An analytic solution is obtained for the head-on collision of strong shock waves with supersonically moving, axisymmetric, slender bodies. A new method of the solution is developed using integral transforms. Pressure distributions on the surface of a cone illustrate the effects of shock strength and body speed. The results are compared with those of Blankenship (1965), which were obtained by numerical methods.

It is also shown that the same analysis can, with certain modifications, be adapted for treating the shock-shock interaction on thin two-dimensional aerofoils of arbitrary shape at small incidence.

The drag experienced by a sphere moving with velocity dependent on a single time scale t0 in an unbounded viscous liquid is considered under the assumption that the Reynolds number is small. It is shown that, unless t0 is sufficiently large, an asymptotic expansion in the Reynolds number becomes invalid for large times. Moreover, when the expansion is valid for large times, the drag can differ considerably from that predicted by the unsteady Stokes equations.

The conservation equations describing steady, incompressible flow in a variable-area duct with mass transfer occurring at its walls are simplified by linearizing the inertial and convective terms. Solutions to a large class of problems can be obtained by means of the general method which is presented. Particular examples considered are entrance flow with heat transfer to an isothermal wall in the presence of mass addition, constant rate of injection of a foreign gas through the wall, vaporization and sublimation of a volatile wall material, and gas-phase combustion of a fuel which enters the duct from its wall. Comparison of the present results with previous work and with new experimental results is discussed for the first of these applications. It is concluded that the present results for velocity and pressure fields are likely to be accurate for small values of the Reynolds number based on wall injection velocity, that the present results for temperature and composition fields are likely to be accurate in the absence of wall mass transfer, and that in the absence of wall mass transfer the linearization technique is likely to exhibit its highest accuracy for flows with uniform entrance conditions.

A thin disk, immersed in a fluid consisting of two superposed homogeneous inviscid liquids, moves in its own vertical plane. A Green's function is found for the potential and a formula for the gravity wave energy is found and evaluated for a specific case.

A general theory is presented to account for the small, free, lateral motions of a flexible, slender, cylindrical body with tapered ends, totally submerged in liquid and towed at steady speed U. For particular shapes of the ends and length of tow-rope, it is shown that the body may be subject to oscillatory and non-oscillatory instabilities for U > 0; at small U, these instabilities correspond to those of a rigid body. At higher U, the system generally regains stability in the above modes, but may be subject to higher-mode, flexural oscillatory instabilities. The critical conditions of stability are calculated extensively and the effect on stability of a number of dimensionless parameters is discussed. It is shown that optimum stability is achieved with a streamlined nose, a blunt tail and a short tow-rope.

Some experiments are described which were designed to test the theory. Rubber cylinders of neutral buoyancy were held in vertical water flow by a nylon ‘tow-rope’. Provided the tail was streamlined and the tow-rope not too short, ‘criss-crossing’, non-flexural oscillations developed at very low flow. Increasing the flow, these oscillations ceased and the cylinder buckled like a column; subsequently higher-mode flexural oscillations developed. However, for a sufficiently blunt tail and short tow-rope, the system was completely stable.

The experimental observations are generally in qualitative agreement with theory. Quantitative comparison of the various instability thresholds and stable zones between experiment and theory, based on estimated values of some of the theoretical dimensionless parameters, is also fairly good.

The effect of finite amplitude on the stable and unstable states of a column of an ideal fluid of circular cross-section under the action of surface tension is studied. The method of solution is a formal extension of the linearized theory; it consists of assuming that the exact solution may be expanded in a power series of a small parameter characterizing the amplitude. The calculation is carried out to the point where the first non-trivial term of the finite amplitude effect is obtained. For the stable states, the result shows that the characteristic wavelength of a disturbance which appears to be stationary with respect to an observer is decreased by the finite amplitude effect. For the unstable states, it reveals that the growth rate depends not only on the wavelength and the magnitude but also on the type of disturbance imposed initially. The last result is a direct consequence of the fact that two independent types of initial disturbance, the disturbance of the velocity field and the disturbance of the free surface, may be imposed simultaneously on the jet.

A stabilizing gradient of solute inhibits the onset of convection in a fluid which is subjected to an adverse temperature gradient. Furthermore, the onset of instability may occur as an oscillatory motion because of the stabilizing effect of the solute. These results are obtained from linear stability theory which is reviewed briefly in the following paper before finite-amplitude results for two-dimensional flows are considered. It is found that a finite-amplitude instability may occur first for fluids with a Prandtl number somewhat smaller than unity. When the Prandtl number is equal to unity or greater, instability first sets in as an oscillatory motion which subsequently becomes unstable to disturbances which lead to steady, convecting cellular motions with larger heat flux. A solute Rayleigh number, Rs, is defined with the stabilizing solute gradient replacing the destabilizing temperature gradient in the thermal Rayleigh number. When Rs is large compared with the critical Rayleigh number of ordinary Bénard convection, the value of the Rayleigh number at which instability to finite-amplitude steady modes can set in approaches the value of Rs. Hence, asymptotically this type of instability is established when the fluid is marginally stratified. Also, as Rs → ∞ an effective diffusion coefficient, Kρ, is defined as the ratio of the vertical density flux to the density gradient evaluated at the boundary and it is found that κρ = √(κκs) where κ, κs are the diffusion coefficients for temperature and solute respectively. A study is made of the oscillatory behaviour of the fluid when the oscillations have finite amplitudes; the periods of the oscillations are found to increase with amplitude. The horizontally averaged density gradients change sign with height in the oscillating flows. Stably stratified fluid exists near the boundaries and unstably stratified fluid occupies the mid-regions for most of the oscillatory cycle. Thus the step-like behaviour of the density field which has been observed experimentally for time-dependent flows is encountered here numerically.

This paper discusses the mathematical properties of similar solutions of the boundary-layer equations in a compressible model fluid, under assumptions first introduced by Stewartson and by Li & Nagamatsu. Assuming a favourable pressure gradient and that backflow is not present, our results include (among other things) a rigorous proof that velocity overshoot occurs in the boundary layer if the wall is heated, and that this is true whether or not suction, blowing or slipping occurs at the wall; while, conversely, velocity overshoot does not occur when the wall is cooled and the amount of slipping at the wall is suitably restricted.

A study is made of the response of a diffusion flame sheet to perturbations of reactant concentration which are introduced, either as changes in the free streams, or as appropriate initial distributions throughout the field. Species’ and energy conservation requirements are approximated by linearized boundary-layer equations; general solutions are derived for species, enthalpy and temperature distributions, as well as for the flame sheet shape, and a number of specific problems are solved.

The axisymmetric, free-surface response of a semi-infinite viscous liquid to either a point impulse or an initial displacement of zero net volume is calculated. The asymptotic disturbance is resolved into three components: (i) a damped gravity wave, which represents a primary balance between gravitational and inertial forces with secondary, but cumulative, modification by viscous forces; (ii) a diffusive motion, which represents a balance between viscous and inertial forces; (iii) a creep wave, which represents a balance between gravitational and viscous forces. Van Dorn has suggested that the results may be relevant to the concentric circular ridges that surround the crater Orientale on the Moon.

It has been shown experimentally that quite large departures occur from the universal inner-law velocity distribution in the presence of severe favourable pressure gradients in turbulent boundary layers and that these departures are associated with the tendency for the turbulent boundary layer to revert to a laminar state. From the measurements a criterion for the onset of reverse transition has been deduced in terms of the mean shear-stress gradient in the wall region of the flow. Experiments in fully developed pipe and channel flows suggest that the proposed criterion may be quite generally applicable to all fully turbulent shear flows.

The hydrodynamic stability is examined of liquids in which the viscosity varies with distance below a free surface. It is assumed that the viscosity becomes indefinitely large with distance from the surface, and that there is an ‘effective depth’ within which most of the motion occurs; but, otherwise, the viscosity distribution is arbitrary. The particular cases of wave-generation by a concurrent air flow at a horizontal liquid surface, and the stability of inclined flow under gravity are treated. It is shown that instabilities may occur which are similar to those known for thin uniform liquid films.

It is shown that as a result of their non-linear interactions, internal gravity waves in an unbounded fluid can be trapped to a layer of finite depth by periodic small variations in either the density gradient or in a weak horizontal steady current. This trapping occurs when the vertical component of the wave-number is half that of the density gradient or of the current variations. The energy density of the wave motion trapped near the ocean surface decreases exponentially with depth over a distance that is inversely proportional to the magnitude of the variations in density gradient or in horizontal current speed.