The nonlinear stability of a viscous film flowing steadily down an inclined plane is investigated by the method of multiple scales. It is shown that the super-critically stable, finite amplitude, long, monochromatic wave obtained by Lin (1969, 1970, 1971) is stable to side-band disturbances under modal interaction if the bandwidth is less in magnitude than to the ratio of the amplitude to the film thickness. Near the upper branch of the linear neutral-stability curve where the amplification rate ci is O(ε2), the nonlinear evolution of initially infinitesimal waves of a finite bandwidth is shown to obey the Landau-Stuart equation, Near the lower branch of the neutral curve, the nonlinear evolution is stronger. An equation is derived for describing this strong nonlinear development of relatively long waves. In practice, disturbance of this type clusters in the form of a hump which cannot be constructed only by the first few harmonics.

The momentum integral method of Klineberg is shown to provide a good description of the major characteristics of two-dimensional laminar viscous-inviscid interactions a t hypersonic speeds. Surface pressure and heat-transfer-rate measurements were made for sharp compression and expansion corners at Mach 12.2 and are compared with the theoretical predictions. The agreement is found to be good for attached, incipient and fully separated flows.

Some theoretical comparisons between methods based on the Klineberg formulation are made which suggest that the full boundary-layer equations are well described using integral methods that incorporate the energy equation. It is further shown that the properties associated with the stability of the governing differential equations are mathematical properties of the analytical model and should not be associated with any physical characteristics of the boundary layer.

A correlation of hypersonic, cold-wall, incipient separation data is also presented.

It is now well known that the turbulence structure of thin shear layers can be strongly affected by the application of extra rates of strain in addition to the shear velocity gradient. Examples of such extra strain rates include lateral divergence or convergence, and streamline curvature in the plane of the mean shear. The changes in Reynolds stress are an order of magnitude larger than would be expected from the explicit extra terms in the Reynolds-stress transport equations, and therefore an order of magnitude larger than predicted by conventional calculation methods. In the present paper, one of a series on ‘complex’ turbulent flows, we show that bulk compression or dilatation (i.e. an extra strain rate div U) also appears to affect turbulent shear layers, typical values of Reynolds stress being increased by compression and decreased by dilatation. The fractional change in Reynolds stress is an order of magnitude larger than the fractional change in volume of a fluid element. The physical mechanism is probably analogous to that responsible for the large effects of divergence or convergence in incompressible flow. Because the phenomenon seems to be of great practical importance we discuss it in the context of engineering calculation methods. An empirical correction formula, analogous to those used to allow for divergence or curvature effects, greatly reduces the large discrepancies found between recent experiments on supersonic boundary layers and calculations by conventional extensions of successful incompressible-flow methods.

It is shown that weak harmonic waves can propagate upstream in a supersonic stream along a plane wall over distances that are very large in comparison with the boundary-layer thickness. These waves are characterized by small reduced frequencies, large wavelengths and small phase velocities. Results are presented for the flow along a single plane wall and also for the flow between two plane parallel plates (channel flow).

The progressive-wave solutions are found by asymptotic expansions for small disturbances, large Reynolds numbers of the basic flow, and small reduced frequencies of the unsteady disturbances. It turns out that, as in the corresponding steady theory, four flow regions have to be distinguished: a middle layer which embraces most of the boundary layer; an inner layer near the wall; the outer flow field; and a transition layer between the middle layer and the outer layer. A quasi-steady treatment of the middle, transitional and outer layers is appropriate. Unsteady effects originate in the inner layer.

The relative importance of viscosity and unsteady effects with regard to the waves is characterized by a dimensionless parameter N which is the product of certain powers of the reduced frequency and the Reynolds number. For N → ∞ and a single wall, Lighthill's steady theory of upstream influence in supersonic boundary layers is recovered as a limiting case of the present theory.

In a drop of liquid which hangs below a horizontal support or a t the end of a tube, the forces due to surface tension, pressure and gravity are in equilibrium. Amongst the many possible equilibrium shapes of the drop, only those which are stable occur naturally. The calculus of variations has been used to determine theoretically the stable equilibria, by calculating the energy change when the liquid in equilibrium experiences axially symmetrical perturbations under physically realistic constraints. If the energy change can be made negative, the drop is unstable. With this criterion, stable equilibria have been identified through which the naturally growing drops evolve until they reach a maximum volume, when they become unstable. These results are illustrated by calculations relating to typical experimental conditions.

The form taken by a finite amplitude internal seiche in a shallow two-layer fluid within a long narrow container is akin to a shock wave in a gas-filled tube; an undular bore or surge is present which reflects back and forth along the length of the container. The internal surge in Loch Ness is sometimes observed to retain its amplitude over several seiche periods, and a resonance with the wind appears possible. This idea is explored by developing a theoretical model and by making a laboratory experiment, but difficulties are encountered in estimating the size of the parameters which characterize the natural phenomenon.

The equations governing the nonlinear development of a centred three-dimensional disturbance to plane parallel flow at slightly supercritical Reynolds numbers are obtained, In contrast to the corresponding equation for two-dimensional disturbances, two slowly varying functions are needed to describe the development: the amplitude function and a function related to the secular pressure gradient produced by the disturbance. These two functions satisfy a pair of coupled partial differential equations. The equations derived in Hocking, Stewartson & Stuart (1972) are shown to be incorrect, Some of the properties of the governing equations are discussed briefly.

The steady flow of an ideal gas past a conical body is investigated by the method of matched asymptotic expansions, with particular emphasis on the flow near the singular ray occurring in linearized theory. The first-order problem governing the flow in this region is formulated, leading to the equation of Kuo, and an approximate solution is obtained in the case of compressive flow behind the main front. This solution is compared with the results of previous investigations with a view to assessing the applicability of the Lighthill-Whitham theories.

We consider the two-dimensional problem of a linearly stratified salt solution contained between two infinite vertical plates. The fluid and the plates are initially at the same temperature. At t = 0, one of the plates is given a step increase in temperature, while the other is maintained at the initial temperature. A time-dependent basic flow is thus generated. The stability of such a time-dependent flow is analysed using an initial value problem approach to the linear stability equations. The method consists of initially distributing small random disturbances of given vertical wavelength throughout the fluid. The disturbances may be in the vorticity, temperature or salinity. The linearized field equations are integrated numerically. The growth or decay of the kinetic energy of the perturbation delineates unstable and stable states. Results have been obtained for a wide range of gap widths. The critical wavelength and the critical Rayleigh number compare favourably with those obtained previously in both physical and numerical experiments.

The limitations of existing one-dimensional experiments on double-diffusive convection are discussed, and a variety of new two-dimensional phenomena are described. We have used the sugar-salt system and shadowgraph photography to make exploratory studies of motions which can arise in a fluid with two smooth, opposing, vertical concentration gradients, with and without horizontal gradients. Many different effects have been observed, the most important of which are the following, (a) In the ‘finger’ case, local disturbances can propagate rapidly as wave motions, which cause a simultaneous breakdown to convection over large horizontal distances. (b) Layers formed in the’ diffusive’ sense overturn locally to produce fingers, but propagate more slowly, as convective rather than wave motions, (c) A series of layers, separated by diffusive interfaces, can become unstable, in the sense that successive layers merge in time as their densities become equal, (d) The presence of horizontally separated sources of water of similar density but different T, S characteristics can lead to the development of strong vertical gradients and extensive quasi-horizontal layering.

Most of our results are qualitative, but it is hoped that they will stimulate further quantitive work on each of the new processes described. It is already clear that much more needs to be done before the mechanism of formation of layers observed in the ocean can be regarded as properly understood.

After making the laboratory approximation of small magnetic Reynolds number, the steady, axisymmetric and purely azimuthal velocity profile that in principle can be generated in an incompressible viscous electrically conducting fluid contained in a fixed infinitely long circular cylinder by a magnetic field transverse to the cylinder axis and uniformly rotating with low frequency is subjected to infinitesimal axisymmetric perturbations. The principle of the exchange of stabilities is assumed to hold and the marginal-stability problem becomes a sixth-order eigenvalue problem involving the magnetic Taylor number and the axial wavenumber. An asymptotic analysis, based on the assumption that the magnetic Taylor number is large, and using solutions of the comparison equation d6y/dz6 = zy, is presented in order to obtain first approximations to the neutral-stability curves of the first and second eigenmodes, and compared with the results of direct numerical integration. It is found that at the onset of instability the secondary motions have a multi-cell structure, the motions in the region, near the cylinder wall, of adversely distributed angular momentum driving through weak viscous action the cells in the interior.

The present series of studies is concerned with low-Reynolds-number flow in general; the main objective is to develop an effective method of solution for arbitrary body shapes. In this first part, consideration is given to the viscous flow generated by pure rotation of an axisymmetric body having an arbitrary prolate form, the inertia forces being assumed to have a negligible effect on the flow. The method of solution explored here is based on a spatial distribution of singular torques, called rotlehs, by which the rotational motion of a given body can be represented.

Exact solutions are determined in closed form for a number of body shapes, including the dumbbell profile, elongated rods and some prolate forms. In the special case of prolate spheroids, the present exact solution agrees with that of Jeffery (1922), this being one of very few cases where previous exact solutions are available for comparison. The velocity field and the total torque are derived, and their salient features discussed for several representative and limiting cases. The moment coefficient CM = M/(8πμω0ab2) (M being the torque of an axisymmetric body of length 2a and maximum radius b rotating at angular velocity ω0 about its axis in a fluid of viscosity μ) of various body shapes so far investigated is found to lie between $\frac{2}{3}$ and 1, usually very near unity for not extremely slender bodies.

For slender bodies, an asymptotic relationship is found between the nose curvature and the rotlet strength near the end of its axial distribution. It is also found that the theory, when applied to slender bodies, remains valid at higher Reynolds numbers than was originally intended, so long as they are small compared with the (large) aspect ratio of the body, before the inertia effects become significant.