This paper studies the effect of profile thickness on the propulsive forces generated by the swimming of a two-dimensional fish. Comparison of numerical calculations with reported experimental data shows good agreement and demonstrates a decrease of thrust with increasing thickness. Previous two-dimensional linearized theories on fish propulsion dealing with the motion of an infinitesimally thin hydrofoil are included in the present contribution as special cases.

An analytical investigation is carried out to determine the conditions for instability in a viscous fluid contained between rotating coaxial cylinders of arbitrary radius ratio. A solution method is outlined and then applied to cylinders having radius ratios ranging from 0·95 to 0·1. Consideration is given to both cases wherein the cylinders are rotating in the same direction and in opposite directions. Results are reported for the Taylor numbers and wave-numbers which mark the onset of instability. The present results are also employed to delineate the range of applicability of the closed-form instability predictions of Taylor and of Meksyn, which were derived for narrow-gap conditions.

We have made a theoretical and experimental study of the effect of radiative transfer on the onset of convection by means of comparative experiments in dry air and ammonia contained between horizontal aluminium plates maintained at different temperatures.

The first effect of radiative transfer is to distort the linear, diffusive profile in the initial, pre-convective state. A non-grey theory is developed and is shown to predict the non-linear profiles profiles with satisfactory accuracy. The change of radiative flux with pressure of ammonia is also predicted and confirmed. This constitutes the first successful attempt to confirm the methods used in atmospheric studies in controlled experiment.

A non-grey theory of the onset of convection is developed, separating effects associated with the distortion of the static state and the dissipative effect of radiative transfer on temperature fluctuations.

Experiments indicate that the critical Rayleigh number in ammonia can be greatly increaesd over the best value for air (Rac = 1786); the maximum observed being Rac = 4870. Theory and experiment are shown to agree within reasonable limits.

This work confirms the validity of methods which are of great importance to astrophysics and atmospheric physics.

The stability of Couette flow is discussed in the case in which the cylindes rotate in opposite directions by an asymptotic method in which the Taylor number is treated as a large parameter. On assuming the principle of exchange of stabilities to hold, the problem is then governed by a sixth-order differential equation with a simple turning point. It is shown how the solutions of this equation can be represented asymptotically in terms of the solutions of the comparison equation yvi = xy. The solutions of this comparison equation have recently been tabulated and we thus have an explicit representation of the solution of the stability problem in terms of tabulate functions. Detailed results for the critical Taylor number and wave-number at the onset of instability and the associated eigenfunctions are given for the limiting case μ → − ∞, where μ = Ω2/Ω1, and Ω1 and Ω2 are the angular velocities of the inner and outer cylinders respectively. In this limiting case it is found that there exists and infinite number of cells between the cylinders, but that the amplitude of the secondary motion in all but the innermost cell is small.

A simple numerical method is presented for solving the eigenvalue problem which governs the stability of Couette flow. The method is particularly useful in obtaining the eigenfunctions associated with the various modes of instability. When the cylinders rotate in opposite directions, these eigenfunctions exhibit an exponentially damped oscillatory behaviour for sufficiently large values of − μ, where μ = Ω2/Ω1. In terms of the stream function which describes the motion in planes through the axis of the cylinders, this means that weak, viscously driven cells appear in the outer layes of the fluid which, according to Rayleigh's criterion, are dynamically stable. For μ = − 3, for example, four cells are present, the amplitudes of which are in the ratios 1·0:0·0172:0·013:0·00125.

The instability to small two-dimensional disturbances of an electrically conducting fluid of variable density is investigated. The viscous fluid is bounded between two vertical parallel planes normal to which a magnetic field of constant intensity is applied. Significant parameters upon which the behaviour of the Rayleigh number at neutral stability depends are the Hartmann number M and the wave-number α which is associated with a periodic disturbance with periodicity in the unbounded horizontal direction.

The solution may be sought by considering basic disturbances which are either symmetric or antisymmetric about the median plane parallel to the boundary planes. It is found that for a given magnetic field strength the critical Rayleigh number governing stability is associated with an antisymmetric disturbance of zero wave-number. The least stable symmetric disturbance which arises when the wave-number is zero is less easily excited. This trend is seen again in the purely hydrodynamic case (M = 0) where, corresponding to a finite wave-numbe value, the more unstable mode at neutral stability is found to be an antisymmetric one.

The most unstable situation occurs when both the Hartmann number and the wave number zero. In this case the result of Wooding (1960) that the minimum critical Rayleigh number is zero and is associated with a symmetric disturbance is reobtained.

It is shown that the solution of the problem of scattering of long waves by a finite barrier in a rotating system may be obtained directly from the solution of the electromagnetic scattering problem for a finite strip. It is shown that the effect of rotation is to produce on both sides of the barrier, but away from the ends, a wave which is propagated without attenuation parallel to the barrier.

The sedimentation of small clusters of uniform spheres, falling freely through a viscous liquid, has been studied with Reynolds numbers (based on diameter of the sphere and its velocity of free fall in the unbounded fluid) of individual spheres ranging from 10−4 to 10. The fall velocity of a cluster is, in all cases, greater than that of individual spheres, the more so when the spheres are closer together. Two spheres falling side-by-side rotate inwards and separate as they fall if Re > 0·05, but no rotation nor separation is observed for Re < 0·03. When equal-sized spheres of Re > 1 fall vertically one behind the other, the rear sphere is accelerated into the wake of the leader, rotates, round it and separates from it when the line of centres is horizontal. If two spheres of unequal size but the same individual terminal velocity fall together, the smaller always travels faster than the larger. When three similar equally spaced spheres are dropped in a horizontal line, they interchange positions but do not separate when 0·06 < Re < 0·16. But, if 0·16 < Re < 3, one sphere is always left behind; which sphere depends critically upon the initial spacing. If three to six equal spheres, of 0·06 < Re < 7, start falling as a compact cluster, they eventually draw level and arrange themselves in the same horizontal plane at the vertices of a regular polygon. The polygon expands at a decreasing rate during fall. When three spheres are arranged initially in a horizontal isosceles triangle, the spheres oscillate about their equilibrium positions but eventually the spheres form a stable equil triagnle. If Re > 7, or the cluster contains 7 or more equal spheres, it shows no tendency to form a regular polygon but breaks up into two or more groups. A regular heptagon, and a hexagon with an additional sphere at its centre, are also unstable.

A theoretical study is made of the behaviour of clusters of spheres falling in a viscous fluid under the assumptions that: (a) intertial effects are negligible, (b) the distance between any two spheres is larg compared with their radii. The equations of motion are derived and solved for a number of particular cases and the results compared with the experimental observations of the same motions reported in the preceding paper (Jayaweera, Mason & Slack 1964). For three or four spheres, initially in a horizontal line, the theory is in general agreement with the experiments. Three spheres forming an isosceles triangle are shown to oscillate about the horizontal and about the equlateral shape, so that this theory is unable to explain the observed tendency for three to six spheres to form a regular horizontal polygon. The stability of the steady configuration of n spheres at the vertices of a regular horizontal polygon is examined and it is found that the configuration is only stable for n < 7, which explains why this configuration is not observed for more than six spheres.

The Hartmann air jet generator (Hartmann 1939) is sound wave generator in which the sound is generated by oscillations of a shock developed in an over-expanded air jet by means of a blunt body: a resonator. A theory for the instability mechanism is advanced, and it is found that the result calculated from the theory are in good agreement with the experimental observations.

Derived herein is a set of partial differential equations governing the propagation of an arbitrary, long-wave disturbance of small, but finite amplitude. The equations reduce to that of Boussinesq (1872) when the assumption is made that the disturbance is propagating in one direction only. The equations are hyperbolic with characteristic curves of constant slope. The initial-value problem can be solved very readily by numerical integration along characteristics. A few examples are included.