The effect of thermal radiation is investigated for the axisymmetric flow over the blunt body associated with a given paraboloidal shock wave. Radiative transfer is treated by means of the differential approximation, which applies to multidimensional flow and is valid throughout the entire range of temperature and optical thickness. The gas is assumed to be perfect and optically grey, and molecular-transport processes are neglected. A semi-analytical solution for the flow and radiation fields is obtained by the method of series truncation.

Results are presented, in the strong-shock approximation, for various values of the appropriate dimensionless variables. In general, radiation is found to have a significant influence on temperature and density, moderate effect on velocity, and little effect on pressure. The stand-off distance between the shock wave and the body is found to decrease significantly with increasing radiation; the body shape is less affected. The anomalous behaviour of the gas temperature on the body streamline as obtained by earlier investigators in the optically thin case does not appear in the present work. The results thus show correct physical behaviour throughout the flow field for all values of optical thickness. The detailed flow quantities exhibit a number of features of multidimensional radiating flow. They also provide a check on the special assumptions made in other, more approximate treatments. Similarities between radiating flow and non-equilibrium reactive flow over blunt bodies are apparent.

The slow motion of fluid exterior to a cylinder lying on a wall is considered for a variety of boundary conditions. In particular, the solution is obtained for the case when the motion far from the cylinder is one of uniform shear. Calculations are made for the force and the moment exerted by the fluid on the cylinder. The asymptotic form of the flow both far from the cylinder and near the cusped corners is presented. The flow sufficiently near a cusp consists of a sequence of eddies of rapidly diminishing strength, and the solution of another boundary-value problem supports the view that the nature of this eddy system is independent of conditions far from the cusp. The nature of inviscid flow with uniform vorticity in cusped corners is also considered.

In experiments concerning the instability of free shear layers, oscillations have been observed in the downstream flow which have a frequency exactly half that of the dominant oscillation closer to the origin of the layer. The present analysis indicates that the phenomenon is due to a secondary instability associated with the nearly periodic flow which arises from the finite-amplitude growth of the fundamental disturbance.

At first, however, the stability of inviscid shear flows, consisting of a non-zero mean component, together with a component periodic in the direction of flow and with time, is investigated fairly generally. It is found that the periodic component can serve as a means by which waves with twice the wavelength of the periodic component can be reinforced. The dependence of the growth rate of the subharmonic wave upon the amplitude of the periodic component is found for the case when the mean flow profile is of the hyperbolic-tangent type. In order that the subharmonic growth rate may exceed that of the most unstable disturbance associated with the mean flow, the amplitude of the streamwise component of the periodic flow is required to be about 12 % of the mean velocity difference across the shear layer. This represents order-of-magnitude agreement with experiment.

Other possibilities of interaction between disturbances and the periodic flow are discussed, and the concluding section contains a discussion of the interactions on the basis of the energy equation.

Laminar incompressible flow past a semi-infinite flat plate is examined by using the method of series truncation (or local similarity) on the full Navier-Stokes equations. The first and second truncations are calculated at points on the plate away from the leading edge, while only the first truncation is calculated at the leading edge. The solutions are compared with the results from other approximate methods.

A new method using a matched asymptotic expansions technique is presented for obtaining the Stokes flow solution for a rigid sphere of radius a moving uniformly in a direction parallel to a fixed infinite plane wall when the minimum clearance ea between the sphere and the plane is very much less than a. An ‘inner’ solution is constructed valid for the region in the neighbourhood of the nearest points of the sphere and the plane where the velocity gradients and pressure are large; in this region the leading term of the asymptotic expansion of the solution satisfies the equations of lubrication theory. A matching ‘outer’ solution is constructed which is valid in the remainder of the fluid where velocity gradients are moderate but it is possible to assume that ε = 0. The forces and couples acting on the sphere and the plane are shown to be of the form (α0+α1ε) log ε + β0 + O(ε) where α0, α1 and β0 are constants which have been determined explicitly.

A theory of the generation of dispersive waves by travelling forcing effects, that may be steady, oscillatory or transient in character, is given for a general homogeneous system. Small disturbances to the system are supposed stable, and governed by a linear partial differential equation with constant coefficients which admits solutions in the form of plane waves satisfying an, in general, anisotropic dispersion relation P(σ, k) = 0 between frequency σ and wave-number vector k.

If the forcing region, supposed of limited extent, travels with constant velocity U, then oscillatory forcing terms of frequency σ0 (which would be replaced by 0 in the limiting case of a steady forcing effect, while taking, for a typical transient one of duration T, values from 0 to about 10/T) produce waves of frequency σ0 + U. k (the Doppler effect). For any such waves, the wave-number k satisfies the equation P(σ0 + U.k,k) = 0, representing a surface in wave-number space here called S(σ0), and their position relative to that of the forcing region is determined by having been generated when that region was in an earlier position, and having subsequently progressed with the group velocity. This implies the rule, also derived analytically in §§2 and 3, that waves with a particular value of k on S(σ0) are found in a direction, stretching out from the forcing region, which is one of the directions normal to S(σ0) at k, namely the one pointing towards S(σ0 + δ). This rule is supplemented by results on wave amplitudes and shapes of crests.

The theory is applied (§§4 and 5) to Rossby waves excited in a beta-plane ocean by travelling patterns of wind stress. If a steady wind-stress pattern moves westward, semicircular waves of length 2π √(U/β) trail behind it, but signals are found also directly ahead, consisting of the disturbance integrated in the west-east direction and subjected to a ‘low-pass filter’ with respect to its north-south components of wave-number. An eastward-travelling pattern, by contrast, produces only a wake-like disturbance, calculated in detail in §4. The waves generated for intermediate directions of travel are identified, and the strong tendencies in all cases for westward intensification of transient currents are noted.

For example, a wind-stress pattern travelling 30° N. of E. leaves a trailing wedge of currents from W. to 30° S. of W. in the steady case. The influence on this conclusion of a finite duration T of such a pattern is investigated in §5 by Fourier analysis in time. The fate of Fourier components of frequency σ0 depends on the ratio L =σ0/√(Uβ). If this is less than 1 for all (T, up to about 10/T, then the disturbance retains its trailing character; on the other hand, any components with L > 1 have a much greater directional spread. Tidal terms make fairly small changes to the results, except that (in an ocean of depth H) they make the directional spread disappear for L greater than about √(βgH/4f2U).

Excitation of gravity waves in non-rotating fluid is briefly considered, includ-ing generation on deep water by a travelling oscillating disturbance (§6), and generation in a uniformly stratified fluid by a vertically moving obstacle (§7). The predicted wave shapes in the latter case, with cusps at a finite distance behind the obstacle, agree excellently (figures 7 and 8) with experiments by Mowbray (1966).

An exceptional case, in that part of S(σ0) is doubly covered, is generation by steady (σ0 = 0) motion of an obstacle along the axis of uniformly rotating homo-geneous fluid, the surface S(0) being a sphere and two coincident planes. Whereas waves corresponding to points on the sphere trail behind the obstacle, the appro-priate normals on the two planes point in opposite directions inside the sphere (§8), permitting the well-known formation of the ‘Taylor column’ ahead of the obstacle at low Rossby numbers.

Still more complicated, because fully three-dimensional, is the case when the obstacle moves at right angles to the axis of rotation (§9). At finite though small Rossby number it is impossible for the Taylor column formed near the body to extend to large distances from it, where on the contrary the disturbance is shown to take the form of slightly trailing cones, shown in cross-section in figure 12, containing waves whose crests have cusps on the boundaries of the cones. An estimate of the length of the Taylor column, as body dimension divided by Rossby number (for small enough kinematic viscosity), is made by considering the fit between the Taylor-column and wave-cone regions.

The axisymmetric flow of fluid, confined within a rapidly rotating annulus and caused by a horizontally applied temperature gradient, is discussed under the assumptions of small thermal Rossby number and small Taylor number. It is initially assumed that the transfer of heat is purely conductive, and the flow is calculated for when the upper fluid surface is either rigid or free. The analysis of the problem in which the upper surface is rigid is extended so as to allow con-vective as well as conductive transfer of heat.

The velocity field generated in a fluid of viscosity, v, by impulsively starting at time t = 0, a sphere of radius a spinning with angular velocity Ω about a diameter is described using a new expansion variable 2 √vt/r. It is first shown how the standard time-dependent boundary-layer equations can be modified to give series solutions satisfying all the boundary conditions. Next, that these new solutions are relevant when the Reynolds number R = a2Ω/v goes to infinity in such a way that $R^{\frac{1}{3}} \Omega t$ is large. Lastly, solutions are given, applicable at small times for non-zero Reynolds numbers. These last expansions show that the velocity components decay algebraically rather than exponentially at large distances.

A method for obtaining analytic solutions to the problem of blunt bodies in the supersonic stream of an ideal gas is presented. The solutions are written in terms of power series whose coefficients are elementary functions. These solutions are approximate, but the approximation is rational, i.e. any higher approximation can, in principle, be obtained. Some of these higher approximations have been calculated. Examples are presented for various free-stream conditions and prescribed body shapes. These are compared with results from standard numerical procedure and with available experimental measurements.

Equations of motion are derived for long waves in water of varying depth. The equations are for small amplitude waves, but do include non-linear terms. They correspond to the Boussinesq equations for water of constant depth. Solutions have been calculated numerically for a solitary wave on a beach of uniform slope. These solutions include a reflected wave, which is also derived analytically by using the linearized long-wave equations.