Small regions in a flow where the density is different from that of the surrounding gas do not exactly follow accelerated motions of the latter, but move faster or slower depending on whether their density is smaller or larger than that of the main flow. This behaviour cannot be quantitatively explained by treating a gas ‘bubble’ as a hypothetical solid particle of the same density, because a gas bubble cannot move relative to the surrounding gas without being transformed into a vortex which absorbs part of the energy of the relative motion.

To illustrate the acceleration effect, the flow velocity behind known pressure waves in a shock tube is compared with the observed velocity of a bubble produced by a spark discharge. The displacement of such a bubble by a wave exceeds that of a flow element by more than 20%, but the bubble density is not known. If the spark discharge is replaced by a small jet of another gas, a pressure wave cuts off a section of this jet which then represents a bubble of known density.

A theory is developed which permits computing the response of such bubbles to accelerations. The ratio of the bubble velocity to the velocity of the surrounding gas depends on the density ratio for the two gases and on the shape of the bubble, but not on the acceleration. Experimental results with H2, He, and SF6 bubbles in air, accelerated by shock waves of various strength, are presented and agree well with the theoretical predictions. The results apply regardless of whether accelerations are produced by pressure waves in a non-steady flow or by curvature of streamlines in a steady flow. Various aspects of the experimental observations are discussed.

The interaction of a simple wave, in steady supersonic flow, with a two-dimension mixing region is treated by applying Fourier analysis to the linearized equations of motion. From asymptotic forms for the Fourier transforms of physical quantities, for large wave-number, the dominant features of the resulting flow pattern are predicted; in particular it is found that a shock wave, incident on the mixing region, is reflected as a logarithmically infinite ridge of pressure. For two particular Mach-number distributions in the undisturbed flow, numerical solutions are obtained, showing greater detail than the results predicted by the asymptotic approach. A method is given whereby the linear theory may be improved to take into account some non-linear effects; and the reflected wave, for an incident shock wave, is then seen to consist of a shock wave, gradually diminishing in strength, followed by the main expansion wave.

This paper is concerned with the determination, on the basis of Oseen's equations, of the flow past a quarter-plane with its leading edge normal to, and its side edge parallel to, a uniform incident stream. The solution is completed, except for a region in the vicinity of the corner, correct to order v1/2 for small kinematic viscosity ν.

Away from the vicinity of the side edge the flow will approximate to the two-dimensional flow past a semi-infinite plate. This two-dimensional flow can be built up successively, if we like to think in terms of boundary conditions at the plate rather than at the edge of the boundary layer, from the potential flow associated with the uniform stream, a shear layer introduced to remove the tangential slip and a potential flow to remove the normal velocity at the plate associated with the shear layer. In the vicinity of the plate the three together give the usual boundary-layer solution.

We start our solution from this same basis, namely, the potential flow associated with the uniform stream and the shear layer to restore the no-slip condition. As a first approximation, neglecting the effects of the edges, this will be the same as for the two-dimensional problem. The normal velocity introduced by this shear layer has to be compensated by a potential flow (see §4). This potential flow in turn (and here our problem diverges significantly from the two-dimensional problem) introduces tangential velocities with components parallel to both leading and side edges which require the introduction of a further shear layer. Over the main body of the plate this secondary shear layer is of a conventional form (§5) but requires special examination near the edges. In §6 it is shown how Carrier & Lewis's (1949) solution can be modified to give the flow near the leading edge away from the tip and in §7, the core of the paper, the flow near the side edge is determined.

In the vicinity of the side edge the extra potential flow has no component in the direction of that edge and so the solution given by Howarth (1950) for the corresponding unsteady problem is applicable. What emerges from the present calculations, however, is that Howarth's application of Rayleigh's analogy to give the excess skin friction is seriously incomplete. For, whilst this argument gives correctly the local increase of order ν in skin friction in the immediate vicinity of the side edge, it omits the widespread effects of the secondary shear layer. These are found to be of the same order in ν as the local effects.

The cross-flow in the side edge region has features of special interest. Its determination depends on a knowledge of the potential flow associated with the primary shear layer and so it depends, for instance, on the shape of the leading edge and is not, as appears to have been assumed up to now, determined completely by local conditions. This is further exemplified by the fact that it cannot be expressed in terms of what would be regarded as the natural boundary-layer variables but involves quite separately the distance from the leading edge.

When a thin film of viscous fluid is produced by passing it through a small gap between a roller or spreader and a flat plate, it often presents a waved, or ribbed, surface. An analysis is given here in terms of lubrication theory to show why in many cases flow leading to a uniform film is unstable. Account is taken of surface tension which proves to be a stabilizing factor. The most unstable values of the wave-number, n (characterizing the disturbance), are calculated as functions of the dimensionless variable T/μU0, and of the geometry of the system; T is the surface tension, μ the viscosity and U0 a representative velocity of the fluid. For the particular case of a spreader in the form of a wide-angled wedge, these predictions are compared with experimental observations. Agreement is obtained for values of T/μU0 between about 10 and 0.1, but for smaller values of T/μU0 it is clear that other considerations, involving only viscous and pressure forces, determine the nature of the secondary flow.

Two time-dependent magnetohydrodynamic flow problems are discussed. In Part I we consider the situation in which a semi-infinite flat plate is moved impulsively in its own plane into an electrically conducting viscous fluid. The ambient magnetic field has the same direction as the motion of the plate; it is found that when $\mu H^2_0| \rho U^2_0 \; \textless \; 1$, the flow pattern approaches asymptotically the steady flow found earlier (Greenspan & Carrier 1959). When $\mu H^2_0| \rho U^2_0 \; \textgreater \; 1$, the asymptotic state is one in which the fluid accompanies the plate in a rigid body motion as was anticipated in the earlier work.

In Part II, an infinite plate is moved impulsively in its own plane in the presence of an ambient magnetic field which is perpendicular to the plane of the plate. It is shown that the problem is not uniquely set until one specifies what three-dimensional problem reduces in the limit to the two-dimensional problem so defined. The answers in the conceptually acceptable limit case investigated here (the plate being a pipe of very large radius) have an asymmetry which at first sight is unexpected.

Approximate equations of motion, continuity and mass transport are given for a viscous liquid of variable density moving very slowly between vertical and impermeable parallel planes. These equations are used to calculate approximate stability criteria when the liquid is at rest under a vertical density gradient. The results are applicable to the problem of the stability of a viscous liquid of variable density to two-dimensional disturbances in a porous medium.

An exact stability analysis for the liquid between parallel planes is also given, and expansions in powers of the disturbance wave-number are obtained for the critical Rayleigh number at neutral stability. The previous approximate results are found to correspond to the leading terms of the series expansions. For the most unstable type of disturbance, the velocity distribution closely resembles plane Poiseuille flow, which was the form assumed in the approximate equations.

An asymptotic expansion is derived for the critical Rayleigh number at neutral stability in a long vertical channel, or duct, the cross-section of which is a thin rectangle. The typical neutral disturbance possesses a ‘boundary layer’ at each end of the cell cross-section, and this has a small stabilizing effect.

The critical Rayleigh number for a long vertical channel of rectangular cross-section is found experimentally by comparing the density gradient of the liquid in the channel at neutral stability with the corresponding density gradient in a vertical capillary tube. There is better agreement with the exact theory than with the approximate theory, the experimental result being about 4% higher than the value predicted by the ‘exact’ asymptotic expression, and about 10% higher than the value predicted by the simple approximate theory.

When a hollow circular cylinder with its axis horizontal is partially filled with water and rotated rapidly about its axis, an almost rigid-body motion results with an interior free surface. The emotion is analysed assuming small perturbations to a rigid rotation, and a criterion is found for the stability of the motion. This is confirmed experimentally under varying conditions of water depth and angular velocity of the cylinder. The modes of oscillation (centrifugal waves) of the free surface are examined and a frequency equation deduced. Two particular modes are considered in detail, and satisfactory agreement is found with the frequencies observed.

The instability of the accelerated interface between a liquid (methanol or carbon tetrachloride) and air has been investigated experimentally for approximate sinusoidal disturbances of wave-number range from well below to well above the cut-off. The growth rates are measured and compared with theoretical results. A third-order theory shows the phenomena of overstability which is found in the experimental results. Some measurements of later stages of growth agree moderately well with the available theory and disclose some additional phenomena of bubble competition, Helmholtz instability with transition to turbulence, and jet instability with production of drops.

This paper is concerned with the dispersion of a material quantity in the steady flow of a viscous fluid through a random network of capillaries (which is a useful model of a porous medium), for the case in which molecular diffusion and macroscopic mixing, due to the randomness of the streamlines, are both important. A Lagrangian correlation function is introduced and the longitudinal and lateral effective diffusivities are thereby calculated for all values of Ul/κ less than some large value. Here, l denotes the length of a capillary, U the mean velocity of the fluid, and κ the molecular diffusivity of the material quantity. The theory is compared with experimental observations of dispersion in flow through granular beds.

This paper describes an attempt to verify experimentally the wavemaker theory for a piston-type wavemaker. The theory is based upon the usual assumptions of classical hydrodynamics, i.e. that the fluid is inviscid, of uniform density, that motion starts from rest, and that non-linear terms are neglected. If the water depth, wavelength, wave period, and wavemaker stroke (of a harmonically oscillating wavemaker) are known, then the wavemaker theory predicts the wave motion everywhere, and in particular the wave height a few depths away from the wavemaker.

The experiments were conducted in a 100 ft. wave channel, and the wave-height envelope was measured with a combination hook-and-point gauge. A plane beach (sloping 1:15) to absorb the wave energy was located at the far end of the channel. The amplitude-reflexion coefficient was usually less than 10%. Unless this reflexion effect is corrected for, it imposes one of the most serious limitations upon experimental accuracy. In the analysis of the present set of measurements, the reflexion effect is taken into account.

The first series of tests was concerned with verifying the wavemaker theory for waves of small steepness (0.002 ≤ H/L ≤ 0.03). For this range of wave steepnesses, the measured wave heights were found to be on the average 3.4% below the height predicted by theory. The experimental error, as measured by the scatter about aline 3.4% below the theory, was of the order of 3%. The systematic deviation of 3.4% is believed to be partly due to finite-amplitude effects and possibly to imperfections in the wavemaker motion.

The second series of tests was concerned with determining the effects of finite amplitude. For therange of wave steepnesses 0.045 ≤ H/L ≤ 0.048, themeasured wave heights were found to be on the average 10% below the heightspredictedfrom the small-amplitude theory. The experimental error was again of the order of 3%.

It is considered that these measurements confirm the validity of the small-amplitude wave theory. No confirmation of this accuracy has hitherto been given for forced motions.

In the steady flow of an incompressible, inviscid, conducting fluid past a magnetized sphere, the first-order effects of the magnetic field and the conductivity are studied. Paraboloidal wakes of vorticity and magnetic intensity are formed, the former being half the size of the latter. The vorticity, generated by the non-conservative electromagnetic force, is logarithmically infinite on the sphere. For the case of a dipole of moment M at the centre of a sphere of radius a, the drag coefficient is $C_D = \frac {144 \mu^{\prime 2}}{5(2\mu + \mu^{\prime})^2} \beta R_M,$ where μ and μ′ are the permeabilities of the fluid and sphere, respectively, β is the ratio of the representative magnetic pressure μM2/2a6 to the free-stream dynamic pressure, and RM is the magnetic Reynolds number.

This paper is intended to give some indication of the impact forces of a water wave on a wall. The effect of gravity forces in the small time interval of impact considered will be small and is neglected. The shape of the wave before impact is considered to be a two-dimensional wedge which is assumed to strike a wall at right angles to its path. The wedge is assumed to be infinite in extent and to have uniform translational velocity V before impact. The choice of a wedge shape enables the problem to be formulated in terms of similarity variables x/Vty/Vt, where the origin of the x, y plane is at the initial point of contact of the vertex of the wedge with the wall. The solution presented here can be easily adapted to the problem of an axi-symmetric cone of water striking a wall, but this is not pursued in the present paper.

A systematic analysis is made of published measurements of the magnitude of temperature fluctuations in the atmospheric boundary layer. These cover a wide range of height, wind speed, and thermal stratification. Within appropriate ranges of the variables, there is evidence for the existence of a dominantly forced convection régime, and also one wherein the predictions of the similarity theory of free convection are fairly closely approached. Subject to the limitations set by the recording systems used, regression relations are obtained for the magnitude of the fluctuations in terms of height and vertical temperature gradient or heat flux.

The Stokes flow problem is concerned with fluid motion about an obstacle when the motion is such that inertial effects can be neglected. This problem is considered here for the case in which the obstacle (or configuration of obstacles) has an axis of symmetry, and the flow at distant points is uniform and parallel to this axis. The differential equation for the stream function ψ then assumes the form L2−1ψ = 0, where L−1 is the operator which occurs in axiallysymmetric flows of the incompressible ideal fluid. This is a particular case of the fundamental operator of A. Weinstein's generalized axially symmetric potential theory. Using the results of this theory and theorems regarding representations of the solutions of repeated operator equations, the authors (1) give a general expression for the drag of an axially symmetric configuration in Stokes flow, and (2) indicate a procedure for the determination of the stream function. The stream function is found for the particular case of the lens-shaped body.

Explicit calculation of the drag is difficult for the general lens, without recourse to numerical procedures, but is relatively easy in the case of the hemispherical cup. As examples illustrative of their procedures, the authors briefly consider three Stokes flow problems whose solutions have been given previously.

A study was made of the transition of a two-dimensional jet. In the region where laminar flow becomes unstable, two kinds of sinusoidal velocity fluctuation have been found; one is symmetrical and the other is anti-symmetrical with respect to the centre line of the jet. The fluctuations grow exponentially at first and develop into turbulence without being accompanied by abrupt bursts or turbulent spots.

The response characteristics of laminar jets to artificial external excitation were investigated in detail by using sound as an exciting agent. The effect of excitation was seen to be most remarkable when the frequency of excitation coincides with that of self-excited sinusoidal fluctuations.

Numerical solutions of equation of small disturbances superposed on laminar flow were obtained assuming the Reynolds number as infinity. Theoretical eigenvalues and eigenfunctions are in good agreement with experimental results, thus verifying the existence of a region of linear disturbance in the two-dimensional jet.

When a body moves through a stratified fluid, i.e. one whose density decreases upwards, gravity waves are set up and this causes a resistance to motion. An axisymmetric case is considered in which a body moves steadily and vertically through a fluid whose density decreases exponentially upwards. The fluid is supposed perfect, incompressible, and unbounded in all directions. The equations of motion are linearized, and with a fairly general initial motion of the surrounding fluid, the limit of the solution as t → ∞ is evaluated. Transform methods are used to solve the equation of motion, and the methods of steepest descents and stationary phase are used to obtain approximate solutions.

Streamlines and the distortion of the constant density levels for a spindle-shaped body are shown. The curves of resistance against a function of the velocity for the circular cylinder, the sphere, and a spindle-shaped body are also given. A criterion is given for when the maximum wave resistance for a sphere may be expected, and an estimate of this maximum resistance is made.

The non-linear and linear ‘shallow-water’ theories, which describe long gravity waves on the free surface of an inviscid liquid, are extended to the case of an electrically conducting liquid on a horizontal bottom, in the presence of a vertical magnetic field. The dish holding the liquid, and the medium outside it, are assumed to be non-conducting. The approximate equations are based on a small ratio of depth to wavelength, on the properties of mercury, and on a moderate magnetic field strength. These equations have a ‘magneto-hydraulic’ character, for in the shallow liquid layer the horizontal fluid velocity and current density are independent of the vertical co-ordinate.

Some explicit solutions of the linear equations are obtained for plane flows and for axi-symmetric flows in which the velocity vector lies in a vertical, meridional plane. The amplitudes of waves in a dish, and the amplitudes behind wave fronts progressing into undisturbed liquid, are found to be exponentially damped, the mechanical energy associated with a disturbance being dissipated by Joule heating.

The approximate non-linear equations for plane flow are studied by means of characteristic variables, and it appears that, because of the magnetic damping effect, there is less qualitative difference between solutions of the non-linear and linear approximate equations at large times than is the case when the magnetic field is absent. In particular, the characteristic curves depart only a finite distance from their ‘undisturbed positions’.

The pressure measured at a static pressure hole differs slightly from the true static pressure, by an amount which depends on the hole size and shape. The present investigation extends the range of previous work to determine the error in static pressure measurement in incompressible turbulent flow. The observed static pressure was always greater than the true static pressure. The results are presented in dimesionless form as a function of the Reynolds number based on hole diameter and friction velocity.

An experimental investigation of the hydrodynamic stability of the laminar hypersonic boundary layer was carried out with the aid of a hot-wire anemometer. The case investigated was that of a flat surface at zero angle of attack and no heat transfer.

The streamwise amplitude variation of both natural disturbances and of disturbances artifically excited with a siren mechanism was studied. In both cases it was found that such small fluctuations amplify for certain ranges of frequency and Reynolds number Rθ, and damp for others. The demarcation boundaries for the amplification (instability) zone were found to resemble the corresponding limits of boundary-layer instability at lower speeds. A ‘line of maximum amplification’ of disturbances was also found. The amplification rates and hence the degree of selectivity of the hypersonic layer were found, however, to be considerably lower than those at the lower speeds. The disturbances selected by the layer for maximum amplifications have a wavelength which was estimated to be about twenty times the boundary-layer thickness δ.

It is shown that in Stokes's flow the perturbation field, due to the addition of one more sphere to a shear flow of a fluid containing a number of non-interacting spheres, has the property that the total additional shearing force, acting on any plane normal to the direction of velocity change, is zero. However, the perturbation velocity, integrated over such a plane, takes a constant value, positive if the plane lies on one side of the sphere and negative if it lies on the other side. It follows that the effect of all the spheres is not to alter the shearing stress at all, but to reduce the mean shear by a factor 1 – 2·5c, where c is the concentration. This suggests that Einstein's viscosity law should be altered to η = η0/(1 – 2·5c) when c is not small.