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.

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.

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.

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.

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’.

In certain cases of steady motion of a conducting fluid in a magnetic field, the primitive equations may be integrated once, yielding a second-order partial differential equation in the stream function. This equation is highly non-linear in general, but for certain choices of basic flow and magnetic fields it is tractable. Several arbitrary functions of integration have to be evaluated to make the analysis useful. This may be done in a region that remains undisturbed. A short discussion is given to suggest a procedure for deciding in a special case whether this undisturbed region is ‘upstream’ or ‘downstream’.

Approximate solutions are found for the fluid flow and heat transfer in a heated cylinder, closed at the bottom and opening at the top into a reservoir of cool fluid, which has been tilted at a small angle to the vertical.

One solution is found for large Rayleigh number when the boundary layer does not fill the tube, and another for small Rayleigh number when the boundary layer fills the tube. In both cases tilting causes a small increase in heat transfer which is proportional to the square of (l/a) tan ϕ, where l/a is the length–radius ratio and ϕ the angle of tilt.

A general numerical procedure is described whereby the details of the steady inviscid flow of a mixture of perfect gases about a blunt body, including the effects of finite dissociation and recombination rates, may be calculated. Several numerical examples are calculated in order to apply the numerical procedure to specific cases and also to show the effects of finite reaction times on the flow about a blunt body.

A simple method is presented in this paper for calculating

1. the secondary velocities, and

2. the lateral displacement of total pressure surfaces (i.e. the ‘displacement effect’)