Parallel flows with step function velocity and density profiles can support waves which have negative energy, in the sense that exciting them lowers the total energy of the system. A number of instabilities can occur because of the coexistence of positive and negative energy waves, or because of the damping of negative energy waves; some particular examples are discussed to show how appreciation of this role of negative energy waves allows one to predict the existence of instability before doing any detailed analysis, and to gain insight into the instability mechanism.

Cairns (1979) has recently shown how the concept of negative wave energy used in plasma physics can be exploited to obtain insights into the linear and nonlinear instability of certain parallel flows. We here analyse the linear stability and nonlinear three-wave resonance in a three-layer flow with step-wise velocity and density profiles. The results are in complete agreement with Cairns’ qualitative predictions. In particular, the existence is confirmed of an ‘explosive’ instability in which all three waves of a resonant triad grow simultaneously while total wave energy is conserved. Such nonlinear instabilities, previously undetected in fluid flows, may well be important in the ocean and atmosphere.

This paper studies the effect of alternating or rotating magnetic fields on containers of conducting fluid. The magnetic Reynolds number is assumed small. The frequency of alternation or rotation is rapid so the magnetic field is confined to a thin layer on the surface of the container. A boundary-layer analysis is used to find the rate of vorticity generation due to the Lorentz force. When the container is an infinitely long cylinder of uniform cross-section, alternating fields normal to the generators or fields rotating about an axis parallel to the generators generate vorticity at a constant rate. For containers of any other shape the rate of vorticity generation includes both constant and oscillatory terms. A perturbation analysis is used to study the flow induced in a slightly distorted circular cylinder by a rotating field. Complex flows develop in the viscous-magnetic boundary layer which may be unstable.

A constitutive equation appropriate for flow of cohesionless granular materials at high deformation rates and low stress levels is proposed. It consists of an extension and a reinterpretation of the theory of Goodman & Cowin (1972), and accounts for the non-Newtonian nature of the flow as evidenced by Bagnold's (1954) experiments. The theory is applied to analyses of gravity flows in inclined chutes and vertical channels. Experiments were set up in an attempt to generate two-dimensional shear flows corresponding to these analyses. Velocity profiles measured by a technique which makes use of fibre optic probes agree qualitatively with the theoretical predictions, but direct comparison is inappropriate because of unavoidable side-wall friction effects in the experiments. The existing measure of agreement suggests that the most prominent effects have been included in the proposed constitutive relations. Tests in the inclined chute revealed the possible existence of surge waves and granular jumps analogous to hydraulic jumps.

The weakly reactive state leading to the ignition of a steady laminar boundary-layer flow of a combustible mixture over a hot, isothermal, non-permeable, non-catalytic flat plate is studied both numerically and using matched asymptotic analysis in the realistic limit of large activation energy. It is shown that the flow consists of a locally similar diffusive-reactive region next to the plate and a non-similar diffusive-convective region outside it; that the analytic solution obtained reproduces the lower half of the S-shaped ignition-extinction response curve such that ignition is expected to occur when a suitably defined Damköhler number, which increases with the streamwise distance, reaches unity; and that at the point of ignition the heat transfer from the wall vanishes identically. An explicit expression for the minimum distance for ignition to occur is also derived.

We present a simple model of a steady barotropic eddy superimposed on a current, with varying Coriolis force, to study the structure of a nonlinear deformed eddy. We also develop the linearized time-dependent formulation of the problem. The resulting solutions are stable perturbations, with scales smaller than the eddy.

The propagation of linearized waves on a non-uniform slowly varying potential current is studied by converting the equations of flow into a Schrödinger ordinary differential equation in the complex plane. This equation, which is solved by the WKB method, indicates the existence of current barriers which allow partial energy transmission while reflecting the complementary part. The classical result of total transmission (Longuet-Higgins & Stewart 1961) as well as that of complete reflexion (Peregrine 1976) are recovered as limiting cases by the present, more general approach.

The cross-stream migration of a deformable drop in a unidirectional shear flow of a second-order fluid is considered. Expressions for the particle velocity due to the separate effects of deformation and viscoelastic rheology are obtained. The direction and magnitude of migration are calculated for the particular cases of Poiseuille flow and simple shear flow and compared with experimental data.

The asymptotic theory for the laminar, incompressible, separating and reattaching flow past the bluff body is based on an extension of Kirchhoff's (1869) free-streamline solution. The flow field (only the upper half of which is discussed since we consider a symmetric body and flow) consists of two basic parts. The first is the flow on the body scale l*, which is described to leading order by the Kirchhoff solution with smooth inviscid separation, but with an $O(Re^{-\frac{1}{16}})$ modification to explain fully the viscous separation (here Re ([Gt] 1) is the Reynolds number). The influence of this $O(Re^{-\frac{1}{16}})$ modification is determined for the circular cylinder. The second part is the large-scale flow, comprising mainly the eddy and the ultimate wake. The eddy has length scale O(Rel*), width O(Re½l*) and is of elliptical shape to keep the eddy pressure almost uniform. The ultimate wake is determined numerically and fixes the eddy length. The (asymptotically small) back pressure from the eddy acts (on the body scale) both in the free stream and in the eddy, and it has a marked effect at moderate Reynolds numbers; combined with the Kirchhoff solution, it predicts the pressure drag on a circular cylinder accurately, to within 10% when Re = 5 and to within 4% when Re = 50. Other predictions, for the eddy length and width, the front pressure and the eddy pressure, also show encouraging agreement with experiments and Navier-Stokes solutions at moderate Reynolds numbers (of about 30), both for the circular cylinder and the normal flat plate. Finally, an analysis in the appendix indicates that, in wind-tunnel experiments, the tunnel walls (even if widely spaced) can exert considerable influence on the eddy properties, eventually forcing an upper bound on the eddy width as Re increases instead of the O(l* Re½) growth appropriate to the unbounded flow situation.