Resonances are discussed between electrons and protons and waves of whistler and hydromagnetic types. The conditions of resonance are investigated. Approximations for the effects on the particles are obtained. The occurrence of a ‘trapping’ phenomenon is demonstrated and the effect on velocity distributions is discussed.

This study concerns the stability of the steady laminar boundary-layer flow of a homogeneous fluid which occurs in a rotating system when the relative flow is slow compared to the basic speed of rotation. Such a flow is called an Ekman boundary-layer flow after V. W. Ekman who considered the theory of such flows with application to the wind-induced drift of the surface waters of the ocean.

Ekman flow was produced in a large cylindrical rotating tank by withdrawing water from the centre and introducing it at the rim. This created a steady-state symmetrical vortex in which the flow from the rim to the centre took place entirely in the shallow viscous boundary layer at the bottom. This boundary-layer flow became unstable above the critical Reynolds number $Re_c = vD|v = 125 \pm 5$ where v is the tangential speed of flow, $D = (v| \Omega)^{\frac {1}{2}}$ is the characteristic depth of the boundary layer, v is the kinematic viscosity, and Ω is the basic speed of rotation. The initial instability was similar to that which occurs in the boundary layer on a rotating disk, having a banded form with a characteristic angle to the basic flow and with the band spacing proportional to the depth of the boundary layer.

The present paper discusses the hydrodynamics and propulsive properties when a jet of fluid is ejected from the opening of a tube. Formulas for the calculation of the thrust are provided and the basic equations for the horizontal rectilinear motion of a rigid torpedo-like body are studied in some detail. The results may be applied to investigate in an elementary way the locomotion of certain aquatic animals.

An explicit determination is made of the velocity potential for the small-amplitude time-periodic excitation of a bottomless, heavy, incompressible fluid which results from an internal point source, assuming that the equilibrium free surface lies in a horizontal plane and taking account of the presence within the fluid of a thin rigid plane with a vertical straight edge. The surface-wave component of the potential is expressed by means of single quadratures for any relative disposition of the source and observation points and, apart from exponential factors involving the depths of the respective points, proves to have the same functional character as the steady-state velocity potential for the acoustic (or compressional) motion which is sustained by an infinite line source parallel to the straight edge of a thin rigid half-plane.

The effect of harmonic oscillations in the magnitude of the surface temperature on the free-convection laminar velocity and temperature boundary layers on a flat plate is analysed. Low-and high-frequency solutions are developed separately. The results obtained are in striking contrast to the corresponding results for forced-convection flows.

The principal object of study is plane flow over bodies with a sharp apex at Mach numbers greater than unity. The magnetic field is assumed to be uniform, rectilinear, and parallel to the undisturbed stream. Flow behaviour near the apex of a wedge is investigated by the method of characteristics. It is found that for small wedge angles an attached shock attenuates initially with distance from the apex, but for larger wedge angles the shock grows stronger.

The structure of a slow magneto-gasdynamic shock is investigated for the case of strong magnetic field and small electrical conductivity. The streamlines are displaced within the shock although the initial and final flow directions are the same. An ordinary gasdynamic shock may occur on the upstream side of the transition. The shock structure theory gives a solution for the flow near the apex of a certain class of bodies.

For the study of slow shock structure, it is shown that the transition is described by a curve in the (F, H)-plane. F is the sum of pressure and momentum flux in the direction of variation; H is the sum of enthalpy and kinetic energy due to the velocity component in the direction of variation. General properties of the (F, H)-plane are found for a gas whose equation of state obeys the conditions of Weyl (1949). Flow behaviour on the transition curve is then determined. The theory of the (F, H)-plane can be used in the study of other one-dimensional processes in magneto-gasdynamics.

The problem of the turbulent boundary layer in two-dimensional, incompressible flow, which is continuously critical, i.e. on the point of separating, over part of its length is solved by combination of the Buri separation criterion with the boundary-layer momentum equation. In the latter an attempt is made to allow for transverse pressure variation and Reynolds stresses by over-estimating the contribution of skin friction. The Buri constant is taken to have a value of Γ =−0·04. Whereas Stratford and Townsend assumed that an initially noncritical boundary layer could be transformed instantaneously into a critical state by the application of an infinite pressure gradient at a point, in the present approach it is considered necessary to induce a critical state over a finite length, this state then being maintained downstream. These solutions are thus not comparable near the starting-point of the flow, but fair agreement is suggested at large distances downstream, with Stratford and Townsend's theoretical solutions and with Stratford's experimental data.

Comparison is also made with an ideal pressure distribution suggested by Stuart and the basis of optimization is discussed. It is suggested that the optimum flow might be similar in form to the continuously critical flow. The thesis that a continuously critical boundary-layer flow is ideal for large extents of the suction surface of an aerofoil is denied.

The paper concludes that the Buri criterion is valid but that in this case the Buri constant has a value of -0·04 rather than the accepted value of -0·06. Previous work by the author in connexion with cascade aerofoils is justified. The analysis predicts theoretical minimum diffusion lengths for given pressure rises provided that the initial boundary layer has achieved a critical condition.

It is shown that the turbulent wake of a self-propelled body moving in a fluid with a vertical density gradient is considerably different than of the same body moving in a fluid having no density gradient. In the uniform density case, the turbulent mixed fluid behind the body expands into an irregular conical shape. In the case of a density gradient, the initial expansion of the mixed fluid is quickly followed by a collapse in the vertical direction which is accompanied by a further spreading in the horizontal direction. This phenomenon is caused by the force of gravity. The volume of fluid behind the self-propelled body has a more or less constant density due to mixing. Thus, it is forced to seek its own density level in the undisturbed fluid.

The collapsing vertical wake is shown to be an efficent generator of internal waves, many of high order. These manifest themselves in surface movements.

The assumption that the internal waves are damped only by viscosity, not by turbulence, leads to results in general accord with the observations.

It is shown that the optimum design of duct in a magnetohydrodynamic generator is close to the one in which the flow Mach number remains constant. This constant-Mach-number generator is analysed in some detail and it is shown that the optimum Mach number can be defined to within a few percent. For a γ of 1·25, this optimum is near 0·85. For very short ducts, the maximum power output is obtained near matched-load conditions but for rather longer ones maximum total power output is obtained by working as close to short-circuit conditions as is practicable. Against this, the minimum compressor requirements are found by working as close to open-circuit conditions as is practicable, and so a compromise must be reached for optimum overall generator design as far as load conditions are concerned. This will probably give an internal ohmic loss in the fluid of about one-third of the total output. Curves are presented which enable the optimum Mach number to be determined with greater precision when the optimum load conditions have been selected.

The wall jet with suction or injection is investigated; an analysis under conditions corresponding to similar flows is shown to reduce to an eigenvalue problem. Asymptotic solutions valid far from the surface are used to initiate the integration and circumvent the usual iteration associated with the two-point boundary-value problem. Typical solutions for various rates of suction and injection are obtained. It is found that the skin friction decreases with increasing rate of suction. Representative thermal solutions are obtained for Prandtl and Lewis numbers equal to one, under the special condition that the surface temperature is equal to the ambient temperature or that the enthalpy varies monotonically from the surface value to the ambient value.

The flow of liquid helium II at subcritical velocities can be described by hydrodynamic equations which assume the possibility of independent motions of the two fluid components, but difficulties arise in supercritical flows with forces of mutual friction between the component fluids. The extensive evidence that mutual friction is caused by scattering of the thermal excitations in the velocity fields of the quantized vortex lines suggests that the equations should include mutual forces which are large near the vortex-lines and negligible elsewhere. In their original theory, Hall & Vinen (1956) did assume this, and here the implications of the assumption are examined in detail. Serious difficulties are found in reconciling localization of the mutual friction with fully developed flow in channels, and it is suggested that, although the mutual friction arises from scattering near the vortex lines, the force on the superfluid is distributed uniformly over distances comparable with the distance between adjacent vortex lines.

The motion of a viscous, electrically conducting fluid past a finite two-dimensional obstacle is investigated. The magnetic field is assumed to be uniform and parallel to the velocity at infinity. By means of a perturbation technique, approximations valid for large values of the Hartmann number M are derived. It is found that, over any finite region, the flow field is characterized by the presence of shear layers fore and aft of the body. The limit attained over the exterior region represents the two-dimensional counterpart of the axially symmetric solution given by Chester (1961). Attention is focused on a number of nominally ‘higher-order’ effects, including the presence of two distinct boundary layers. The results hold only when M [Gt ] Re; Re = Reynolds number. However, a generalization of the procedure, in which the last assumption is relaxed, is suggested.

In this paper, we consider the elementary processes involved in the appearance of a simple twin wake in the two-dimensional steady aligned flow of an electrically conducting viscous incompressible fluid in a uniformaly applied magnetic field. Fundamental relations between the twin wake and its elementary process are generally discussed, and the corresponding initial-value problem is formulated. Then, the solution of the initial-value problem corresponding to the above simple twin wake is obtained, and two limiting cases with the magnetic Prandtl number near one and zero (or infinite) are discussed in detail. In both cases, it is found that negative vorticity appears in between the main positive vorticity associated with the two Alfven spots. This fact gives some clue to the understanding of the property of twin wakes. As an application, the crossed-flow case is also discussed, and a conjecture upon the vorticity distribution is made.

A normalized formulation of the problem of isothermal bubble growth, dominated by viscosity and diffusion, reveals that in its full generality the problem involves eight dimensionless parameters. Limiting cases for extreme values of each of these parameters are investigated.

Blunt-body solutions for suspersonic flow usually concern closed body surfaces. This paper reports on an experimental investigation of a two-layer shock cap and indicates the existence of a predictable contact surface separating the layers. The inner layer was generated by injecting air through a contoured axisymmetric channel on a blunt body so as to simulate a hemispherical contact surface in a Mach number 4.8 flow.

Results show the existence of the contact surface and the influence of a range of mass-injection rates upon the displacement of the bow shock and contact surface from the body.

From the conditions of energy and momentum conservation it is shown that all four interaction coefficients of the elementary quadruple interactions discussed in Part I of this paper are equal. A further conservative quantity is then found which can be interpreted as the number density of the gravity-wave ensemble representing the random sea surface. The well-known analogy between a random linear wave field and a mass particle ensemble is found to hold also in the case of weak non-linear interactions in the wave field. The interaction conditions for energy transfer between waves to correspond to the equations of energy and momentum conservation for a particle collision, the final equation for the rate of change of the wave spectrum corresponding to Boltzmann's equation for the rate of change of the number density of an ensemble of colliding mass particles. The stationary wave spectrum corresponding to the Maxwell distribution for a mass particle ensemble is found to be degenerate. The question of the irreversibility of the transfer process for gravity waves is discussed but not completely resolved.

The general character of the magnetohydrodynamic flow past a non-conducting flat plate in the presence of transverse magnetic fields is analysed in some detail. The appropriate extension of the Rayleigh problem to the magnetohydrodynamic case is shown to yield solutions which correctly predict some features of the steady flow past a semi-infinite flat plate; in addition, it is shown that the knowledge of these significant features permits an easy evaluation of their magnitudes in other extensions of the Rayleigh problem. The flow past a semi-infinite flat plate is analysed by two methods. First, by linearizing the governing equations and incorporating the assumption of a low ratio of viscous to magnetic diffusivity, the results for skin friction and the normal component of magnetic field at the plate are obtained and are shown to be useful in interpreting the character of these low conductivity flows. Secondly, the complete set of governing equations is formulated as a finite difference problem and solved numerically on a digital computer. The results obtained, in addition to demonstrating feasibility of the numerical calculations, show that the disturbance produced by the plate is no longer confined to a thin viscous layer if the ratio of viscous to magnetic diffusivity is greater than 10−2, but that an appreciable Alfvén type disturbance is excited.

The method of characteristics is used to calculate numerical solutions for the head-on collision of a shock wave with a centred rarefaction wave in the onedimensional unsteady flow of a perfect gas. It is found that, for sufficiently strong shock waves, the rate of increase in the strength of the shock as it propagates through the rarefaction is so great that the temperature behind the shock reaches extreme values.

The problem of the stability of plane Couette flow to infinitesimal disturbances is carried numerically to larger Reynolds numbers than heretofore. The flow is definitely stable up to R = 1430, although at Reynolds numbers in this vicinity plane Couette flow has been observed experimentally to be turbulent. Observations on boundary-layer instability suggest this to be the source of turbulence in plane Couette flow channels.

This paper describes a series of experiments carried out in a high-enthalpy stream of argon on materials that are known to sublime. The results confirm that an axisymmetric Teflon model ablates to a stable shape which is independent of the initial nose profile. The effect of changing the total enthalpy of the gas is simply to alter the recession rate of the nose. The experimental results show poor agreement with a simple theory which ignores the effects of mass transfer in the boundary layer.