A study is made of the asymptotic equations governing gravity waves on water which cause a transition from one surface level to a slightly higher one and approach a steady wave-form as time increases, at least at the head of the wave. A two-parameter family of limit processes is surveyed in each of which the time scale and the horizontal length scale tend to infinity in a definite relation to the amplitude, as that tends to zero. Small-amplitude linearization is shown to be possible at most during a transitory stage of the wave development. Arbitrarily close approach to steadiness at the head of the wave is found to imply that a substantial part of the transition wave must be ultimately governed by a nearsteady variant of the non-linear equation of Korteweg & de Vries (1895) and must take the form of a train of cnoidal waves characterized by a parameter which changes slightly from crest to crest.

There are stationary solutions of finite amplitude convection in a layer of fluid heated from below which show increasing heat transport with decreasing Rayleigh number in the neighbourhood of its critical value. It is shown that those solutions are unstable and that convection with periodic time dependence can occur in these cases, when the heat flux is the given parameter instead of the temperature difference between the boundaries of the layer. The time dependence has been calculated explicitly for the case of convection with temperature variation of the material properties.

This paper is an analysis of the steady incompressible, two-dimensional flow of conducting fluids through ducts of arbitrarily varying cross-section under the action of a strong, uniform, transverse, magnetic field. More precisely, the flow is such that the velocity is given by ${\bf u} = (u_x(\tilde{x}, \tilde{y}), u_y(\tilde{x}, \tilde{y}), 0)$, the position of the duct walls by $\tilde{y} = f_t(\tilde{x}), f_b(\tilde{x})$ and $\tilde{z} = \pm b $, where b [Gt ] ft−fb, and the magnetic field by B0 = (0, B0, O). It is assumed that the magnetic field is strong enough to satisfy the conditions that the interaction parameter, N(=M2/R) [Gt ] 1, where M is the Hartmann number and R is the Reynolds number, and also that M [Gt ] 1 and Rm [Lt ] 1, where Rm is the magnetic Reynolds number.

We examine the flow in three separate regions:

1. the ‘core’ region in which the pressure gradient is balanced by electromagnetic forces;

2. Hartmann boundary layers where electromagnetic forces are balanced by viscous forces;

3. thin layers parallel to the magnetic field in which electromagnetic forces, inertial forces, and the pressure gradient balance each other. These layers which have thickness $O(N^{-\frac{1}{3}})$ occur where the slope of the duct wall changes abruptly.

By expanding the solution as a series in descending powers of N we calculate the velocity distribution for regions (i) and (ii) for finite values of N attainable in the laboratory.

The Hartmann oscillator problem is studied using the hydraulic analogy between compressible gasdynamics and incompressible flows with a free surface. Extensive photographs, taken of the oscillator in order to visualize the periodic flow, show that qualitative features of the flow agree well with the observations made with a cavity in an air jet.

The construction and operation of the water channel built and used at the University of California at Los Angeles for this study is also described. Continuous ‘shooting’ water flow was found to be analogous to supersonic isentropic gas flow; a static depth of approximately ¼ in. of water appeared to be satisfactory. It seems that the most valuable aspect of the hydraulic analogy is its ability to easily present excellent visualization of flow phenomena, both steady and unsteady.

The stress in a suspension of incompressible deformable particles exceeds that which would exist in the pure incompressible liquid undergoing the same flow by the product of the volume concentration of the particles and the tensor $\overline{p}_{ik} - 2\eta_0 e^{-(1)}_{ik}$ where η0 is the viscosity of the pure liquid, $\overline{p}_{ik}$ is the average stress and $e^{-(1)}_{ik}$ the average rate of strain in particles in the flowing suspension. For a dilute suspension of viscoelastic spheres these tensors can be determined by using an adaptation of Jeffery's solution of the problem of an isolated rigid ellipsoid. In steady laminar flow, the material of each sphere is continuously deformed and rotates within an ellipsoidal boundary of fixed dimensions and orientation. Approximate expressions are obtained for the steady-rate viscosity and normal stress differences in terms of the dynamic viscosity and dynamic rigidity functions of the suspension. These are valid either when the rate of shear is sufficiently small or when the ratio of the dynamic viscosity of the spheres to ηo is sufficiently large. The three normal stress components are all unequal. In slow steady elongation of the suspension, the spheres suffer static deformation into prolate spheroids so the elongational viscosity depends only on their static elastic properties. It appears from the investigation of special cases, however, that this static deformation is not possible for rates of elongation above a critical value.

Dynamically passive transfer of heat across a layer of liquid supporting a progressive, periodic surface wave is approached from a Lagrangian viewpoint. The model layer considered is the region between two constant pressure surfaces of a Gerstner wave and the thermal boundary conditions are that the average temperature of any surface particle remains constant and that there is horizontal homogeneity of the average temperature field.

It is shown that fluctuations in the temperature of any particle are negligibly small for ordinary liquids and a uniformly valid approximation to the average temperature of each particle is presented.

The extent to which the flux of heat through the layer is augmented is computed for typical cases and it is shown to be at most doubled. Indication is given of extensions of the method to other kinds of progressive waves and to situations in which the boundary conditions are unsteady and spatially inhomogeneous.

A stably stratified shear flow has the velocity profile V(1 - e−y/h) and the density profile log(ρ/ρ) = σ(1 − e−y/h) in 0 < y < ∞ and is bounded by the rigid plane y = 0. It is proved that small disturbances with respect to this basic flow are stable for all wavelengths and Richardson numbers. The eigenvalues for neutral disturbances (internal gravity waves) are enumerated. The results are applicable to the atmosphere and to an infinitely deep ocean.

Careful plume-rise observations by photographic means have been made on smoke plumes from the Lakeview Generating Station (Ontario) and compared with some existing theoretical formulae. Supporting data in considerable detail on stack parameters were available.

A linearly rising portion of the plume was found beyond a specific transition point in a neutral atmosphere. Also, the previously established simple power law $z \sim F^{\frac{1}{3}} U^{-1} x^{\frac{2}{3}}$, was found to agree quite well with observation up to the transition point.

In an unstable atmosphere the plume was sometimes above and sometimes below a corresponding plume in neutral conditions, under the opposing influences of increased dilution and the direct effect of instability in promoting plume rise.

An extension of numerical methods described in previous papers is used to analyse the time-dependent rotationally-symmetric motion of an incompressible viscous fluid contained between two concentric spheres having a common axis of rotation. The motion is governed by a pair of coupled non-linear partial differential equations in three independent variables, with singular end conditions. The computational process is described, and numerical solutions are presented for cases in which one (or both) of the spheres is given an impulsive change in angular velocity-starting from a state of either rest or uniform rotation. Reynolds numbers lie in the range 10-1500.

The possible occurrence of a viscous region near a surface from which there is rapid efflux of gas, accompanied by large heat transfer, is postulated and investigated theoretically. Such a viscous region, denoted as a boundary shock wave, may occur in the case of a large high-speed meteor entering the earth's atmosphere, when a very high rate of vaporization induces translational non-equilibrium. The conditions across a boundary shock wave and its structure are calculated from the appropriate macroscopic equations reduced to closed-form expressions under the restrictions of a perfect gas, flowing at constant total enthalpy.

A model of airflow over a mountain is treated mathematically in this paper. The fluid is inviscid, incompressible and of variable density. The flow is in a long channel, bounded above by a rigid horizontal lid and below by an obstacle. The variation with height of the horizontal velocity and of the density is specified far upstream. The details of flow are examined for particular conditions upstream which lead to a linear vorticity equation, although the non-linear inertial terms in the Euler equations of motion are exactly represented. In this case the flow is described by the superposition of solutions of some diffraction problems. Classical techniques of diffraction theory are then used to demonstrate the existence and some general properties of solutions for steady flow. Thus a steady solution is always possible if no restriction is placed on the amount of energy available to drive the flow, that is to say there is no critical internal Froude number (measuring the dynamical effect of buoyancy) for the existence of a steady flow. Finally the flows past a dipole and a vertical wall are computed.

Flow at high Hartmann number along the annular channel between two nonconducting cylinders of circular cross-section is examined. The core is divided and wakes appear at the boundaries of the various regions. In §§ 1-4 the case in which the cylinders are concentric is studied. In §5 the cylinders are eccentric, and it is shown that in this case there is a net flow of current around the annulus.

It is shown that a deformable body can move persistently from rest through a perfect fluid without having to produce vorticity in the fluid.

A general theory is given for the effect of solid walls on a translating and rotating particle in the limiting case of zero Reynolds number. Both the force and couple on the body are found as an expansion in terms of a parameter k = a/d, assumed small, where a is a characteristic particle size and d a characteristic distance of the particle from walls. It is shown how such expansions may be used in specific examples. The theory is then extended to include the general motion of a small particle in an arbitrary Stokes flow field in which solid or other boundaries are present. Finally the motion of two small bodies of arbitrary shape in an arbitrary Stokes flow field is considered.