The influence of hydrodynamic dispersion on thermal convection in porous media is studied theoretically. The fluid-saturated porous layer is homogeneous, isotropic and bounded by two infinite horizontal planes kept at constant temperatures. The supercritical, steady two-dimensional motion, the heat transport and the stability of the motion are investigated. The dispersion effects depend strongly on the Rayleigh number and on the ratio of grain diameter to layer depth. The present results provide new and closer approximations to experimental data of the heat transport.

An undisturbed geostrophic density current flows along a vertical wall (the coast) with the free streamline (the front) located at a distance L from the wall which is comparable to the Rossby radius of deformation. Finite amplitude perturbations with downstream wavelengths much larger than L are discussed, and it is shown that the slope of the front in the horizontal plane increases with time. Some perturbations tend to ‘break’ seaward by developing large transverse velocities away from the coast. The temporal evolution of some perturbations is such as to completely ‘block’ the upstream flow, but the subsequent behaviour is beyond the scope of the theory. We also discuss the propagation of the nose of the intrusion when a density current debouches from a coastal source and then flows along the coastal boundary.

Lake & Yuen (1978) have suggested that in very steep wind waves the modulation-frequency of the wave amplitude may correspond to the frequency of the fastest-growing subharmonic instability of a uniform train of waves whose amplitude equals the mean wave amplitude $\overline{a}$. The approximate theory of Benjamin & Feir (1967) gives this frequency as $(\overline{a}k)f_d$, where κ is the wavenumber and fd the frequency of the unperturbed waves. This expression applies strictly only to very small values of the wave steepness $\overline{a}k$.

More recently (Longuet-Higgins 1978) the present author calculated accurately all the normal-mode instabilities of steep gravity waves on deep water. In this note these calculations are used to determine the frequency of the fastest-growing sub-harmonic instabilities precisely. When compared with the experimental data of Lake & Huen, these frequencies show even closer agreement.

The Craik–Leibovich (CL) equations for Langmuir circulations are shown to be an Eulerian approximation to an exact theory of the generalized Lagrangian mean (GLM) due to Andrews and McIntyre. Derivation of the CL equations using the GLM formalism is decisively simpler than the original method. The CL theory is then compared to other wave–current interaction theories of Langmuir circulations, notably those of Garrett and of Moen.

The stability of natural convection in a vertical annular enclosure has been studied by the linear theory. It was found that for all Prandtl numbers the instability sets in as a wave travelling upward. For low Prandtl numbers, the larger the curvature the more stable the flow; the reverse is true for high Prandtl numbers. The theoretical predictions of the mode of instability were verified for air. A multicellular flow pattern was observed to drift upward with the predicted wave speed. The measured wavelength of the cells is in good agreement with the linear analysis.

This paper presents the first ‘exact’ solutions to the creeping-flow equations for the transverse motion of a sphere of arbitrary size and position between two plane parallel walls. Previous solutions to this classical Stokes flow problem (Ho & Leal 1974) were limited to a sphere whose diameter is small compared with the distance of the closest approach to either boundary. The accuracy and convergence of the present method of solution are tested by detailed comparison with the exact bipolar co-ordinate solutions of Brenner (1961) for the drag on a sphere translating perpendicular to a single plane wall. The converged series collocation solutions obtained in the presence of two walls show that for the best case where the sphere is equidistant from each boundary the drag on the sphere predicted by Ho & Leal (1974), using a first-order reflexion theory, is 40 per cent below the true value when the walls are spaced two sphere diameters apart and is one order-of-magnitude lower at a spacing of 1.1 diameters.

Exact solutions are presented for the three-dimensional creeping motion of a sphere of arbitrary size and position between two plane parallel walls for the following conditions: (a) pure translation parallel to two stationary walls, (b) pure rotation about an axis parallel to the walls, (c) Couette flow past a rigidly held sphere induced by the motion of one of the boundaries and (d) two-dimensional Poiseuille flow past a rigidly held sphere in a channel. The combined analytic and numerical solution procedure is the first application for bounded flow of the three-dimensional boundary collocation theory developed in Ganatos, Pfeffer & Weinbaum (1978). The accuracy of the solution technique is tested by detailed comparison with the exact bipolar co-ordinate solutions of Goldman, Cox & Brenner (1967a, b) for the drag and torque on a sphere translating parallel to a single plane wall, rotating adjacent to the wall or in the presence of a shear field. In all cases, the converged collocation solutions are in perfect agreement with the exact solutions for all spacings tested. The new collocation solutions have also been used to test the accuracy of existing solutions for the motion of a sphere parallel to two walls using the method of reflexions technique. The first-order reflexion theory of Ho & Leal (1974) provides reasonable agreement with the present results for the drag when the sphere is five or more radii from both walls. At closer spacings first-order reflexion theory is highly inaccurate and predicts an erroneous direction for the torque on the sphere for a wide range of sphere positions. Comparison with the classical higher-order method of reflexions solutions of Faxen (1923) reveals that the convergence of the multiple reflexion series solution is poor when the sphere centre is less than two radii from either boundary.

Solutions have also been obtained for the fluid velocity field. These solutions show that, for certain wall spacings and particle positions, a separated region of closed streamlines forms adjacent to the sphere which reverses the direction of the torque acting on a translating sphere.

Experimental results for the release of a fixed volume of one homogeneous fluid into another of slightly different density are presented. From these results and those obtained by previous experiments, it is argued that the resulting gravity current can pass through three states. There is first a slumping phase, during which the current is retarded by the counterflow in the fluid into which it is issuing. The current remains in this slumping phase until the depth ratio of current to intruded fluid is reduced to less than about 0.075. This may be followed by a (previously investigated) purely inertial phase, wherein the buoyancy force of the intruding fluid is balanced by the inertial force. Motion in the surrounding fluid plays a negligible role in this phase. There then follows a viscous phase, wherein the buoyancy force is balanced by viscous forces. It is argued and confirmed by experiment that the inertial phase is absent if viscous effects become important before the slumping phase has been completed. Relationships between spreading distance and time for each phase are obtained for all three phases for both two-dimensional and axisymmetric geometries. Some consequences of the retardation of the gravity current during the slumping phase are discussed.

The well-known Whitham theory may be applied to shocks diffracting over concave corners, provided that the diffraction results in Mach reflexion. This paper compares the theory with data obtained during experiments with diffracting shocks. If an incident shock is classified as weak or strong in the strict sense defined by von Neumann, then it is found that the Whitham theory accurately determines the Mach number of the Mach stem at the wall for both the weak and the strong cases. The theory also has some further value for strong shocks but not for weak; it is not applicable when the diffraction is regular.

A backwash vortex has been observed under waves running up a sloping bed. The vortex is formed at the edge of the swash zone when backwash turns back into an oncoming wave bore. The vortex eroded a bed consisting of glass beads, and formed a step which induced the formation of sand ripples.

Autorotation of an elliptic cylinder about an axis fixed perpendicular to a parallel flow is explained in this paper by means of numerical solutions of the Navier-Stokes equations. Potential-flow theory predicts, for constant angular velocity, half a period in which a torque supports rotation and half a period in which it opposes rotation, with vanishing torque in the average. This balance is disturbed by viscous-flow effects in such a way that, for a given angular velocity, vortex shedding either damps rotation or, under certain conditions, favours rotation. The proper interplay of those conditions, which include synchronization of vortex shedding and rate of rotation, results in auto-rotation. The numerical results for Re [les ] 400 are compared with experimental data for Re = 90000 from the literature. The agreement of the force coefficients and the large-scale flow patterns is surprisingly good.

A set of self-similar solutions for blast waves associated with the deposition of variable energy at the front is presented. As a consequence of self-similarity, the results are applicable when the ambient atmosphere into which the wave front propagates is at a negligibly low pressure and temperature. Besides the class of (1) blast waves associated with energy gain that covers a regime bounded on one side by the well-known solution for adiabatic strong explosion waves (ASE) and, on the other side, by the solution for waves having the Chapman–Jouguet condition established immediately behind the front, included within the scope of our analysis are two others: (2) blast waves associated with energy loss that occupy a regime between the ASE solution and the case of infinite density ratio across the front, and (3) a non-unique class of solutions for blast waves associated with energy deposition that may have either locally sonic or supersonic flow immediately behind the front, extending over the regime between the waves headed by the Chapman-Jouguet detonation and the case of infinite rate of energy deposition. Specific results for a number of representative cases are expressed in terms of integral curves on the phase plane of reduced blast wave co-ordinates, as well as in the form of particle velocity, temperature, density, and pressure profiles across the flow field.