The stability of steady, plane, one-dimensional, trans-Alfvénic shocks to small normal disturbances (i.e. those in which the perturbed quantities are functions only of time and the distance from the plane of the shock wave) is discussed. The magnetic diffusivity of the ambient gas is taken to be very much greater than each of the viscous diffusivities and the thermal diffusivity.

It is confirmed in detail that all plane-polarized trans-Alfvénic shocks, except the 2–3 type (which is the only type that has no steady-state structure), are unstable to disturbances in those components of the magnetic field and velocity which are transverse to the plane of polarization. An incident Alfvénic wave, consisting of a weak, diffusing current-sheet would initially cause the shock profiles of these transverse quantities to grow linearly with time, while outside this shock region steady, uniform states would be reached. An integral condition is obtained which, together with the relevant boundary conditions; determines the asymptotic shock profiles of the transverse quantities whenever the disturbance is such that a steady state is reached. This removes the puzzling arbitrariness of these profiles.

It is also shown that the ‘1–4’ trans-Alfvénic shock is unstable to magnetoacoustic waves and contact fronts. A qualitative description of how it may be broken up is given. If the disturbance is of finite extent, a steady state is reached. An integral equation is obtained which, together with the relevant boundary conditions, determines the asymptotic steady-state shock-profiles for this case. This removes the apparent arbitrariness of these profiles.

The behaviour of ‘2–3’ trans-Alfvénic shocks and of switch-on and switch-off shocks is discussed.

The three-dimensional equations for a general inertial ocean current are transformed so that the temperature rather than the vertical co-ordinate appears as an independent variable. A downstream power series expansion is made of the equations and boundary conditions, which involves an expansion about the mean potential vorticity. A general first-order solution is obtained for a boundary current between two level surfaces, one of no motion and one of uniform temperature. The case of constant potential vorticity is treated for arbitrary inviscid boundary conditions; it is found that the current can exist as a boundary layer only if the open ocean geostrophic drift is westward everywhere in the depth interval. This result is extended to arbitrary potential vorticity distributions by an asymptotic analysis in physical space.

This paper considers the barotropic stability of an asymmetric zonal current on a rotating earth. The current is of hyperbolic tangent form in latitude. For this stability problem the neutral wave solutions are found and estimates of the growth rates for the unstable waves are obtained from the neutral solutions as in earlier investigations by the writer. It is again found that the beta effect which is due to the earth's curvature and rotation tends to stabilize the basic flow.

The stability of the basic flow is examined for a special case. For this case the current is centered at 35 degrees latitude, has a total shear of 30 m sec−1 and a half-width of 550 km. The most unstable waves are found to be wave-numbers 6 and 7 which amplify by a factor of e in 2·9 days. In addition, wave-numbers 5, 8 and 9 are also unstable. The stability of the symmetric jet is also examined for a comparable case. It is found that a wider band of wave-numbers is unstable. The most unstable wave is wave-number 8 which amplifies by a factor of e in 2·7 days.

In conclusion it is noted that these growth rates are slower than the amplification rates for the unstable waves associated with the baroclinic stability problem.

The flow in the wake of a two-dimensional model with a blunt trailing edge was examined at Reynolds numbers (based on model chord) between 1·4 × 105 and 2·56 × 105. The ratio of total boundary-layer thickness at the trailing edge to model base height was approximately 0·5. Measurements were taken of base pressure and vortex shedding frequency together with traverses of the wake using a hot-wire anemometer. Traverses carried out along the wake showed a peak in the root-mean-square velocity-fluctuation at a distance equal to one base height from the model rear face. The position of the peak is referred to as the position of the fully formed vortex. The investigation was extended to a model fitted with splitter plates up to four base heights long. For each plate tested, a position of the fully formed vortex was found, and its distance from the model base was discovered to be inversely proportional to the base pressure coefficient. The flow about the model with splitter plates is described as being separated into five régimes of flow.

An analysis is made of the effect of a longitudinal gravity field on two-dimensional supercavitating flow past wedges. Under the assumption that the flow is both irrotational and incompressible, a non-linear theory is developed for steady flow. By utilizing conformal mapping in combination with the Schwarz reflexion principle, the mathematical problem is reduced to a three-parameter, non-linear integral equation with one constraint. The equation is derived by reflecting the flow about the rigid boundaries; the constraint is obtained by requiring the net singularity strength inside the cavity-wedge system to be zero. A successive-approximation procedure is used to obtain a numerical solution of the integral equation. Typical results are presented in graphs and tables, and the results of the present work are compared to those of Acosta's linear theory.

The adequacy of direct one-step chemical kinetics for describing ignition and extinction in initially unmixed gases is studied through the particular case of inviscid axisymmetric stagnation-point flow. Oxidant is assumed to blow from upstream infinity at a non-gaseous reservoir of pure fuel at its boiling (or sublimating) temperature. Before reaching the reservoir the oxidant reacts with gaseous fuel flowing in the opposite direction to form product and release heat. This heat is in part conducted and diffused to the reservoir interface to transform more fuel into the gaseous state and continue the steady-state burning. Second-order Arrhenius kinetics for Lewis-number unity is examined. A critical parameter characterizing the phenomenon is shown to be the first Damkohler similarity group D1, the ratio of a time characterizing the flow to a time characterizing the chemical activity.

For small D1 the reactants convect away heat without releasing the energy stored in their chemical bonds. Regular perturbation about chemically frozen flow establishes this condition as the weak burning limit. For large D1 singular perturbation describes a narrow region of intense chemical activity. For infinite D1 (indefinitely fast rate of reaction) the region is reduced to a surface of discontinuity (the thin-flame kinetics of Burke & Schumann).

For intermediate D1 numerical techniques establish that a solution describing burning of moderate intensity joins the two previously mentioned asymptotic limits. It is suggested that sudden transition of the system between the various branches in this domain of intermediate D1 accounts for the phenomena of ignition and extinction of burning.

The analysis presented here is a second-order analysis of the resonant interactions triads of waves with wavelengths in the capillary-gravity and pure capillary ranges. The analysis is not a power series perturbation analysis, and one of the objects is the removal of the secularity which arises through power series perturbations. It is further found that the interactions are energy-conserving to the order considered here. Suitable modifications are made to accommodate the inevitable viscous attenuation associated with these small wavelengths. A start is made toward describing more completely the various spectra of random seas in wave-number and frequency regions where these interactions are dynamically significant.

Howard's semicircle theorem and a variational principle for the instability of unidirectional flow of an inviscid fluid are applied to the non-uniform vortex sheets discussed in a previous paper (Hocking 1964). Certain results of §5 of that paper are shown to be wrong and the correct results are obtained.

The asymptotic solution for the incompressible viscous flow past a semi-infinite flat plate constructed by Goldstein (1956, 1960) can be valid only if the solutions of certain ordinary differential equations obey certain constraints (given in Goldstein 1956, 1960). In this paper, we construct the solutions of these equations, show that the necessary constraints are met, and hence establish the validity of the asymptotic solution up to the order considered. The manner in which undetermined constants appear in the solution are discussed.

A model equation is constructed whose analysis reveals the same features, including stable hexagonal cells, as analysis of genuine approximate equations for the thermal convection problem. Taking advantage of the relative simplicity of the calculations an attempt is made to clarify certain procedures customarily used in non-linear stability theory; in particular, the basis for the usual formal expansions and the appropriateness of neglecting terms of fourth and higher order are discussed. It is demonstrated that a whole class of equations leads to hexagonal cells, thereby giving confidence that results on convection cells found elsewhere in idealized situations will remain valid when more realistic situations are studied.

It is predicted that, at a temperature difference a little less than that at which motion starts according to linear stability theory, a steady hexagonal convective pattern will develop from finite-amplitude instabilities in a horizontal layer of fluid heated from below. This is because the first disturbances to start growing must be the triplet of two-dimensional ‘rolls’ which form angles of 60° with each other and whose amplitudes and phases first fall in certain critical ranges. The growth of these disturbances stabilizes all other disturbances and is such that ultimately the right phases and amplitudes occur to give hexagonal cells. If the temperature difference is increased somewhat beyond its critical value, the hexagonal pattern becomes unstable and a two-dimensional roll pattern is predicted. In an intermediate temperature range, rolls are unstable but transport more heat than hexagons. ‘Free–free’ boundary conditions, a viscosity which varies with temperature, and a fixed disturbance wave-number are assumed in this extension of the work of Palm (1960) and Segel & Stuart (1962). Other theoretical results and some experimental results are compared with the present predictions.