In this study, we revisit Rayleigh’s visionary hypothesis (Rayleigh, Proc. R. Soc. Lond., vol. 29, 1879a, pp. 71–97), that patterns resulting from interfacial instabilities are dominated by the fastest-growing linear mode, as we study nonlinear pattern selection in the context of a linear growth (dispersion) curve that has two peaks of equal height. Such a system is obtained in a physical situation consisting of two liquid layers suspended from a heated ceiling, and exposed to a passive gas. Both interfaces are then susceptible to thermocapillary and Rayleigh–Taylor instabilities, which lead to rupture/pinch off via a subcritical bifurcation. The corresponding mathematical model consists of long-wavelength evolution equations which are amenable to extensive numerical exploration. We find that, despite having equal linear growth rates, either one of the peak-modes can completely dominate the other as a result of nonlinear interactions. Importantly, the dominant peak continues to dictate the pattern even when its growth rate is made slightly smaller, thereby providing a definite counter-example to Rayleigh’s conjecture. Although quite complex, the qualitative features of the peak-mode interaction are successfully captured by a low-order three-mode ordinary differential equation model based on truncated Galerkin projection. Far from being governed by simple linear theory, the final pattern is sensitive even to the phase difference between peak-mode perturbations. For sufficiently long domains, this phase effect is shown to result in the emergence of coexisting patterns, wherein each peak-mode dominates in a different region of the domain.

The flows caused by a uniformly expanding circular cylinder or sphere in a perfect gas at rest at various (from small to very large) expansion velocities are analysed by a method of successive approximations in which the first approximation represents the incompressible flow. The shock waves which form around the bodies are also treated. The results for a sphere, even up to the third approximation, agree closely with Taylor's (1946) calculations for all the expansion rates.

Convection driven by radial and normal gravity forces in a rotating, horizontal cylinder is examined. The cylinder is subjected to uniform volumetric heating and constant-temperature wall cooling. The parameters are the radial-gravity and normal-gravity Rayleigh numbers, Rar and Rag (with Rar, Rag ≤ 106), the rotational Reynolds number, Re = 2Ω r02/v (0 ≤ Re ≤ 250), and the Prandtl number (Pr = 7). Critical conditions for the radial-gravity rest state correspond to a two-cell flow in the azimuthal plane with Rar,c = 13738. Finite-amplitude transient and steady flows are obtained with a Galerkin finite element method for Rayleigh number ratios in the range 0.1 ≤ Rar/Rag ≤ 100. When radial gravity dominates the flows tend to be multicellular and, during transients, initial high-wavenumber forms evolve to lower-wavenumber forms. When normal gravity dominates the flows are bicellular. When radial and normal gravity forces are comparable, in the presence of rotation, complex time-dependent motions occur and the largest rates of fluid circulation and heat transfer are observed.

In this paper transonic small-disturbance theory is applied to a simplified model of the flow near the front of a ducted body. The body is assumed to consist simply of two parallel flat plates which extend from the inlet station to infinity downstream. The velocity far upstream is sonic, and the velocity far downstream in the duct, which is assumed to be known, is slightly subsonic. Air is therefore ‘spilled’ around the intake edges. An analytic solution is found for the resulting flow field up to the ‘limiting Mach wave’, and asymptotic solutions are found for the supersonic flow and for the shock wave far from, and near, the intake edges. The pressure distribution along the outside walls is then known at both ends, and its computation is completed by an empirical procedure. Distributions of pressure along the centre-line and along the inside and outside walls are shown. These results may be used to compute the drag of sharp-edged intakes with a very small frontal area.

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.

A modification is made to the traditional Lagrangian approach to the derivation of the mean EMF of turbulent fluids which allows for finite conductivities. Consideration is confined to the case of homogeneous, isotropic but generally mirrornon-invariant and compressible turbulence. The eddy magnetic diffusivity and the coefficient α of the alpha-effect are expressed in terms of statistical moments of displacements of adjacent particles which undergo convective transport and microscopic diffusion in a turbulent flow. These expressions, being valid for arbitrary conductivities, reproduce known results in the cases of both very large and very small magnetic Reynolds numbers. Difficulties and advantages of the use of the results obtained for evaluations of the mean EMF are discussed.

A theoretical expression for the slow part (the nonlinear fluctuation part) of the pressure–strain rate is compared with simulations of anisotropic homogeneous flows. The purpose is to determine the quantitative accuracy of the theory and to test its qualitative predictions that the generalized Rotta coefficient, a non-dimensionalized ratio of slow term to kinetic energy anisotropy, varies with direction and can be negative (this is counter to isotropy return). Comparisons are made between theoretical and simulated values of the slow term and of the generalized Rotta coefficients. Also compared to simulations is an extension of the theory to account for non-stationary turbulence fields. The implication of the comparison for two-point closure theories and for Reynolds stress modelling is pointed out.

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.

Effects of product dissociation and preferential diffusion on the structure, propagation and diffusional-thermal instability of the classical one-dimensional laminar flame have been studied by using activation energy asymptotics. Analytical expressions as functions of dissociation and diffusion parameters have been obtained for such bulk flame parameters as the flame temprature reduction and the propagation rate eigenvalue, and for the dispersion relation governing flame stability. Results on flame propagation show that while under most situations the flame speed is reduced due to product dissociation, it can attain values in excess of the non-dissociative limit for highly mobile product species which can preferentially back diffuse to the upstream portion of the reaction zone where they recombine and release the associated recombination heat. Results on flame stability show that it is promoted in the presence of product dissociation which has a moderating influence on the flame temperature fluctuations, and for highly-mobile product species because of the enhanced burning rate and curvature-induced concentration modification.

The effect of small-amplitude, time-periodic, free-stream disturbances on an otherwise steady axisymmetric boundary layer on a circular cylinder is considered. Numerical solutions to the problem are presented, and an asymptotic solution to the flow, valid far downstream along the axis of the cylinder is detailed. Particular emphasis is placed on the unsteady eigensolutions that occur far downstream, which turn out to be very different from the analogous planar eigensolutions. These axisymmetric eigensolutions are computed numerically and also are described by asymptotic analyses valid for low and high frequencies of oscillation.