A two-dimensional, moving, incompressible and electrically conducting fluid having trapped magnetic field (i.e. field threading the fluid) is confined by a uniform vacuum magnetic field. The fluid-vacuum field interface is horizontal and assumed to be free from instabilities. Dissipation by viscosity and resistivity is neglected, hence, for steady motion, the trapped field may be regarded as frozen into the fluid, in the sense that both move together. A pressure disturbance is introduced into the system, and it is found that a harmonic stationary wave is formed upstream provided certain criteria hold.

The mean-velocity field over monochromatic, 1·96 Hz, deep-water waves was measured by means of hot-wire anemometers for a range of wind speeds (relative to wave speed) of 0·4 to 3·0. The mean-velocity profile, over waves 0·64 cm in amplitude, was the same as that over a rough plate; that is, the mean velocity varied as the logarithm of the height above the mean-water level, except very close to the water, where the effect of the viscous sublayer became important. The wave-induced perturbation-velocity field and its associated Reynolds stresses were also measured and compared with numerical solutions of various linear equations governing shearing flow over a wavy boundary. The comparison showed that the measured velocity field was not well predicted by these theories.

The development of the boundary layer on the upper surface of a horizontal flat plate in a non-diffusive, stratified flow is described. It is shown that the flow can be characterized by two basic parameters, the Reynolds (RL) and Russell (RuL) numbers, and that, depending on the relative magnitude of these two parameters, three different régimes of flow can be defined. The delineation of these régimes and the description of the flow in each of them is obtained by deriving a uniformly valid first approximation to the Boussinesq equations of motion for a flow contained in the two-dimensional parameter space RuL > 0, RL > 1. The critical stratification for the self-blocking of a horizontal boundary layer is shown to be given by the condition RuL = O(RL½).

The existence of elementary, exponential solutions of the linear Boltzmann equation for gases is examined. Using the hard-sphere model of scattering, it is found that, in problems involving velocity perturbations, there are no discrete non-zero eigenvalues. Thus the relaxation to the asymptotic distribution is non-exponential and is described by the continuum eigenfunctions. For temperature perturbations, however, we find two non-zero discrete eigenvalues whose values are ±0·975 in units of the minimum scattering cross-section. Relaxation to the asymptotic distribution is therefore exponential, although still very rapid.

The conclusions stated above are based upon a truncation of the scattering kernel and a subsequent numerical solution of the resulting integral equations.

The boundary layer on the upper surface of a horizontal plane in a diffusive, stratified flow is examined. The analysis shows that density diffusion increases the role of the buoyancy forces and causes a significant change in the properties of the boundary layer when compared to the non-diffusive case. A uniformly valid first approximation for moderate Russell numbers is derived, and the effects of buoyancy and diffusion are evaluated by solving the resulting equations numerically.

An energy principle is presented which gives necessary and sufficient conditions for exponential stability for a large class of dissipative systems. The maximal growth rate Ω of an unstable system is shown to be the least upper bound of a certain functional, giving a variational expression for Ω. These results are used to discuss the gravitational stability of incompressible viscous fluids and resistive magnetofluids in arbitrary geometries.

In a horizontal convecting layer several distinct transitions occur before the flow becomes turbulent. These are studied experimentally for several Prandtl numbers from 1 to 104. Cell size plan form, transitions in plan form, transition to time-dependence, as well as the heat flux, are measured for Rayleigh numbers from 103 to 105. The second transition, occurring at around 12 times the critical Rayleigh number, is one from steady two-dimensional rolls to a steady regular cellular pattern. There is associated with this a discrete change of slope of the heat flux curve, coinciding with the second transition observed by Malkus. Transitions to time-dependence will be discussed in part 2.

An experimental investigation has been made of the flow patterns at the initiation of convection in a layer of a high Prandtl number liquid confined between rigid, horizontal surfaces and heated from below. The experiment was designed to overcome the limitations of earlier experiments and to correspond closely to the conditions of the theory. In particular, the upper and lower rigid surfaces which enclosed the layer were made of copper which has a high thermal conductivity. To aid in the analysis of the experimental results some supplementary observations of the flow patterns were made using a glass upper plate. For small fluid depths and large temperature differences between the upper and lower surface the initial flow was in the form of hexagonal cells as predicted theoretically. With increasing Rayleigh number the cellular flow appeared to transform into rolls as predicted. For large fluid depths and small temperature differences only circular plan-form rolls were observed. This is in agreement with the results of other experimenters. It is tentatively proposed that this non-appearance of an initial cellular flow is due to the shape of the test chamber having a dominating influence on the flow pattern when the temperature gradient in the fluid is small. Measurements were also made of the development time for the flow patterns and the critical Rayleigh number.

An extension to Coles's (1956) ‘law of the wall–law of the wake’ formulation for incompressible unblown boundary layers with momentum thickness Reynolds number Reθ > 6000. is made for Reθ < 6000. It is found that κ = 0·40, the von Kármán constant for Reθ > 6000, is replaced by Ω = 0·40 (Reθ/6000)⅛ for Reθ < 6000. Based upon the data of Simpson (1967) this formulation is extended to injection and undersucked (dθ/dx > 0) flows in ‘law of the wall’ and ‘velocity-defect’ representations. This law of the wall for the logarithmic turbulent region and Reichardt's sublayer variation of εM/ν are used to obtain a continuous expression for εM/ν as a function of U+, V+w, and Reθ for the wall region. This expression is in reasonable agreement with the generated εM/ν blowing results and in less agreement with the unblown and suction results. Eddy viscosity and mixing length results confirm that εM/δ*U∞ ∝ Ω2 and l/δ ∝ Ω for the outer region and that εM/δ*U∞ and l/δ are substantially independent of blowing and moderate suction, as also reflected by the velocity defect representation for injection and suction.

Several generalizations of theorems of the types originally stated by Helmholtz concerning the dissipation of energy in slow viscous flow have been given recently by Keller, Rubenfeld & Molyneux (1967). These generalizations included cases in which the fluid contains one or more solid bodies and drops of another liquid assuming the drops do not change shape. Some further extensions are given herein which allow for drops which may be deformed by the flow and include the effect of surface tension. The admissible boundary conditions have also been extended and particular theorems applicable to infinite domains, spatially periodic flows and to flows in infinite cylindrical pipes are derived. Uniqueness theorems are also proved.

Measurements of the velocity distribution close to the bed have been made under laminar flow conditions in a wave tank. The classical solution for the velocity distribution was found to be valid when the bed was smooth, but considerable deviations between theory and experiment were observed with beds of sand. It is suggested that these deviations were caused by vortex formation around the grains of sand. The similarity between the velocity profiles obtained in these tests and those reported by other writers under supposedly turbulent conditions suggests that even at high Reynolds numbers vortex formation may continue to be the dominant effect in oscillatory boundary layers of this sort.

The Rayleigh number at which steady convective flow changes to time-dependent flow is determined experimentally for several fluids with Prandtl numbers from 1 to 104. The time dependence is of two forms: (i) a slow tilting of the cell boundary, with time scale of the vertical thermal diffusion time, (ii) an oscillation with a faster time scale determined by the orbit time of the fluid around the cell. The nature of this oscillation is one of hot (or cold) spots advected with the original cellular motion. At a fixed point in the fluid this produces a time periodic oscillation of the temperature. A discrete change of slope of the heat flux curve accompanies this transition. As the Rayleigh number is increased, transition to disorder is seen to result from an increase in the frequency and number of these oscillations.

The Korteweg–de Vries equation is shown to govern formation of solitary and cnoidal waves in rotating fluids confined in tubes. It is proved that the method must fail when the tube wall is moved to infinity, and the failure is corrected by singular perturbation procedures. The Korteweg–de Vries equation must then give way to an integro-differential equation. Also, critical stationary flows in tubes are considered with regard to Benjamin's vortex breakdown theories.

The study of natural convection in a rectangular cavity whose side walls are maintained at different fixed temperatures can be regarded as one of the classical problems of thermal convection (see Batchelor 1954; Elder 1965 and Gill 1966). One aspect of this problem is the question of stability of the laminar flow solution for given values of the three governing parameters, the Rayleigh number A = γgΔTL3/κν, the aspect ratio h = H/L and the Prandtl number σ = ν/κ. Here H is the height and L the width of the cavity, T is the temperature difference between the two walls and g the acceleration due to gravity. The fluid filling the container has coefficient of expansion γ, thermal diffusivity κ and kinematic viscosity ν. Instead of studying the stability of the exact laminar flow solution, which is not a parallel flow, the stability of the following solution of the governing eauations is examined instead:

The three-dimensional steady flow of a shallow viscous liquid with non-uniform surface tension has been considered when the variation in surface tension results from the presence of an insoluble chemical contaminant on the surface. Similarly solutions for the particular problem of a channel flowing into a semi-infinite lake have been obtained, the depth and surface concentration at infinity being specified.

Non-linear Kelvin–Helmholtz instability, of two parallel horizontal streams of inviscid incompressible fluids under the action of gravity, is studied theoretically. The lower stream is denser and there is surface tension between the streams. Some progressing waves of finite amplitude are found as the development of a slightly unstable wave of infinitesimal amplitude. In particular, the non-linear elevation of the interface between the fluids is calculated. The finite amplitude of the waves does not equilibrate to a constant after a long time, but varies periodically with time. In practice, slight dissipation should lead to equilibration at an amplitude close to a value given by the present theory.

This paper considers inviscid film flows that originate by injection of fluid through the bounding surface. In the first part of the paper plates of an average inclination are considered. Power-law injection rates are studied in some detail. Later it is shown how the analysis has to be modified for almost horizontal plates.

A study is made of the laminar flow in the neighbourhood of the trailing edge of an aerofoil at incidence. The aerofoil is replaced by a flat plate on the assumption that leading-edge stall has not taken place. It is shown that the critical order of magnitude of the angle of incidence α* for the occurrence of separation on one side of the plate is $\alpha^{*} = O(R^{\frac{1}{16}})$, where R is a representative Reynolds number, for incompressible flow, and α* = O(R−¼) for supersonic flow. The structure of the flow is determined by the incompressible boundary-layer equations but with unconventional boundary conditions. The complete solution of these fundamental equations requires a numerical investigation of considerable complexity which has not been undertaken. The only solutions available are asymptotic solutions valid at distances from the trailing edge that are large in terms of the scaled variable of order R−⅜, and a linearized solution for the boundary layer over the plate which gives the antisymmetric properties of the aerofoil at incidence. The value of α* for which separation occurs is the trailing-edge stall angle and an estimate is obtained from the asymptotic solutions. The linearized solution yields an estimate for the viscous correction to the circulation determined by the Kutta condition.

An experiment is reported, in which turbulent shear-stresses as well as mean velocities have been measured in a three-dimensional turbulent boundary layer approaching separation. It is shown that even very close to the wall the stress vector does not align itself with the mean velocity gradient vector, as would be required by a scalar ‘eddy viscosity’ or ‘mixing length’ type assumption. The calculation method of Bradshaw (1969) is tested against the data, and found to give good results, except for the prediction of shear-stress vector direction.

The Stokes creeping flow, induced by the passage of a uniform current parallel to the axis of a stationary non-conducting ellipsoid of revolution in an incompressible viscous fluid occupying, apart from the ellipsoidal region, the whole space, is investigated. The magnetic field, which is due to the distortion of the uniform current by the ellipsoid, is zero all over the surface of the ellipsoid. The induced flow field is symmetric with respect to the axis, and also with respect to a plane through the centre perpendicular to the axis of the ellipsoid. The case of a non-conducting circular disk, with its plane perpendicular to the direction of the undisturbed current, is deduced from that of a planetary ellipsoid.