An expansion scheme is developed to describe the steady axisymmetric flow of a thin tubular liquid film of varying radius; the necessary small parameter is provided by the ratio between the characteristic film thickness and the characteristic tube radius. The co-ordinate system used is an orthogonal one based on the fluid interface and the fluid streamlines. The differential equations that arise thus treat the metric as an unknown set of variables. The method is restricted to situations dominated by viscous forces. Reference is made to numerical solutions that have been obtained in connexion with an industrial polymer-film-blowing process.

The problem of magnetohydrodynamic free convection of an electrically conducting fluid in a strong cross field is investigated. It is solved by using a singular perturbation technique. The solutions presented cover the range of Prandtl numbers from zero to order one. This includes both the important cases of liquid metals and ionized gases. A general examination is given of the role of the important parameters: Hartmann, Grashof and Prandtl numbers of the problem. This provides clear insight into its singular character and yields the correct expansion parameters. The boundary-layer approximations are derived from the complete Navier-Stokes and energy equations. The conditions for these approximations to be valid will be explicitly stated. Attention is given to ‘power law’ wall-temperatures and magnetic fields, and an assessment is given of the range of application.

The stability of viscous time-dependent rotational Couette flow, induced by an impulsively started inner cylinder was experimentally investigated. The ratio of the radius of the inner cylinder to that of the outer cylinder was $\frac{1}{10}$. The space between the cylinders was filled with distilled water. The time-dependent instabilities were observed by means of dye injection. They appeared as a series of disks more or less evenly spaced along the inner cylinder. The spacing and growth of these instabilities were recorded using a motion picture camera. From the motion pictures the critical time (which is the time from the impulsive start to the first onset of instability) and spacing of the instabilities were experimentally determined, and a marginal stability curve (Reynolds number versus critical time) was constructed.

The phenomenon under investigation occurs in various flow systems characterized by being dispersive to small-amplitude waves, but such that phase and group velocities approach the same finite limit at extreme wavelengths: for example, water in an open channel, density-stratified fluid flowing between horizontal boundaries, and rotating fluid contained in a tube. It is well known in each of these examples that, when a solid body is moved steadily at a subcritical velocity (i.e. less than the long-wave limit) relative to the undisturbed fluid, the body experiences resistance accountable to the continual development of a pattern of waves on the leeward side. But in these circumstances there is a second effect, upstream influence, consisting of a disturbance in the form of a uniform long wave that extends to a steadily increasing distance ahead of the body; and this has for some time been the subject of a controversy, particularly with regard to the question whether or not it is always present. The aim of the paper is to establish the principle that this second effect is inevitably coexistent with the first, being an essential component of the mechanism of wave resistance.

Measurements are presented of the reflected wave field produced by a plane oblique shock wave impinging on a turbulent boundary layer at an initial Mach number of 2·5. The outgoing waves are either a single shock, with the same deflexion as the incident shock, or a shock of approximately 10° deflexion followed by a region of compression in which is embedded an expansion fan having the same turning as the incident shock. The transition between these two types of wave field was not studied, but it is fairly abrupt and appears to be closely linked to the onset of boundary-layer separation. The observed wave systems broadly agree with the suggestions of a number of previous workers, but not with a recent theoretical treatment. Surface-pressure measurements and oil flow photographs are used to determine the onset of separation, and from these it is found that the overall pressure rise associated with incipient separation is rather smaller than previous work would suggest.

In the present paper the stability of plane Couette flow is studied. The stream function and distribution of vorticity and the Reynolds stresses for the linearized solution are computed. The stability effect of the non-linear terms are also discussed and it is found that for small amplitudes, the non-linear terms are destabilizing. A neutral curve determining the necessary amplitude in order to get instability, is found. The convergence of the expansion in the latter case is, however, somewhat uncertain and the result should therefore only be considered as a first, rough approximation.

The method of matched asymptotic expansions is employed to calculate the mass transport velocity due to small amplitude oscillatory waves propagating in conditions of density and viscosity discontinuities. For progressive waves in a two-layer system, it is found that the velocity at the interface is in the direction of wave propagation; when the uppermost surface is free, the velocity there is in the direction opposite to that at the interface. If the difference in the densities is small, the calculated transport velocity associated with an internal wave can be of more importance than that associated with the surface wave as obtained from the work of Longuet-Higgins (1953).

The decay of perturbations in a radiating gas is analyzed by first carrying out complete solutions for the decay of initial sinusoidal perturbations in the temperature, gas velocity, and pressure. These sinusoidal perturbations are superposed to yield solutions for the decay of initial ‘step’ temperature profiles consisting of constant initial temperature perturbations inside finite planar, cylindrical and spherical regions, with zero initial temperature perturbations outside. In contrast to the sinusoidal case, which may be described by a single radiation parameter, the decay of the step profile is determined by both the optical depth of the initial profile and the Boltzmann number, which is inversely proportional to the blackbody radiative flux. As the limits of zero and infinite Boltzmann numbers are approached, constant-density and constant-pressure cooling expressions are recovered. For a broad range of intermediate and small Boltzmann numbers the cooling proceeds in time from a constant-density process to a constant-pressure process. This transition is produced by gasdynamic waves generated near the profile edges by the radiative cooling. The temperature near the profile centre may increase during the transition period.

The train of capillary waves which appears on the forward face of a steep gravity wave is discussed by considering the capillary waves as stationary waves on a slowly varying running stream, using the non-linear capillary wave solution of Crapper (1957) and the recent method due to Whitham (1965a, b). Then energy input and damping are introduced into an energy equation which becomes a non-linear ordinary differential equation for the capillary wave steepness. Numerical solutions for various gravity wavelengths and steepnesses are discussed, and some light is thrown on the problem of the breaking of gravity waves.

The trailing edge region of a finite flat plate in laminar, incompressible flow is examined for the limit of high Reynolds numbers.

It is shown that the trailing edge region is an elliptic region of O(R−¾) and therefore a correct mathematical description must be based upon the full Navier–Stokes equations.

The ‘method of series truncation’ is used to reduce the full Navier–Stokes equations, written in parabolic co-ordinates, to an infinite set of non-linear, coupled, ordinary differential equations. Two sets of asymptotic boundary conditions, called simplified and exact boundary conditions, are determined by matching the Navier–Stokes region downstream with Goldstein's near wake solution.

By numerical integration the solutions for the first and second truncations are obtained for both sets of asymptotic boundary conditions. The results confirm that the size of the trailing edge region is of O(R−¾).

The flow of an elastico-viscous liquid contained between two concentric spheres which are rotating with the same angular velocity about axes passing through the centre of the spheres is considered. The angle between these axes is small. The solution is obtained by expanding the velocity components in terms of a small parameter α2, which is usually associated with problems involving oscillatory flows. The analysis is shown to have a direct application to the Balance Rheometer. In particular, it is shown that inertial effects in this rheometer are likely to be very small.

Consideration is given to the flow of viscous and elastico-viscous liquids contained between two infinite parallel plates which rotate with the same angular velocity Ω about axes normal to the plates but not coincident. In the viscous case, it is shown that a simple exact solution of the relevant equations can be obtained. In the elastico-viscous case, certain formulae are derived which should facilitate the interpretation of experimental results obtained from the Maxwell orthogonal rheometer.

The problem of flow from a large reservoir through a circular orifice is formulated by considering the velocity potential and Stokes's stream function as the independent variables and the radial and axial dimensions as the dependent variables, and a finite difference solution is obtained to the resulting boundary-value problem. This inverse formulation has the advantage over a finite difference solution in the physical plane that the region of flow is rectangular and consequently well adapted for minimum logic in programming a digital computer. The inverse finite difference solution is more readily obtained than a comparable solution in the physical plane, even though the inverse partial differential equation and associated boundary conditions are non-linear. The results from the inverse finite difference solution are in close agreement with other most recent results from approximate solutions to this problem.

The inverse method of solution is applicable to other free streamline as well as confined axisymmetric potential flow problems. The essential difference in other problems will be in the boundary conditions.Keywords: Orifice, Finite Differences, Non-linear Partial Differential Equation, Potential Flow.