphere. The influence of density gradients within a spherical container is investigated and it is found that for a certain class of exceptional slow modes with westward phase speed, a bottom-heavy density gradient is destabilizing. This surprising behaviour is explained in terms of a new branch of the stability curve developed by Eltayeb & Kumar (1977).

The stability of an electrically conducting Boussinesq fluid which is confined between two horizontal planes a distance d apart is investigated. The fluid is heated from below, cooled from above and the whole system rotates rapidly with angular velocity Ωc about a vertical axis. A weak non-uniform horizontal magnetic field, whose strength is measured by the Alfvén angular velocity ΩM [[Lt ] Ωc, see (1.2)] permeates the fluid and corresponds to the flow of a uniform electric current parallel to the rotation axis. When the modified Rayleigh number R [see (2.1)] is greater than zero and q = κ/λ < 1, where κ and λ are the thermal and magnetic diffusivities respectively, instability sets in as a westward-propagating wave with a low frequency of order κ/d2.

When R = 0 and ΩM > 2(ν/λ)½ Ωc, where ν is the viscosity, Roberts & Loper (1979) have isolated an exceptional class of unstable fast inertial waves which grow on the magnetic diffusion time scale τλ = d2/λ. When R < 0 and Γ = τλΩ2M/τc, exceeds some value dependent upon q, a class of unstable slow waves also exists for a range of negative values of R. These waves propagate eastwards (westwards) when q is less (greater) than unity. In this case the fluid is stably stratified and the energy for the disturbance is taken from the magnetic field. The resulting description of the stability boundary for R < 0 in the Γ, R plane extends and clarifies the results of Roberts & Loper (1979), which are valid when both Γ and q are large.

Numerical results are presented for planar sinusoidal waves (amplitude α, wave-number k). The average swimming speed and power consumption are computed for a wide range of the parameters. The optimal sine wave for minimizing power consumption is found to be a single wave with amplitude αk ≈ 1. The power consumption is found to be relatively insensitive to changes in the flagellar radius. The optimal flagellar length is found to be in the range L/A = 20–40. The instantaneous force distribution and flow field for a typical organism are presented. The trajectory of the organism through one cycle shows that a wave of constant amplitude may have the appearance of increasing amplitude owing to the yawing motion of the organism.

The results are compared with those obtained using resistance coefficients. For organisms with small cell bodies (A/L = 0.05), the average swimming speed predicted by Gray-Hancock coefficients is accurate to within 10%. For large cell bodies (A/L = 0.2), the error in swimming speed is approximately 20%. The relative error in the predicted power consumption is 25–50%. For the coefficients suggested by Lighthill, the power is consistently underestimated. The Gray-Hancock coefficients underestimate the power for small cell bodies and overestimate it for large cell bodies.

The unsteady large amplitude linearized motion of a flexible slender wing in inviscid incompressible fluid is analysed. This extension to large amplitudes is made by assuming small local angles of incidence during motion along a finite amplitude trajectory. As a specific example, the periodic propulsion produced by a flexible slender wing is analysed for large amplitude harmonic motion with small local angles of attack and results for the available thrust and efficiency are presented.

The axisymmetric flow of an incompressible fluid through a pipe (of radius a) suffering a severe constriction is studied for large Reynolds numbers R, the features of symmetric channel flows being virtually the same. Here ‘severe’ refers to a constriction whose typical dimensions are finite, and the oncoming velocity profile is taken to be of a realistic type, i.e. with no slip at the wall. The study adopts (Kirchhoff) free-streamline theory, which, for the mostly inviscid description, affords a rational basis consistent with viscous separation. The major (triple-deck) separation takes place on the constriction surface and is followed by a downstream eddy of length O(aR). Another, less familiar, separation is predicted to occur at a distance 0.087a In R + O(a) ahead of the finite obstacle. Free-streamline solutions are found in the two main extremes of moderately severe and very severe constriction. In both extremes, and in any slowly varying constriction, the major separation is sited near the maximum constriction point. The upstream separation point is also derived, to O(a) accuracy in each case. The upstream separation can be suppressed, however, if the constriction has no definite starting point and decaysslowly upstream, but then the upstream flow response extends over a much increased distance. Comparisons with Navier-Stokes solutions and with experiments tend to favour the predictions of the free-streamline theory.

Seeley, Hummel & Smith (1975) reported the results of experiments to study the dynamics of flow around spheres at intermediate Reynolds numbers using a nondisturbing flow-visualization technique. The flow patterns were recorded on ciné photographs and the information stored was processed in order to obtain the velocity field. The position of fluid elements shown by the photochromic indicator traces were estimated by eye on a projection screen. In this paper, a new set of results based on the same films has been reduced and computed using the ‘POLLY’ film-reading system described by Esmail, Smith & Hummel (1976). Some numerical boundary-layer solutions are included to show the reliability of the data, and comparisons with the results previously reported by Seeley et al. (1975) are presented.J. W. Smith will be pleased to send a complete set of experimental data on request.

A model equation is derived for the combined longitudinal and lateral dispersion of a buoyant contaminant in open-channel flow. The central hypotheses are that the water is shallow and that to a first approximation the effluent is vertically well mixed. The model includes allowance for the reduction in turbulent intensity due to weak vertical stratification, a buoyancy-driven secondary flow, and the redistribution of longitudinal momentum by the secondary flow. For a plume the theoretical results for the excess spreading due to buoyancy agree well with Prych's (1970) experimental results.

A container of viscous incompressible liquid is bounded by rigid parallel planes and rotates steadily about an axis normal to these planes. A rigid sphere moves steadily parallel to the rotation axis and the Rossby and Ekman numbers characterizing the motion are both small. The drag on the sphere is calculated in the case when the length of the Taylor column is comparable to the axial dimension of the container. Viscous effects are allowed for in the boundary of the Taylor column, but the Ekman layers on the sphere and on the bounding planes are shown not to affect the drag to leading order. The determination of the drag involves solving dual integral equations. This is done numerically and, for the limiting cases of long and short containers, analytically. The interaction of the Taylor column and the ends of the container leads to an increase in the drag over its value in an unbounded fluid, but the increase is smaller than that measured by Maxworthy (1970).