This paper deals theoretically with the problem of the hydrodynamic stability of a stratified flow of a viscous fluid. The primary flow consists of two laminar streams of viscous fluids of different densities flowing in opposite directions between two parallel inclined planes under the action of gravity. The effect of surface tension at the interface of the two fluids is included in the formulation of the problem.

Since instability can be expected to occur at low Reynolds numbers when the inclination is nearly vertical, the solution of the Orr–Sommerfeld equations is developed as a power series in the transverse space co-ordinate. It is shown that for the vertical case, the flow is unstable for all values of the Reynolds number. Surface tension is found to influence both the direction and celerity of the disturbance. Results are also given for inclinations slightly away from the vertical, where small critical Reynolds numbers do exist.

Problems of the propagation of shock waves and strong detonation waves through ducts of variable cross-section and ducts with porous walls, and the interaction of a rarefaction wave with a shock and a contact surface in one-dimensional unsteady, or two-dimensional steady, flow are solved using a simple rule. Tables enclosed in this paper permit efficient calculation of several problems of non-uniform shock motion.

The object of this paper is to present a unified account of the distributions of velocity, shear stress and enthalpy in the compressible turbulent boundary layer on a flat plate. As a start, the set of velocity profiles measured over a range of heat-transfer conditions at Mach numbers between 5 and 8 by Lobb, Winkler & Persh (1955) is examined. It is found that by plotting in terms of the Howarth variable $\eta = \int ^y_0 (\rho|\rho_ 0)dy$, the outer parts of the profiles for different Mach numbers are brought together on a single curve of the approximate form u/u∞ = (η/Δ)1/n, Δ being the transformed boundary-layer thickness. By evaluating the reference density ρ0 and kinematic viscosity ν0 at the so-called ‘intermediate’ enthalpy (Eckert 1955) the innerparts of the profiles can also be collapsed, although less completely, to fit a ‘law of the wall’ u/uτ = A log (ηuτ/ν0 - c) + B. Here uτ = (τw/ρ0)½, and A, B and c are the same constants as in incompressible flow.

These properties provide a physical starting point from which the remaining features of the mean flow can be calculated. By substitution of appropriate stream functions in the equation of motion the distribution of shear stress r in inner and outer regions is found; this approximates to the form $\tau |\tau_w = 1 - (u|u_\infty)^{(n+2)}$ over the whole layer. A relation between the distributions of enthalpy and shear stress is then found from the energy equation, using a turbulent Prandtl number a which is assumed constant across the layer to relate eddy conductivity to eddy viscosity. The final expression is similar in form to Crocco's integral for the laminar boundary layer with α taking the place of the laminar Prandtl number σ, but contains two extra terms proportional respectively to (α − σ)cf and (α − σ)c½j, which represent the effect of the inner viscous regions.

The enthalpy integral is evaluated using the stated velocity-shear relation, and an expression which agrees well with the available experimental data is found for the heat-transfer coefficient as a function of recovery factor and skin-friction coefficient. It is also found that the usual quadratic enthalpy-velocity relation, exact for α = σ = 1, remains an acceptable approximation for Prandtl numbers considerably different from unity.

Two parallel infinite plane disks, between which is contained a viscous fluid, oscillate torsionally about a common axis. Of specific interest are the cases where (i) one disk only is in motion, and (ii) the disks oscillate 180° out of phase, but with the same frequency and amplitude. The basic parameter is found to be the Reynolds number derived from the frequency, the kinematic viscosity and the disk separation. Solutions of the Navier-Stokes equations for both small and large Reynolds numbers are developed and the transverse and radial-axial flows investigated.

An experimental investigation has been made of the effect of ejecting nitrogen and helium coolant gases at the nose of a bluff body at a nominal Mach number of 5·8. The gases were ejected with ‘swirl’, to encourage them to flow tangentially to the model surface at ejection, and also straight out. Measurements were made of the pressure and temperature on the surface of the model at incidences of 0, 4, 8°, and for a range of coolant gas flows. From these measurements the flow patterns and distributions of heat flux were deduced.

It was found that ejection with swirl did not in fact lead to an easing of the heating problem because the high tangential velocity with which the coolant was injected into the boundary layer increased the wall shear stress, and hence, by the Reynolds analogy, the heat flux, so that it predominated over the reduced driving temperature difference associated with the cooled boundary layer.

With straight-out ejection it was found that the heat alleviation capabilities of the ejected coolant were reduced considerably if the flow rate was sufficiently large that the bow shock wave was bulged out. However, provided that the external flow is not disturbed, straight-out ejection provides an effective way of reducing the heat flux.

When a concentrated pressure travels with constant velocity over the free surface of water, it carries with it a familiar pattern of ship waves. Let viscosity and surface tension be neglected, let the free-surface condition be linearized, let the depth of water be assumed infinite, and let initial transient effects be ignored. Then, as is well known, the wave motion everywhere can be found by standard methods in the form of a double integral. The wave pattern at a great distance behind the disturbance can be found by an application of the ordinary method of stationary phase, which shows that the wave amplitude is considerable inside an angle bounded by the two horizontal rays θ = ± θc from the disturbance, where $\theta_c = \rm {sin}^{-1} {\frac{1}{3} \eDot 19{\frac{1}{2}\deg$. But the method fails in two regions, near the track θ = 0 of the pressure point, and near the critical lines θ = ±θc.

These two regions are treated in the present paper. It is shown that near θ = 0 the linearized surface elevation oscillates with indefinitely increasing amplitude and indefinitely decreasing wavelength. (This result holds only when the pressure is concentrated at a point and applied at the free surface.) Near the critical lines the surface elevation at a greater distance behind the pressure point can be expressed in terms of Airy functions, and this expression goes over into the known wave pattern inside the critical angle. It is shown that near the critical lines the crest length increases as the cube root of the distance, and that the separation between crests remains constant. Contour maps of the wave surface are given for three distances behind the moving pressure point.

The flow of a viscous incompressible electrically conducting fluid past a sphere is studied; the uniform ambient flow field is colinear with the ambient uniform magnetic field. The force exerted on the sphere is computed for various conductivities and Reynolds numbers; of particular interest is the distinction in behaviour between the flow with ambient particle speed greater than ambient Alfvén speed and that with particle speed less than Alfvén speed.

Gravity waves on the surface of an inviscid incompressible fluid of finite depth are considered. The waves are assumed to be periodic in time and in the horizontal direction. The surface profile, potential function, pressure and frequency of the motion are determined (to third order) as series in powers of the amplitude divided by the wavelength. It is found that the frequency increases with amplitude for depths less than a certain multiple of the wavelength and decreases with increasing amplitude for greater depths. Graphs of the surface profile and of the pressure as a function of depth are included.

Exact one-dimensional solutions of the magnetohydrodynamic equations of an incompressible fluid are considered. It is shown that one class of plane wave solutions of the linearized equations is also a possible class of solutions of the general equations including the effect of displacement current. A similar result is also established for the solutions for a horizontally stratified fluid. For the particular case when the viscosity is equal to the magnetic diffusivity an exact solution is obtained for the magnetohydrodynamic Rayleigh problem for a semi-infinite plate. It is shown that this solution may be employed directly to give the solution for liquids of small, but not necessarily equal, viscosity and magnetic diffusivity.

Starting with expressions for viscous stress, heat flux and diffusion flux, we formulate a continuum theory for steady flow of a binary mixture of chemically inert perfect gases through a normal shock of arbitrary strength. For shocks of vanishing strength, a solution by series expansion in Grad's (1952) shcok-strength parameter gives a result essentially the same as found previously by Dyakov (1954). For stronger shocks a straightforward numerical integration, quite analogous to that useful in the simpler pure-gas problem, is laid out.

The resulting problem has eight parameters: shock strength, ratio of specific heats, ration of bulk viscosity to shear viscosity, Prandtl number, Schmidt number, thermal diffusion factor, molecular mass ratio, and initial mixture concentration. A dozen examples were worked out on a simple desk calculator to exhibit the influence of some of these parameters. They involve the gas pairs argon40–argon36, argon–neon, argon–helium, and xenon–helium.

In discussing the results, special attention is paid to the degree of success with which the weak-shock theory may be extrapolated to arbitrary shock strength, and to the question of the accuracy of the Navier-Stokes approximation for a mixture of gases of very different molecular weights.