In the theory of thermal convective instability between two horizontal planes there are many solutions that are periodic in the horizontal co-ordinates, while in experiment convection is observed to take place in cellular patterns. It is often assumed, or decided after insufficient argument, that the periodic solutions of the mathematical model ‘explain’ or correspond to these patterns, but a completely satisfactory discussion of this correspondence has not been given. Indeed, with certain mathematical solutions ambiguities arise as to what cell centres and cellular boundaries are. A detailed discussion has recently become especially necessary because attempts are being made to predict which particular cellular pattern will occur in given experimental conditions.

In this paper the topic is studied afresh and the question is asked: what features, in the mathematical model, correspond to what an experimentalist observes in cellular convective motion? In answer a definition of a cell is formulated which relates certain surfaces in the flow field of the mathematical model to steady vertical cellular boundaries that are observed in experiment, and which shows where the cell centres lie. In particular the classical hexagonal cellular pattern, of the mathematical model, is shown to be the prototype pattern of what is experimentally observed. On the other hand the square and so called ‘rectangular’ cases of linearized theory are shown not to correspond truly to square and rectangular cells at all. The new formulation is especially relevant to theoretical work on the prediction of cell shape and direction of flow in cells, since precise knowledge of the shapes of the cellular boundaries and locations of cell centres is essential if predictions are to be compared with observation.

A solution of the Navier-Stokes equations is obtained for the flow resulting from the steady rotation of a semi-infinite right circular (solid) cylinder about its vertical axis. Incompressible viscous fluid is assumed to fill the space outside the cylinder on one side of a horizontal solid plane. In the proposed method of solution the pertinent physical quantities are expressed as series in positive powers of the Reynolds number Re with space-dependent coefficients. It is shown that the coefficients of (Re)M can be obtained by solving linear partial differential equations which depend on the coefficients of (Re)i, where i < M. A truncated solution, which holds for small Re, is obtained by solving for the first two coefficients. These results indicate that at the flat end of the cylinder the pressure distribution is nearly constant, yet along the adjacent bounding plane it rises with the radial direction.

An investigation is carried out to determine the conditions marking the onset of convective motion in a horizontal fluid layer in which a negative temperature gradient occurs somewhere within the layer. In such cases, fluid of greater density is situated above fluid of lesser density. Consideration is given to a variety of thermal and hydrodynamic boundary conditions at the surfaces which bound the fluid layer. The thermal conditions include fixed temperature and fixed heat flux at the lower bounding surface, and a general convective-radiative exchange at the upper surface which includes fixed temperature and fixed heat flux as special cases. The hydrodynamic boundary conditions include both rigid and free upper surfaces with a rigid lower bounding surface. It is found that the Rayleigh number marking the onset of motion is greatest for the boundary condition of fixed temperature and decreases monotonically as the condition of fixed heat flux is approached. Non-linear temperature distributions in the fluid layer may result from internal heat generation. With increasing departures from the linear temperature profile, it is found that the fluid layer becomes more prone to instability, that is, the critical Rayleigh number decreases.

The object of this work was to investigate experimentally the structure of the early shear layer of high-speed jets and its relation to the mechanism of noise generation. Of special interest was the question of the existence of periodic fluctuations in the velocity field. The experimental investigation is divided in three parts.

1. Optical observation of the jet flow by means of the shadowgraph technique.

2. Measurement of mean Mach number and temperature profiles.

3. Survey by means of hot-wire of the component of the fluctuating velocity field in the mean flow direction.

The shadowgraphs show very interesting features of the breakdown process of free shear layers, but fail to show any propagation of strong acoustic disturbances in the near pressure-field.

Mean profile measurements show that the flow field in the range 1 [les ] x/d [les ] 4 develops in a conical fashion, i.e. the mean profiles in Mach number and temperature can be expressed in terms of a single conical variable η = (2r–d)/2x. The fluctuating velocity field is described in terms of the intensity of turbulence, its spectral distribution, and two-point space-time correlation functions. Similarity laws are given for the power spectra and the space-time correlation functions. On the cylinder r = ½d, the convection speed of the turbulent field is different for the different eddy sizes, varying from the local mean speed for small eddies to ½Uexit for the large eddies. Measurements of the angular correlation function are reported which show no correlation of the fluctuations across the jet diameter.

When I was first asked to give a general lecture to the 5th British Theoretical Mechanics Colloquium (held at Liverpool University, 2–5 April 1963), I put up the title ‘Shock Waves’, thinking that I might run over a wide field of present-day research, pointing out some of the unanswered problems. In the intervening months, however, I came across the new book by Dr Bradley (1962) of Liverpool University, The Physics and Chemistry of Shock Waves, and as recently as last November there appeared a summarizing article by Pain bt Rogers (1962) of London University in Reports on Progress in Physics. The first of these deals in great detail with the modern physical and chemical aspects of the subject—with real gas effects and experimental techniques-and the second summarizes the general classical properties and gives an extended account of recent work, for example, on real gas effects and on magnetohydrodynamics. I also bore in mind that magnetogasdynamics received a majestic treatment at the 4th Colloquium from Dr Shercliff. In the end, then, I decided to confine my remarks to one particular problem, namely the reflexion and diffraction of shock waves, and to concentrate mainly on developments during the past ten years. This paper is the text of the lecture.

The flow of an electrically conducting fluid up a hot vertical plate in the presence of a strong magnetic field normal to the plate is considered. A solution is developed based on the idea of matching ‘outer’ and ‘inner’ solutions in the moving layer of fluid. An approximate Pohlhausen method of solution is also given which yields results in fairly good agreement with the exact analysis.

The problem of flow of an inviscid, incompressible fluid inside a circular pipe, with a sphere on the axis of the pipe, has been studied by Lamb (1936) (irrotational flow), Long (1953) and Fraenkel (1956) (swirling flow). Because of the difficulty of satisfying all the boundary conditions in the problem, only approximate solutions, valid for spheres of small diameter (compared with that of the pipe) have been obtained. In this paper, it is found that by introducing a vortex sheet over a segment of the diameter of the sphere, flow patterns can be obtained by an inverse method for the case of large spheres. Four different types of flow are considered: (1) irrotational flow, (2) swirling flow with constant axial and angular velocities far upstream, without lee waves, (3) swirling flow with constant axial and angular velocities far upstream, with lee waves, and (4) rotational flow with a paraboloidal velocity distribution far upstream.

An exact solution is given for the flow in a convergent or divergent channel of a class of anisotropic fluids in which the fluid has a preferred direction.

A linearized theory is developed for the oscillations of a slender body which is floating on the free surface of an ideal fluid, in the presence of incident plane progressive waves. Green's theorem is used to represent the velocity potential and the first-order slender-body potential is developed from asymptotic approximation. The general theory is valid for arbitrary slender bodies in oblique waves, and detailed results are presented for a body of revolution.

It is suggested that the use of prolate spheroidal co-ordinates in certain problems involving slender bodies may lead to results which not only are more likely to be uniformly valid for blunt bodies, but in many cases require less complicated analysis than results obtained by standard methods which use cylindrical co-ordinates. The method is developed for a simple problem in potential theory and is then applied also to a problem in Stokes flow, yielding a procedure for obtaining the Stokes drag on a slender body of arbitrary shape. For comparison purposes, consideration is also given to the use of both cylindrical and di-polar co-ordinates, and as a by-product of the comparison of results on cylindrical and spheroidal systems some new simple formulae involving Legendre polynomials are obtained heuristically, and then rigorously proved.