The classical theory of Brownian motion applies to suspensions which are so dilute that each particle is effectively alone in infinite fluid. We consider here the modifications to the theory that are needed when rigid spherical particles are close enough to interact hydrodynamically. It is first shown that Brownian motion is a diffusion process of the conventional kind provided that the particle configuration does not change significantly during a viscous relaxation time. The original argument due to Einstein, which invokes an equilibrium situation, is generalized to show that the particle flux in probability space due to Brownian motion is the same as that which would be produced by the application of a certain ‘thermodynamic’ force to each particle. We then use this prescription to deduce the Brownian diffusivities in two different types of situation. The first concerns a dilute homogeneous suspension which is being deformed, and the relative translational diffusivity of two rigid spherical particles with a given separation is calculated from the properties of the low-Reynolds-number flow due to two spheres moving under equal and opposite forces. The second concerns a suspension in which there is a gradient of concentration of particles. The thermodynamic force on each particle in this case is shown to be equal to the gradient of the chemical potential of the particles, which brings considerations of the multi-particle excluded volume into the problem. Determination of the particle flux due to the action of this force is equivalent to determination of the sedimentation velocity of particles falling through fluid under gravity, for which a theoretical result correct to the first order in volume fraction of the particles is available. The diffusivity of the particles is found to increase slowly as the concentration rises from zero. These results are generalized to the case of a (dilute) inhomogeneous suspension of several different species of spherical particle, and expressions are obtained for the diagonal and off-diagonal elements of the diffusivity matrix. Numerical values of all the relevant hydrodynamic functions are given for the case of spheres of uniform size.

The transition of homogeneous turbulence from an initially isotropic three-dimensional to a quasi-two-dimensional state is simulated numerically for a conducting, incompressible fluid under a uniform magnetic field B0. The magnetic Reynolds number is assumed to be small, so that the induced fluctuations of the magnetic field are small compared with the imposed magnetic field B0, and can be computed from a quasi-static approximation. If the imposed magnetic field is strong enough, all variations of the flow field in the direction of B0 are damped out. This effect is important e.g. in the design of liquid-metal cooling systems for fusion reactors, and the properties of the final state are relevant to atmospheric turbulence. An extended version of the code of Orszag & Patterson (1972) is used to integrate the Navier-Stokes equations for an incompressible fluid. The initial hydrodynamic Reynolds number is 60. The magnetic interaction number N is varied between zero and 50. Periodic boundary conditions are used. The resolution corresponds to 323 points in real space. The full nonlinear simulations are compared with otherwise identical linear simulations; the linear results agree with the nonlinear ones within 3% for about one-fifth of the large-scale turnover time. This departure is a consequence of the return-to-equilibrium tendencies caused mainly by energy transfer towards high wavenumbers. The angular energy transfer and the energy exchange between different components are smaller, and become virtually zero for large values of N. For N ≈ 50 we reach a quasi-two-dimensional state. Here, the energy transfer towards high wavenumbers is reduced for the velocity components perpendicular to B0 but relatively increased for the component parallel to B0. The overall behaviour is more similar to three-than to purely two-dimensional turbulence. This finding is of great importance for turbulence models of the atmosphere. The realization of a purely two-dimensional state does not seem to be possible for decaying turbulence. The magnetic field causes highly intensified pressure fluctuations, which contribute to the redistribution of the anisotropic Lorentz forcing.

The viscous boundary layer on a finite flat plate in a stream which reverses its direction once (at t = 0) is analysed using an improved version of the approximate method described earlier (Pedley 1975). Long before reversal (t < −t1), the flow at a point on the plate will be quasi-steady; long after reversal (t > t2), the flow will again be quasi-steady, but with the leading edge at the other end of the plate. In between (−t1 < t < t2) the flow is governed approximately by the diffusion equation, and we choose a simple solution of that equation which ensures that the displacement thickness of the boundary layer remains constant at t = −t1. The results of the theory, in the form of the wall shear rate at a point as a function of time, are given both for a uniformly decelerating stream, and for a sinusoidally oscillating stream which reverses its direction twice every cycle. The theory is further modified to cover streams which do not reverse, but for which the quasi-steady solution breaks down because the velocity becomes very small. The analysis is also applied to predict the wall shear rate at the entrance to a straight pipe when the core velocity varies with time as in a dog's aorta. The results show positive and negative peak values of shear very much larger than the mean. They suggest that, if wall shear is implicated in the generation of atherosclerosis because it alters the permeability of the wall to large molecules, then an appropriate index of wall shear at a point is more likely to be the r.m.s. value than the mean.

This work is concerned with the dispersion of a buoyant solute in a straight horizontal pipe of circular cross-section where dispersion is affected by molecular diffusion, the laminar flow along the pipe and density currents. Erdogan & Chatwin (1967) have derived an equation for the mean concentration $\overline{C}$ of a buoyant solute using a relatively simple asymptotic model, and have predicted that the dispersion induced by buoyancy effects depends on the Péclet number of the flow. In this part of this study an approximate expression is derived for $\overline{C}$ from Erdogan & Chatwin's equation, and an asymptotic form is obtained for the second moment of distributions of buoyant solutes. The examination of the second moment leads to a simple, but important, result: the dispersion induced by density currents at large times is small compared with the dispersion induced by density currents at times when transient effects are significant.

Chatwin (1970) has described the approach to normality of a cloud of solute which is injected into a pipe containing solvent in steady laminar flow. This paper is concerned with the modification of Chatwin's theory when there is a small density difference between the solvent and the dissolved solute. Asymptotic series are derived for the induced density currents and for the distribution of solute in the case when the molecular diffusivity is constant throughout the pipe's cross-section. These series lead to the asymptotic forms of the moments of the distribution, thereby describing some additional deviations from normality caused by buoyancy effects at large times. The theory predicts that the additional dispersion due to buoyancy effects is proportional to the square of the Rayleigh number and depends on the Péclet number of the flow. There is excellent agreement between the results and those previously obtained by the author (1976) from Erdogan & Chatwin's (1967) model of dispersing buoyant solutes. The results confirm that Erdogan & Chatwin's intuitive theory correctly models the significant features of the situation for large Schmidt numbers.

A constant-pressure axisymmetric turbulent boundary layer along a circular cylinder of radius a is studied at large values of the frictional Reynolds number a+ (based upon a) with the boundary-layer thickness δ of order a. Using the equations of mean motion and the method of matched asymptotic expansions, it is shown that the flow can be described by the same two limit processes (inner and outer) as are used in two-dimensional flow. The condition that the two expansions match requires the existence, at the lowest order, of a log region in the usual two-dimensional co-ordinates (u+, y+). Examination of available experimental data shows that substantial log regions do in fact exist but that the intercept is possibly not a universal constant. Similarly, the solution in the outer layer leads to a defect law of the same form as in two-dimensional flow; experiment shows that the intercept in the defect law depends on δ/a. It is concluded that, except in those extreme situations where a+ is small (in which case the boundary layer may not anyway be in a fully developed turbulent state), the simplest analysis of axisymmetric flow will be to use the two-dimensional laws with parameters that now depend on a+ or δ/a as appropriate.

The large negative Soret coefficient of 1N-LiI gives rise in a Rayleigh-Bénard experiment to a density distribution which is observed to stabilize the fluid layer for values of the Rayleigh number as large as 196 times the value of 1708 for the onset of convection in a pure fluid. The Soret transport also affects the convective heat flux. A power law relating heat flux and temperature difference is found with the same exponent as is found in pure fluids but with a lower value of the multiplicative constant. The Rayleigh number at the onset of power-law behaviour depends on the Soret coefficient. Three types of oscillation are seen: transient oscillations at onset, low frequency fluctuations at low Rayleigh number, and higher frequency oscillations similar to those observed in pure water. The intermediate state found after onset in NaCl solutions is not found in LiI.

Analytic solutions are obtained for non-equilibrium dissociating flow of an inviscid Lighthill-Freeman gas after a curved shock, by dividing the flow into a thin reacting layer near the shock and a frozen region further downstream. The method of matched asymptotic expansions is used, with the product of shock curvature and reaction length as the small parameter. In particular, the solution gives expressions for the reacting-layer thickness, the frozen dissociation level, effective shock values of the frozen flow and the maximum density on a stream-line as functions of free-stream, gas and shock parameters. Numerical examples are presented and the results are compared with experiments.

The two-dimensional theory of lunate-tail propulsion is extended to motions of arbitrary amplitude, regular or irregular, so that an accurate comparison may be made with the actual lunate-tail propulsion of scombroid fishes and cetacean mammals. There is no restriction at all on the amplitude of motion but the tail's angle of attack relative to its instantaneous path through the water is assumed to remain small. The theory is applied to the regular finite amplitude motion of a thin aerofoil with a rounded leading edge to take advantage of the suction force and a sharp trailing edge to ensure smooth tangential flow past the rear tip. This can represent the vertical motions of the horizontal lunate tails of large aspect ratio with which cetacean mammals propel themselves or the horizontal undulations of the vertical lunate tails of certain fast fishes. The dependence of the thrust, the hydromechanical propulsive efficiency and the energy wasted in churning up the eddying wake on the reduced frequency, the angle of attack and the amplitude of motion is exhibited.

The frequencies of the first four sloshing internal wave modes in two superposed fluid layers contained in a circular channel are calculated for two positions of the free surface and for various ratios of the depths of the two layers. Flow patterns are given for the first four sloshing modes for the case in which the fluids occupy a semicircular space and the depth of the upper layer is one-quarter of the radius.

It is hoped that the results obtained will provide a guide for estimating the frequencies of sloshing internal wave modes in long lakes.