The weakly nonlinear, two-dimensional problem for the disturbance due to a slender obstacle in a uniformly stratified, Boussinesq fluid moving past the obstacle with constant basic horizontal velocity U, is considered up to second order in the amplitude ε of the disturbance. Analogous rotating problems are also treated. Particular attention is given to calculating explicitly the columnar-disturbance strengths upstream and downstream of the obstacle, both in the stratified and in the rotating problems, with a view to discussing the truth or otherwise of Long's hypothesis (LH).

Whether or not columnar disturbances are found far upstream, violating LH, depends, interalia, on whether or not the flow is externally bounded by rigid horizontal planes (or by a tube or annulus, in the rotating problem), and on whether the problem is made determinate by means of an ‘inviscid transient’ formulation, or by means of a ‘viscous’ one.

The inviscid, transient, bounded problem, for time-development of lee waves from a state of no initial disturbance, always exhibits columnar disturbances oforder ε2 somewhere in the fluid. They are generated, not near the obstacle, but in the ‘tails’ or transient terminal zones of the lee-wave trains. The columnar-disturbance strengths are largely independent of how the flow is set up from an initially undisturbed state. I n all but one instance the effect is non-zero far up-stream. The exception is the singly-subcritical stratified (or narrow-gap rotating) case, in which the excitation has modal structure sin(2z), the fluid region being 0 [les ] z [les ] π in this case the only columnar disturbance that can penetrate up-stream has structure sinz and so is not excited.

A completely different result holds for ‘viscous’ formulations for unseparated, bounded régimes (with steady lee waves spatially attenuated by effects of small molecular diffusion). The strengths of all columnar disturbances, upstream and downstream, vanish in the limit of small diffusivity.

In the inviscid, transient, unbounded problem, the upstream influence is, likewise, evanescent, being O(ε2t−2) as time t → ∞.

The basic expansion in powers of ε will be invalid for times ∝ ε−1 or greater, because of resonant-interactive instability of the lee waves.

The dispersion considered consists of a large number of identical small rigid spheres with random positions which are falling through Newtonian fluid under gravity. The volume fraction of the spheres (c) is small compared with unity. The dispersion is statistically homogeneous, and the axes of reference are chosen so that the mean volume flux across any stationary surface is zero. The problem is to determine the mean value of the velocity of a sphere (U). In §3 there is described a systematic and rigorous procedure which overcomes the familiar difficulty presented by the occurrence of divergent integrals, essentially by the choice of a quantity V whose mean value can be found exactly and which has the same long-range dependence on the position of a second sphere as U so that the mean of U – V can be expressed in terms of an absolutely convergent integral. The result is that, correct to order c, the mean value of U is U0(1 – 6.55 c), where U0, is the velocity of a single sphere in unbounded fluid. The only assumption made in the calculation is that the centres of spheres in the dispersion take with equal probability all positions such that no two spheres overlap; arguments are given in support of this assumption, which is expected to be valid only when the spheres are identical. Calculations which assume a simple regular arrangement of the spheres or which adopt a cell model of the hydrodynamic interactions give the quite different result that the change in the mean speed of fall is proportional to $c^{\frac{1}{3}}$, for reasons which are made clear.

The general procedure described here is expected to be applicable to other problems concerned with the effect of particle interactions on the average properties of dispersions with small volume fraction of the particles.

The purpose of this paper is to consider theoretically how the homogeneous suction from a slightly concave permeable wall will affect the instability of the laminar boundary layer to the onset of longitudinal vortices. The curves of neutral stability and of several growth factors of the vortices are given. The critical values of the Görtler parameter Gc, and the wavenumber σc, based on the momentum thickness of the boundary layer, are found: Gc = 1·17 and σc = 0·22.

The hydrodynamic stability of plane Poiseuille flow of second—order fluids to finite amplitude disturbances is examined using the method of Stuart, and Watson as extended by Reynolds & Potter. For slightly non-Newtonian fluids subcritical instabilities are predicted. No supercritical equilibrium states are expected if the entire spectrum of disturbance wavelengths is present. Possible implications with respect to the Toms phenomenon are discussed.

The importance of fluctuating turbulent stresses in the flow over a wave is examined. It is shown that anisotropic stresses, which are most likely to be turbulent Reynolds stresses, are essential to the process of energy flow to the wave. Two fundamentally different methods of predicting fluctuating turbulent Reynolds stresses are examined. One method makes use of a phenomenological closure of the conservation equation for the turbulent Reynolds stresses and is similar to the turbulent boundary-layer calculation scheme of Bradshaw, Ferriss & Atwell (1967). The second method is based on the assumption that the turbulent stresses are determined by the recent history of velocity shear experienced by a fluid parcel and results in a viscoelastic constitutive relation for the turbulence; in the limit of shortest’ memory’ this relation becomes the eddy viscosity model proposed by Hussain & Reynolds (1970). Comparison of predicted and measured values of surface pressure indicates that the eddy viscoelasticity model can explain measured pressure distributions but the comparison is not conclusive. Suggestions for further measurements are made.

A simple model is suggested to explain the flow mechanism at the upstream edge of a square-edged broad-crested weir. The separation bubble that may be seen to occur at this point is treated as an area of constant static head, while the main flow outside the bubble is deemed to be irrotational and divided from the bubble by a free streamline. If this model is accepted, the flow pattern will be such that energy requirements will be met along the free surface and the streamline bounding the bubble while, within these boundaries, the Laplace equation will be satisfied at every point. Accordingly, a solution satisfying these conditions is established by the use of a relaxation technique.

In practice, it is likely that the cavity flow within the bubble will be bounded not by a single streamline but by a turbulent mixing zone and that there will be some increase in pressure near the point of re-attachment. Nevertheless, the surface profile and flow pattern observed in experiments show fair agreement with those predicted using the simple model. Whilst acknowledging, then, that the bubble and the zone bounding it are in fact of a more complex character, we may say that this simplified treatment affords a sound model of the main flow and so permits a better understanding of the action of the square-edged broad-crested weir.

It is shown that for a given body and a given orientation g there is always a position of the centre of mass which produces a stable falling motion in a very viscous fluid with g vertical and, in general, with a spin about the vertical axis. The corresponding terminal settling speed is bounded by means of several variational principles.

Relations between the terminal speeds for falls with different downward directions and between the terminal speed and the geometry of the body are deduced. In particular, it is proved that for a large class of slender bodies the first approximation to the drag obtained from the slender-body theory of Burgers (1938) is correct. It follows that the ratio of the terminal speeds for falls with the long axis vertical and horizontal is near two.

To solve the kinetic equation for particles of a monodisperse two-phase mixture the method of successive approximations is developed; this resembles in its main features the well-known Chapman-Enskog method in the kinetic theory of gases. This method is applicable for a mixture whose state differs slightly from the equilibrium, i.e., when time and space derivatives of the dynamic variables describing the mean flow of both phases of the mixture are sufficiently small. Accordingly, the solution obtained is valid when the time and space scales of the mean flow exceed considerably those for random pseudo-turbulent motion of particles and a fluid. The conservation equations for determination of all the dynamic variables are formulated in approximations which have the same meaning as those of Euler and Navier & Stokes in hydromechanics of one-phase media.

Velocity distributions within the boundary layer of a swirling flow of incompressible fluid in a convergent conical nozzle have been investigated. Theoretical calculations with boundary conditions more appropriate to physically existent situations discounted the existence of 'super-velocities’ within the boundary layer. Parallel experimental investigations demonstrated an interdependence of core and boundary-layer flows which precluded the maintenance of the flow conditions required by the analysis.

The shock propagation theory of Brinkley & Kirkwood (1947) is extended to provide a uniformly valid analytic solution of point-explosion problems both when the undisturbed medium is uniform and when it is stratified. This is achieved mainly by selecting the parameter expressing a similarity restraint in this theory such that initially it gives precisely the Taylor–Sedov solution, while asymptotically, in the weak regime, still retaining the well-known Landau–Whitham–Sedov form of the solution for shock overpressure. The shock overpressure, as calculated by the present method for spherical and cylindrical blast waves in the entire regime from the point of explosion to where they have become very weak, shows excellent agreement with that from the exact numerical solutions of Lutzky & Lehto (1968) and Plooster (1970). The solution for a spherical shock propagating in an exponential atmosphere stratified by a constant acceleration due to gravity also shows a good agreement with the exact numerical solution of Lutzky & Lehto.

Equations are derived for the variation along a straight coastline of Kelvin and continental-shelf waves. It is assumed that the effects of nonlinearity and dispersion are of the same order in a small parameter ε defined by the equation ε2 = f2=L2/gH, in which f is the Coriolis parameter, L the shelf width and H the water depth beyond the shelf. Kelvin waves are found to satisfy the Korteweg–de Vries equation, while continental-shelf waves satisfy a closely related equation. An approximate rule is derived for the variation along a real coastline of the maximum wave height for fully developed nonlinear Kelvin waves.

When water is slightly stratified, internal gravity waves are considerably shorter than surface waves of comparable frequency. Here, this fact is exploited in demonstrating that an internal wave is unstable when it forms part of a resonant triad with a surface wave and another internal wave whose wave number is approximately equal to that of the original internal wave. It is suggested that in a system where there are two classes of waves of comparable frequencies but greatly differing wavelengths the short waves may be expected to generate long waves by this mechanism.