It is shown that in an incompressible fluid in which the magnetic diffusivity λ is much less than the kinematic viscosity ν, certain magnetic field distributions of limited spatial extent (conveniently described as magnetic eddies) can exist on a length scale such that the associated Reynolds number and magnetic Reynolds number are respectively small and large compared with unity. The Lorentz forces are in equilibrium with the dynamic forces associated with the fluid motion. The boundary condition imposed on this motion is that at a large distance from a magnetic eddy the velocity field should be a uniform axisymmetric irrotational straining motion. The eddies are steady in the limit λ → 0, but decay slowly in a fluid of finite conductivity. Two particular eddies are considered in detail: a disk-shaped eddy in a compressive straining motion, and a spherical eddy in an extensive straining motion. Possible applications to turbulence in interstellar gas clouds are qualitatively considered.

A semi-infinite body of liquid is in uniform rotation about a vertical axis for t < 0. A concentrated, vertical displacement of the free surface is imposed at t = 0. The motion of the free surface for t > 0 is calculated in linear approximation with the aid of Hankel and Laplace transformations, together with an intermediate transformation to parabolic co-ordinates. The result appears as an integral superposition of dispersive waves that divides naturally into two parts, corresponding to waves of the first and second class, with angular frequencies that are respectively greater or less than twice the angular speed of rotation. The waves of the first class, which are qualitatively similar to those in the classical (Cauchy–Poisson) problem, are found to dominate the asymptotic representation, as obtained through a stationary-phase approximation. The analysis is carried out in such a way as to separate the effects of Coriolis acceleration and free-surface curvature. Attention is focused on concave surfaces, such as would be realized in a laboratory experiment, but it is pointed out that striking differences exist between the dispersion laws for concave and convex surfaces, especially as regards waves of the second class.

Upper bounds for the heat flux through a horizontally infinite layer of fluid heated from below are obtained by maximizing the heat flux subject to (a) two integral constraints, the ‘power integrals’, derived from the equations of motion, and (b) the continuity equations. This variational problem is solved completely, for all values of the Rayleigh number R, when only the constrains (a) are imposed, and it is thus shown that the Nusselt number N for any statistically steady convective motion cannot exceed a certain value N1(R), which for large R is approximately (3R/64)½. When (b) is included as a constraint, the variational problem is solved for large R, under the additional hypothesis that the solution has a single horizontal wave number; the associated upper bound on the Nusselt number is (R–248)⅜. The mean properties of this maximizing ‘flow’, in particular the mean temperature and mean square temperature deviation fields, are found to resemble the mean properties of the real flow observed by Townsend; the results thus tend to support Malkus's hypothesis that turbulent convection maximizes heat flux.

The small-perturbation theory for steady, inviscid, non-equilibrium flow is extended to include the transonic speed range. The resulting transonic small-perturbation equation for the velocity potential is solved for flow inside a wavy cylinder. It is shown that this solution gives a representation of transition through sonic speed at the narrowest sections of the cylinder. This extension of the theory is applied to non-equilibrium nozzle flow.

Assuming that the Stokes flow past an arbitrary particle in a uniform stream is known for any three non-coplanar directions of flow, then the force on the body to O(R), for any direction of flow, is given explicitly in terms of these Stokes velocity fields. The Reynolds number (R) based on the maximum particle dimension is assumed small. For bodies with certain types of symmetry it suffices merely to know the Stokes resistance tensor for the body in order to calculate this force. In this case the resulting formula is identical to that of Brenner (1961) and Chester (1962). However, for bodies devoid of such symmetry, their formula is incomplete—there being an additional force at right angles to the uniform stream which remains invariant under a reversal of the flow at infinity. As this additional force is a lift force, it follows that the Brenner-Chester formula furnishes the correct drag on a body of arbitrary shape; moreover, this drag is always reversed to at least O(R) by a reversal of the uniform flow at infinity.

Exactly analogous formulae are derived using the classical Oseen equations, and it is shown that although this gives both the correct vector force on bodies with the above types of symmetry and the correct drag on bodies of arbitrary shape, it gives in general an incorrect lift component for completely arbitrary particles.

Finally, the singular perturbation result for the force on an arbitrary body is extended to terms of O(R2log R). This higher-order contribution to the force is given explicitly in terms of the Stokes resistance tensor, and has the property of being reversed by a reversal of the flow at infinity, regardless of the geometry of the body.

These results are collected in the Summary at the end of the paper.

The velocity fields associated with a variety of flows which may be described by perturbations of the Blasius solution are considered. These are flows which, for example, because of localized mass transfer, involve the initial-value problem of boundary-layer theory, or which involve a variable ratio of the viscosity-density product, or finally which involve mass transfer. The perturbation solutions are presented so that in accord with the usual linearization procedures further applications for the determination of first-order effects can be readily made. In addition, each of these perturbations involves a common differential operator whose eigenfunctions form a complete orthogonal set. Accordingly, a procedure for systematically improving each perturbation solution to obtain higher-order effects by quadrature is presented. The results of applications in several cases are given and are compared to more accurate solutions where available.

Experimental measurements were made of the difference in pressure at large distances on either side of a spherical particle settling slowly along the axis of a long circular tube filled with viscous liquid. At low particle Reynolds numbers and for small sphere-cylinder diameter ratios, it was found that the product of the pressure difference and the cross-sectional area of the tube is equal to twice the drag on the particle, in accord with theory.

The motion of a bounded shallow liquid, initially of arbitrary shape, in an arbitrary state of motion and lying on a paraboloid of revolution (including a level surface as a special case) can always be separated into three parts:

1. the motion of the centre of gravity, which is entirely independent of the other motions and is governed by a pair of simple ordinary linear differential equations;

2. an isotropic two-dimensional dilatation and rotation which are also governed by a simple linear differential equation;

3. the motions that remain after removal of the velocity fields associated with the preceding motions; these will be called additional motions.

The additional motions exert a ‘pressure’, determined by their total energy, which tends to increase the spread of the liquid. If the spread does increase then the additional motions lose energy which then appears as energy associated with the dilatation.

The effect of the dilatation and rotation on the additional motions can be described by transformation into a co-ordinate system that rotates and dilates with the liquid. In these co-ordinates, with a properly adjusted time scale, the additional motions satisfy equations that are isomorphic with the original equations of motion; however, the liquid now appears to be lying on a parabola that is always concave upwards.

Lamb's analysis of small-amplitude, shallow-water oscillations in a rotating paraboloid, interpreted by him in the inconsistent context of an approximately plane free surface, is re-interpreted to obtain results that are valid for $0 \le \omega^2l|2g \; \textless \;1$ (ω = rotational speed, l = latus rectum of paraboloid); no equilibrium is possible for ω2l/2g > 1. It is shown that the frequencies of the dominant modes for the azimuthal wave numbers 0 (axisymmetric motion) and 1 are independent of ω for an observer in a non-rotating reference frame and that the frequencies of all other axisymmetric modes are decreased by rotation (Lamb concluded that they would be increased). An axisymmetric mode of zero frequency, which was over-looked by Lamb, also is found.

Exact solutions to the non-linear equations of motion, which reduce to the aforementioned dominant modes for small amplitudes, are determined. The axisymmetric solution is inferred from similarity considerations and is found to contain all harmonics of the fundamental frequency. The finite motion of azimuthal wave-number 1 is a quasi-rigid displacement of the liquid and is found to be simple harmonic except for a second-harmonic component of the free-surface displacement (but the horizontal velocity at a given point remains simple harmonic).

The boundary-layer flow over a semi-infinite flat plate is investigated. For time t < 0 there is the usual steady velocity boundary layer and, neglecting viscous dissipation, no thermal boundary layer. At t = 0 a temperature boundary layer is initiated without altering the velocity, and the subsequent temperature distributionn is studied for large and small t.

The nature of the flow of a dissociating diatomic gas in a convergent-divergent supersonic nozzle is investigated by measuring the distribution of flow parameters along the nozzle. Two parameters are measured, namely the degree of dissociation and the static pressure. The degree of dissociation is measured by a spectroscopic method. The pressure is measured by [xcup ]-tube manometers. The results are compared with Bray's (1959) theory and Logan's (1957) prediction.

A method developed by Roshko (1953a, b) has been applied to the case of flow round a right-angled elbow, in which the fluid separates at the inner corner at a speed which may be higher than the velocity of the separated flow far downstream. Numerical results have been evaluated and are presented for values of the ratio of the channel widths, downstream to upstream of the elbow, equal to 0·1 and 1·0.

The elbow may be regarded as one half of a two-dimensional representation of an axisymmetric plate valve, on which experiments indicate considerable variation in performance, the discharge coefficient being found to depend to a marked extent on the length of the valve land.

Heat transfer by convection from isothermal rotating cones is investigated experimentally by measuring the sublimation rate from naphthalene-coated cones and using the analogy between heat and mass transfer. Measurements are made for a range of conditions from entirely laminar flow to conditions when the outer 70% of the surface area is covered by turbulent flow. Mass-transfer measurements for laminar flow over cones of vertex angles 180°, 150°, 120° and 90° are in good agreement with the theoretical prediction. For turbulent flow, experimental results for cones of the above vertex angles also agree very well with the semi-empirical analogy calculations for the disk case. A different heat- and mass-transfer relationship with the rotational Reynolds number is observed in the measurements on the 60° cone, and is believed to be due to a change of flow characteristics. The instability and the transition of flows over different cone models are also discussed.

The statistical density function is derived for a variable (such as the surface elevation in a random sea) that is ‘weakly non-linear’. In the first approximation the distribution is Gaussian, as is well known. In higher approximations it is shown that the distribution is given by successive sums of a Gram–Charlier series; not quite in the form that has sometimes been used as an empirical fit for observed distributions, but in a modified form due to Edgeworth.

It is shown that the cumulants of the distribution are much simpler to calculate than the corresponding moments; and the approximate distributions are in fact derived by inversion of the cumulant-generating function.

The theory is applied to random surface waves on water. The third cumulant and hence the skewness of the distribution of surface elevation is evaluated explicity in terms of the directional energy spectrum. It is shown that the skewness λ3 is generally positive, and positive upper and lower bounds for λ3 are derived. The theoretical results are compared with some measurements made by Kinsman (1960).

It is found that for free, undamped surface waves the skewness of the distribution of surface slopes is of a higher order than the skewness of the surface elevation. Hence the observed skewness of the slopes may be a sensitive indicator of energy transfer and dissipation within the water.

Previous results on the over-all density ratio of shock waves in CO2, confirming experimentally the theoretical equilibrium value, have been extended to a shock Mach number of 7·3. The discrepancy between our results and earlier Princeton results approaches 18% at a Mach number of 7. Possible reasons for this are discussed, with particular reference to the interferometer technique, but no explanation has been found.

The interaction of a pitching circular disk with the motion induced by the disk in the surrounding fluid is investigated in this paper. MacCamy's (1961) method of simplifying the three-dimensional problem of a circular disk to the two-dimensional problem is found to apply in the present analysis. The integral equation is solved numerically to determine the dependence of pressure, added moment of inertia, and damping coefficient on the frequency of the oscillation.

The increase in hypervelocity launcher velocities, the tendency for models to become more fragile and the use of sabot techniques has made it increasingly more important to reduce peak accelerations during the launch phase. At the same time a high acceleration must be maintained to enable a launcher barrel of reasonable length to be used. This paper considers the theoretical design of a launcher to give constant acceleration to the model. A similarity solution is found for the launcher barrel and the solution is extended to the first stage of the launcher. Departure from ideal gas laws is also discussed and the theory is extended to include the effect of molecular volume. The motion of the driving piston in the first stage is considered in detail and techniques to ensure that the piston motion results in constant base pressure on the model are presented.

This is a study of the longitudinal and vertical turbulent velocities which were measured both near the surface and near the bottom of an estuarine tidal current. The turbulence spectra, representing mainly the energy-containing eddies of the flow, were derived, and it was also possible to determine values for the Reynolds stresses at different depths and to make an estimate of the scales of the turbulence. The results are compared with similar observations in the open sea and certain features are discussed which resemble those of measurements of turbulent flow in laboratory experiments and in the atmosphere.

Vibrationally relaxing flow through a nozzle is examined in the case when the amount of energy in the lagging mode is small. It is shown that there exists a ‘boundary-layer’ region in which relatively large departures from equilibrium occur. The position of this region is given by the type of criterion that has previously been used to predict the onset of ‘freezing’. An analytical solution for the distribution of the vibrational energy in the nozzle is obtained for a particular nozzle geometry, and an expression for the final asymptotic ‘frozen’ value of the vibrational energy far downstream is found. This asymptotic solution can be obtained from conditions at the ‘freezing’ point provided a suitable boundary condition is applied there.