When a semi-infinite flat plate is immersed parallel to an unbounded, plane, steady, asymmetric, constant shear flow of an incompressible viscous fluid, an interaction occurs between the surface-generated vorticity and the external vorticity. A physical assumption was made in a previous paper (Mark 1962) concerning this problem that the pressure field in a thin layer adjacent to the top-side of the plate may be accurately approximated by the undisturbed constant-pressure field—that which exists before the insertion of the plate into the flow. This means that the vorticity interaction is assumed to have no effect on the undisturbed pressure field. That this is a valid first approximation far downstream along the plate where the interaction is intense is given rigorous support in this paper.

The flow below the plate is examined on a heuristic basis. It is found that there is a strong possibility for the flow to separate from the lower surface near the leading edge of the plate. However, far downstream the flow settles down to a Stokes-type flow near the plate.

The results of Ursell (1964) are confirmed by proving that there are no additional contributions to Ursell's integral from singularities of the integrand at infinity. The method consists of proving that the asymptotic expansion of a force coefficient Δ(N) is uniformly valid in a finite sector of the complex N-plane. This in turn requires that the kernel of an integral equation remains small in this sector.

According to the Prandtl–Batchelor theorem for a steady two-dimensional flow with closed streamlines in the inviscid limit the vorticity becomes constant in the region of closed streamlines. This is not true for three-dimensional flows. However, if the variation of the flow field along one direction is slow then it is possible to expand the solution in terms of a small parameter characterizing the rate of variation of the flow field in that direction. Then in the leading-order approximation the projections of the streamlines onto planes perpendicular to that direction can be closed. Under these circumstances the extension of the Prandtl–Batchelor theorem is obtained. The resulting equations turned out to be a three-dimensional analogue of the equations of the quasi-cylindrical approximation.

A simple analysis shows that with a disturbance present the potential jump in a steady flow in a canal is expressed in terms of (1) the effective volume (displaced volume and added mass/density of fluid) and (2) the depth Froude number for either a submerged body or a body with thin waterplane area. For a ship moving in a canal, the expression for potential jump contains a contribution from the line integral term along the intersection between the ship hull and the free surface. When a pressure distribution is given on the free surface, the potential jump can be expressed explicitly in terms of the depth Froude number and the total pressure force, regardless of the shape of the pressure distribution. From the present relations, the added mass of a ship in steady motion in a canal is computed from the potential jump computed previously by the author for various Froude numbers. This added mass plays an essential role in the equation of motion initially when a sudden external force is applied to a steady moving ship. The present analysis is complimentary to that of Newman (1976) and the extension of that to the three-dimensional case. As practical applications of the potential jump, which has had a limited interest, we proposed approximate formulas for speed correction and sinkage of a ship in a towing tank experiment. Also proposed is an approximate formula for the speed correction in a wind tunnel experiment. The present approximate formula is compared with ‘exact’ numerical results obtained by the localized finite element method for both towing tank and wind tunnel experiments. The present speed correction formula is also compared with existing approximate formulas for a wind tunnel experiment. The present formulas compare favourably with the exact numerical results.

The paper presents a new analytical and experimental study of transonic flow around spheres. The results of the analytical study, which employs the method of orthogonal collocation for simultaneous solution of the momentum equations, the equation of continuity and the energy equation, are compared with hitherto unpublished measurements obtained on spheres of various sizes (1·02, 2·54 and 3·81 cm in diameter) in air, in dry steam and in wet steam with free-stream Mach numbers in the transonic range (0·58 < M∞ < 0·97). The relationship θ h = 91·78 + 8·59 M∞ between the attached-shock angle and the free-stream Mach number was obtained by fitting the theoretical pressure distributions to the experimental ones.

An investigation is made into the propagation of a one-dimensional combustion wave, which consists of a flame front and a precursor shock wave which pass down a tube closed at one end, in the presence of a transverse magnetic field in the undisturbed gas at rest. The shock wave is assumed to be of sufficient strength to ionize completely the initially non-electrically-conducting gas and the conditions at the flame front are taken to satisfy the Chapman–Jouguet condition. Details of the solution are compared with the corresponding results for ordinary gasdynamic deflagration.

This note provides numerical values for the long-wave limit of the virtual-mass coefficient relating to the heaving motion of a half-immersed circular cylinder on water of finite depth, found analytically by Ursell in the preceding paper; some preliminary analysis is needed, however.

The effect of a parallel magnetic field upon the stability of the plane interface between two conducting viscous fluids in uniform relative motion is considered. A parameter reduction, which has not previously been noted, is employed to facilitate the solution of the problem. Neutral stability curves for unrestricted ranges of the governing parameters are found, and the approximate solutions of other authors are examined in this light.

An analysis of the structure of the wakes and waves in steady compressible magnetohydrodynamics is presented. No restriction is made on the equation of state of the gas or on the ratios of the various dissipative parameters. An asymptotic solution is obtained which furnishes directly the flow far from a body and which may be used in the construction of the entire flow field. The non-dissipative solutions are obtained as a non-uniform limit for vanishing dissipation; no matter how small the dissipation, one can go far enough from the origin that the flow is essentially dissipative. For non-aligned fields the wave pattern consists of a downstream wake and either two or four standing waves, depending on the flow regime. For aligned fields, two of these waves become wakes, so that the wake is a superposition of three structured layers, with either all downstream or two downstream and one up-stream. It is found that the non-dissipative limit of the wake is non-unique for the aligned fields case. Different limits are obtained depending on how the various dissipative parameters vanish.