Three-dimensional steady flow past a body placed in a uniform stream of viscous, thermally conducting fluid is considered within the framework of the Oseen approximation. Asymptotic forms for the fundamental matrix are obtained for both supersonic and subsonic flow. It is shown how the solution to the flow past a body may be obtained from the fundamental matrix, and that the fundamental matrix itself provides the far field flow.

This paper presents the practical and rigorous solution of the potential flow problem associated with the oscillation of a shallow-draft cylinder of infinite length on a free surface. The problem is three-dimensional to the extent that the amplitude of the cylinder oscillation is periodic along its axis as well as with time. The complementary problem associated with the interaction of the fixed cylinder with an incident wave train aligned at some oblique angle with respect to the cylinder axis is also treated. The use of a Green's function reduces the problem to an integral equation which is solved numerically. Numerical results are computed for pressure amplitude distributions, force coefficients, added mass and damping coefficients, transmission and reflexion coefficients and wave height ratios.

The flow produced by a disk performing small oscillations in a rotating system is considered. Results are obtained for the first-order harmonic velocity and the second-order steady velocity. It is then shown that this mathematical model does not always represent an axially bounded fluid in the limit of infinite separation, and to be general one must allow a steady, azimuthal, perturbation velocity with an arbitrary value to exist at infinity.

The rotating system of an infinite disk beneath an unbounded fluid can exhibit resonance if the disk performs torsional oscillations at a certain frequency. This effect is examined in detail, and the solution is shown to depend crucially upon the existence of a small, steady departure from the basic rotational state in the far field.

The dispersion relationship for plane hydromagnetic waves in a stratified rotating fluid (α) indicates that the well-known analogy between rotating fluids and stratified fluids in regard to their hydrodynamic behaviour does not extend to magnetohydrodynamic behaviour, and (b) lends credence to a certain conjecture made in a previous paper, namely that effects due to density stratification can be neglected when considering the dispersion relationship for free hydromagnetic oscillations of the Earth's core if the Brunt—Väisälä frequency is much less than twice the angular speed of the Earth's rotation.

The structure of a strong blast wave under the influence of an expanding inner contact surface is studied asymptotically in the Newtonian limit: $\epsilon \equiv (\gamma - 1)/2\gamma \ll 1, \epsilon \dot{y}^2_s \gg a^2_{\infty}$. The theory treats the interaction of a shock layer and an inner flow region (the entropy wake) and reduces the problem to an ordinary differential equation for the shock radius. The pressure–volume relation of Cheng et al. (1961) is recovered and extended to a higher order of ε.

It is shown that, depending on the rate of growth of the contact surface, the shock layer may ‘reattach’ to the surface at large time. In a number of cases, the reattachment is approached in an oscillatory manner which leads to a period of non-uniformity. The associated problem of multiple time scales (treated in sequels to this paper) is identified.

The reattachment of a supersonic jet with a turbulent boundary layer abruptly expanding into an axisymmetric parallel diffuser has been experimentally investigated using a surface-flow technique. Measurements were made in the started condition, where the blowing pressure is sufficiently high to establish an oblique shock system in the diffuser. The proposed reattachment criterion correlates present measurements in terms of the diffuser area ratio, and also those of other workers for unconfined flow in terms of the free stream Mach number after separation. As already reported for unconfined flow, it is found that disturbances downstream of reattachment do not affect the upstream region.

The problem of the reflexion of tides in an enclosed sea such as the North Sea at a point at which it either enters the ocean or its width suddenly increases is considered by investigating the reflexion of a Kelvin wave at the open end of a rotating uniform semi-infinite channel.

It is shown that for a given channel, if the wave period is less than a pendulum day, then, according to the linearized theory of long waves in a rotating system, the reflexion coefficient increases with the angular velocity of rotation. It is also shown that there is a resonance effect for certain critical channel widths, namely, those at which extra modes within the channel become possible.

We consider the propagation of waves of small finite amplitude ε in a gas whose internal energy is characterized by two temperatures T (translational) and Ti (internal) in the form e = CvfT + CvfTi, and Ti is governed by a rate equation dTi/dt = (T − Ti)/τ. By means of approximations appropriate for a wave advancing into an undisturbed region x > 0, we show that to order εδ, the equation satisfied by velocity takes the non-linear form \[ \bigg(\tau\frac{\partial}{\partial t}+1\bigg)\bigg\{\frac{\partial u}{\partial t}+\bigg(a_1+\frac{\gamma + 1}{2}u\bigg)\frac{\partial u}{\partial x}-{\textstyle\frac{1}{2}}\lambda\frac{\partial^2u}{\partial x^2}\bigg\}=(a_1-a_0)\frac{\partial u}{\partial x}, \] where a1, a0 are the frozen and equilibrium speeds of sound in the undisturbed region, δ = ½(1 − (a20/a21)), and λ is the diffusivity of sound due to viscosity and heat conduction (λ may be neglected except when discussing the fine structure of a discontinuity). Some numerical solutions of this model equation are given.

When ε is small compared with δ, it is also possible to construct a solution for the flow produced by a piston moving with a constant velocity by means of a sequence of matched asymptotic expansions. The limit reached for large times for either compressive or expansive pistons is the expected non-linear solution of the exact equations. For a certain range of advancing piston speeds, this is a fully dispersed wave with velocity U in the range a0 < U < a1. If U > a1 the solution is discontinuous, and indeterminate in the absence of viscosity; a singular perturbation technique based on λ is then used to determine the structure of the wave head.

The energy theory, giving a sufficient condition for stability, is developed for the motions in a horizontal, heated layer subject to buoyancy and surface tension effects. The free surface is assumed to be non-deformable (Pearson's 1958 model).

It is shown that the equations governing the energy theory are the symmetric part of the time-independent linear theory problem, and that the surface tension terms behave like a bounded perturbation to the Bénard problem. The qualitative behaviour of the optimal stability boundary as a function of its parameters is given. The optimal stability boundary is computed, and compared with previous linear and non-linear stability theories in terms of allowable subcritical instabilities.

An electron beam technique was used to measure density profiles of strong shock waves in argon with high accuracy. The experimental results are compared with the results of theoretical models. Of the models that are available in enough detail for comparison, the best agreement with experiment is shown by the direct simulation Monte-Carlo method (Bird 1968), assuming an inverse 12th power molecular interaction force law. It is shown that the density maximum slope thickness is not sufficient for a detailed description of the shock wave structure.

A Symposium on ‘The flow of fluid-solid mixtures’ was held at the University of Cambridge from 24 to 28 March 1969, under the auspices of the International Union of Theoretical and Applied Mechanics. There were 104 participants, representing 19 countries, and attendance was by invitation only. Since there will be no publication of the proceedings in full, the following condensed account of the developments described at the various sessions has been prepared for publication by three of the participants, all of whom were involved in the organization of the Symposium.

Certain aspects of the transport of solid particles by a turbulent airstream are discussed, namely: the conveyance of particles in a horizontal pipe, including those carrying an appreciable electrostatic charge; the mechanism of deposition onto a solid wall; and the behaviour of fine particles in a shear flow, such as that in a round jet.

Rough estimates of the effect of the particles on the gaseous turbulence are made, and a primitive physical explanation is offered of the observed velocitylag and pressure drop associated with the transport of particles in a horizontal pipe, under conditions where the influence of the particles’ weight is significant.

Attention is drawn to the difficult problem of dynamically scaling a two-phase flow, and to the different types of interaction between the phases which can occur in a pipe according to its size, the gas velocity through it, and the physical characteristics of the particles.

The paper is an annotated version of a survey presented to the I.U.T.A.M. Symposium on ‘Flow of fluid-solid mixtures’ held in Cambridge during March 1969.