The effects of a reduction in momentum flux on the downstream flow characteristics of a steady, two-dimensional flow are investigated. In particular, quantities such as the changes in mean depth, mean fluid velocity, mean kinetic and potential energies and the length of the induced downstream wave are examined. This is done with the use of a fourth-order perturbation expansion in wave slope. The results are compared with the conflicting results that had been obtained previously by different authors. Agreement is found with the second-order theory of Benjamin (1970), but the work of Doctors & Dagan (1980) is found to be in error and is corrected.

In a large-scale process at I.C.I. Mond Division a flocculated suspension of fine solid particles is allowed to settle through an aqueous solution towards the horizontal interface with an underlying denser solution. When the solids have settled into the lower layer they are pumped away and dumped while the clarified upper layer can be safely disposed of or recycled. The passage of the floes through the interface is the rate-determining step in the process, and the analysis of this part of it is the subject of the paper. It is hoped that the analysis may have a much wider application in that the settling process could be used as the basis of a direct laboratory method of measuring properties of floes. At present both the porosity and permeability of flocs are almost impossible to measure directly, but, using the present analysis, the settling of a single floc through two successive density interfaces would provide (in principle) values for both.

The flow of a rotating, two-layer fluid system over long ridges of constant cross-section is considered. Homogeneous incompressible fluids of constant, but different, density are confined between two ‘infinite’ horizontal plane surfaces which rotate at a constant angular velocity about a vertical axis. The ridge is located on the lower surface while upstream of the ridge each fluid is in uniform motion in a direction normal to the ridge. Solutions are obtained for both an f-plane and a β-plane under the following restrictions: E [Lt ] 1, Ro ∼ E½, S ∼ O(1), H/L ∼ O(1), d/L ∼ O(1) and h/L ∼ E½ where E is the Ekman number, Ro is the Rossby number, S is a stratification parameter, H/L is the two-fluid depth to ridge width ratio, d/L is the lower fluid depth to ridge width ratio and h/L is the aspect ratio of the ridge. This set of restrictions assures that viscosity is important in considering the dynamics of the system. Furthermore the restrictions are ones that make laboratory experimentation feasible. Solutions are also presented for the non-viscous case (i.e. E = O), and are compared with their viscous counterparts. The importance of viscosity in this physical system is thus demonstrated.

Strongly nonlinear vortex-Tollmien-Schlichting-wave interaction equations are derived for the case where the undisturbed motion represents the developing flow in a circular pipe. The effect upon the equations of moving the wave input position further downstream is investigated and the development of the flow is found to be accelerated by increasing the size of the wave disturbance. Numerical solutions of the three-dimensional interaction equations are presented and indicate that the form of interaction considered here appears to promote the three-dimensionality as the flow develops downstream. It is shown that one of the interactions considered here can develop within an initially two-dimensional Blasius boundary layer.

In this paper we solve some model problems by the method of matched expansions and deduce some features of Green's function using the reciprocal theorem to demonstrate that scattering by density gradients can enhance the radiation efficiency of quadrupoles to that of dipoles. If a fluid is subjected to body forces close to and on both sides of a density interface, then the source system degenerates to a quadrupole only if the components of the forces perpendicular to the interface are equal in magnitude and opposite in direction and, additionally, the components of the forces per unit fluid density parallel to the interface are equal but opposite. In other cases dipole rather than quadrupole radiation efficiency is achieved. But we also show that, despite published assertions to the contrary, the radiation efficiency is not enhanced further and there is no monopole contribution to the radiated sound.

Laboratory experiments and numerical modelling studies have been performed for a rotating, thermally driven fluid system in a cylindrical annulus with a vertical rotation vector and axis of symmetry. The thermal forcing was through the imposition of an axisymmetric temperature gradient on a thermally conducting lower boundary, with additional heating through the outer sidewall. The upper and inner walls were nominally insulating. Flow patterns were observed in the experiments through the use of small, reflective flakes (Kalliroscope) in the working fluid, which was water. The rotation rate and temperature difference were varied to construct a regime diagram in thermal Rossby number–-Taylor number space. The curve separating axisymmetric flow from wave flow is ‘knee-shaped’, similar to the side-heated and -cooled baroclinic annulus which has been extensively investigated previously. Very near the transition curve, the initial wavenumber persists indefinitely, but well into the wave regime the initial wavenumber is higher than the equilibrated value. Far enough into the wave regime, the initial waves have wavenumbers several times that of the equilibrated value, and the initial disturbances form near the outer wall very early in the experiment. Numerical studies indicate that these waves are effective in distributing heat and that they occur in a region of positive static stability. These waves rapidly grow inward to fill the annulus and reduce in number as weaker waves are absorbed by the stronger ones. The period of transition between these waves and the equilibrated long-wave pattern is characterized by irregular flow. Closer to the transition curve, the temporal transition to longer waves as the flow equilibrates is simpler, with initial waves filling the annulus. In that case, the transition is characterized by a slow process of individual waves weakening and merging with adjacent waves.

The method of matched asymptotic expansions is used to extend the short-wave asymptotics of the transmission coefficient T by the addition of the terms of order 1/N5, (logN)2/N6 and log N/N6 as N → ∞ (where N = wavenumber times cylinder radius). The result is the formula \begin{eqnarray*} T &=& \frac{2{\rm i}}{\pi N^4}\exp (-2{\rm i}N)\left[1+\frac{4\log N}{\pi N}-\frac{4}{\pi N} \bigg(2-\gamma-\log 2+\frac{{\rm i}\pi}{8}\bigg)+\frac{8(\log N)^2}{\pi^2N^2}\right.\\ && \left.-\frac{8\log N}{\pi^2N^2}\bigg(5-2\gamma - \log 4+\frac{{\rm i}\pi}{4}\bigg)\right] + O\bigg(\frac{1}{N^6}\bigg)\quad {\rm as}\;N\rightarrow \infty \end{eqnarray*} (where γ = Euler's constant). The first term above is that derived rigorously by Ursell (1961) using an integral-equation method; the second term is that added by Leppington (1973) using matched asymptotic expansions; and the next three terms are those derived in this paper. Significant agreement between numerical values of T obtained from the completed fifth-order asymptotics and those obtained using Ursell's multipole expansions is demonstrated for 8 [les ] N [les ] 20 (table 2). The extensions of the perturbation expansions for the potential in the various fluid sub-domains (used in the method of matched expansions) provide some interesting cross-checks, between the solutions for potentials occurring later in the series and determined at advanced matching stages, with those for potentials occurring earlier on and determined independently at an earlier stage in the matching process. Some examples are given.

In this paper, we study the velocity field at the density interface of a two-layer stratification system when the flow is forced at the mid-depth of the lower layer by the source–sink forcing method. It is known that, in a sufficiently strong linear stratification, the source–sink forcing in certain configurations produces a single-vortex pattern which corresponds to the lowest eigenmode of the Helmholtz equation (Kanda & Linden 2001). Two types of forcing configuration are used for the two-layer experiments: one that leads to a steady single-vortex pattern in a linear stratification, and one that results in an unsteady irregular state. Strong single-vortex patterns appear intermittently for the former configurations despite the absence of stratification at the forcing height. When the single-vortex pattern occurs at the density interface, a similar flow field extends down to the forcing height. The behaviour is explained as the coupling of the resonant eigenmode at the interface with the horizontal component of the forcing jets. The results show that stratification can organise a flow, even though it is forced by an apparently random three-dimensional forcing.