An Advanced Study Institute sponsored by NATO on “Magnetohydrodynamic Phenomena in Rotating Fluids” was held in Cambridge, England, from 26 June to 4 July 1972. The aim of the meeting was to provide, through invited lectures, a systematic account of those aspects of the geophysics of the earth's core and of astrophysics in which both rotation effects and magnetic effects play an important part. In addition to the invited lectures, shorter contributions and discussions were strongly encouraged. The main Sessions and the Session Chairmen were as follows:

A    The earth's magnetism and planetary magnetism magnetism    Sir Edward Bullard

B    Convection, differential rotation, and magneto-hydrodynamics in the sun    Dr N. O. Weiss

C    Stellar and interstellar magnetism; pulsars    Prof. E. Spiegel

D    Boundary layers and detached shear layers and spin-up problems    Prof. E. R. Benton

E    Waves and instabilities influenced by Lorentz and/or Coriolis forces    Dr R. Hide

F    Dynamo theory    Prof. P. H. Roberts

These Chairmen, together with Professor P. A. Gilman and Dr H. K. Moffatt, who acted as Director of the Institute, constituted the Organising Committee. The summary of the meeting which follows has been substantially written by Dr Weiss (sessions B and C), Dr Hide (A and E), Professor Benton (D) and Professor Roberts (F), whose willing collaboration is gratefully acknowledged.

There were just over a hundred participants, the great majority of whom were accommodated at Trinity College, Cambridge, for the duration of the meeting.

An investigation of the stability of plane Couette flow with viscous heating of a Navier–Stokes–Pourier fluid with an exponential dependence of viscosity upon temperature is presented. Using classical small perturbation theory, the stability of the flow can be described by a sixth-order set of coupled ordinary differential equations. Using Galerkin's method, these equations are reduced to an algebraic eigenvalue problem. An eigenvalue with a negative real part means that the flow is unstable.

Neutral stability curves are determined at Brinkman numbers of 15, 19, 25, 30,40,80 and 600 for Prandtl numbers of 1, s and 50. A Brinkman number of 19 corresponds approximately to the maximum shear stress which can be applied to the system.

The results indicate that four different modes of instability occur: one termed an inviscid mode, arising from an inflexion point in the primary flow; a viscous mode, due to the stratification of viscosity in the flow field and an associated diffusive mechanism; a coupling mode, resulting from the convective and viscous dissipation terms in the energy equation; and finally a purely thermal mode.

A photographic study has been made of Kelvin-Helmholtz waves on thin liquid sheets and it has been found that, contrary to two-dimensional theory, wave growth is critically dependent upon sheet velocity and distance from the origin. This is attributed to boundary-layer separation and to subsequent vortex growth.

A consistent slender-body approximation is developed for the flow past a fish- like body with arbitrary combinations of body thickness and low-aspect-ratio fin appendages, but with the fins confined to the plane of symmetry of the body. Attention is focused on the interaction of the fin lifting surfaces with the body thickness, and especially on the dynamics of the vortex sheets shed from the fin trailing edges. This vorticity is convected by the (non-lifting) flow past the stretched-straight body, and departs significantly from the purely longitudinal orientation of conventional lifting-surface theory. Explicit results are given for axisymmetric bodies having fins with abrupt trailing edges, and calculations of the total lift force are presented for bodies with symmetric and asymmetric fin configurations, moving with a constant angle of attack.

Linearized two-dimensional potential flow theory is applied to an airfoil with an upper surface spoiler. The spoiler wake is modelled as a cavity of empirically given constant pressure, and a sequence of conformal transformations maps the linearized physical plane, with a slit on the real axis representing the airfoil plus cavity, onto the upper half of the plane exterior to the unit circle. The complex acceleration potential is used, and its real part is specified on the real axis, repre- senting the cavity boundary, while its imaginary part is specified on the unit semicircle, representing the wetted surface of the airfoil and spoiler. Solutions are found for both the steady-state lift and the transient lift after spoiler actuation for airfoils of arbitrary camber, thickness and incidence, with and without a simple flap, and with spoilers of arbitrary position, height and angle. The empirical cavity pressure is arbitrary for the steady-state solution, but is assumed to have the free-stream value for the transient solutions. Comparisons are made with the results of wind-tunnel experiments, and, for the steady-state solutions, with predictions of an earlier theory. The agreement of the present theoretical predictions with the experimental results is generally good, and is in most cases somewhat better than that of the earlier theory.

Here, we have studied the propagation of an arbitrary disturbance bounded in space on an arbitrary two- or three-dimensional transonic flow. First we have presented a general theory valid for an arbitrary system of n first-order quasi- linear partial differential equations and then used the theory for the special case of gasdynamic equations. If a disturbance is created in the neighbourhood of a sonic point, only a part of the disturbance stays in the transonic region and it is bounded by a wave front perpendicular to the streamlines. This part of the disturbance is governed by a very simple partial differential equation and the problem essentially reduces to the discussion of one-dimensional waves. The disturbance decays in the neighbourhood of the points where the flow acceler- ates from a subsonic state to a supersonic state and it attains a steady state where the flow is decelerating.

Free-surface deformations accompanying natural convection in shallow liquid pools heated from below by a straight wire are studied. Two-dimensional flow ensues as density gradients are set up in the bulk liquid and surface tension gradients are set up in the interface. Surface profiles are measured experimentally using a reflecting schlieren technique. Profiles concave above the rising warm current are produced in the shallower pools and convex profiles in the deeper pools. The profile for intermediate depths is a hybrid with a shallow crest rising from a shallow trough. A mathematical model of the system is developed and yields numerical results in qualitative agreement with experiment. The import- ance of various fluid properties and system parameters on the surface profile is examined using further calculations based on the model.

A dilute suspension of rigid axisymmetric particles in a time-dependent shear flow is considered in circumstances where the shear flow alignment of the particles dominates small but not unimportant Brownian disorientations. Three cases are studied: stress relaxation on the cessation of a steady shear, the sudden application of a steady shear from a state of rest and the sudden change from one steady shear rate to another. The rheology exhibits effects on two basic time scales provided that the particle aspect ratio r is not extreme: oscillatory features with a frequency proportional to the shear rate γ which are due to the rotation of the particles about their Jeffery orbits, and an exponentially fading memory due to Brownian diffusion with a characteristic time praportional to the inverse diffusion rate D−1. When the particle aspect ratio r becomes large, the oscillation frequency is reduced to γ/r while the diffusion rate is enhanced to Dr2 for some features and to Dr4 for others.

This paper complements earlier papers by Van Dyke (1970) and by Wilson (1971) which have appeared under the same title. Second-order boundary-layer theory is used to examine the region near the entrance to a single channel placed in a uniform stream. It is found that there are additional effects to those present in the three models treated by Van Dyke and Wilson. In particular, the cascade model misses the leading term in the separation force while the irrotational-entry model misses that in the skin friction.

There are also two new effects far downstream: logarithmic terms appear (apparently for the first time in second-order theory); and a resonance with the first eigensolution occurs.

A numerical method for solving the time-dependent Navier–Stokes equations in two space dimensions at high Reynolds number is presented. The crux of the method lies in the numerical simulation of the process of vorticity generation and dispersal, using computer-generated pseudo-random numbers. An application to flow past a circular cylinder is presented.

The propagation of nonlinear gravity–capillary surface waves in a deep slowly varying current is investigated using the conservation equations in the eikonal approximation. Graphical comparisons are made between solutions of the wave- slope and wavenumber equations for infinitesimal waves and finite amplitude waves. Finite amplitude effects are shown to be weaker for small amplitude capillary waves than for gravity waves. The ‘wave barrier’ noted by Gargett & Hughes (1972) for infinitesimal gravity waves on a slowly varying current is seen to be removed by finite amplitude effects.

A boundary layer flows over a flat plate which has on it a small hump situated downstream of the leading edge. The description of the boundary-layer flow, based upon a triple-deck structure, shows how the presence of the hump generates an interaction between the inviscid region just outside the layer and the viscous region near the hump. The pressure force dominant in the boundary layer and the connexion of the local flow with the main stream develop together and are self-perpetuating, and both remain of primary significance for a wide range of hump sizes, even for a hump buried well inside the boundary layer. By consideration of the limiting cases of very small and very large humps, a consistent account of the nature of the disturbances due to the various sizes of hump is produced. The forces and couples on the hump are also evaluated.