The Nusselt number and the energy content of a convective fluid and its container have been measured for two structures formed at low Rayleigh numbers. The results for the energy content are discussed. It is found that this energy is accumulated mainly in the lateral walls of the container, these being parallel to the rolls formed.

A large class of explicit solutions for Hele-Shaw flow with a free surface is presented. The results are valid when surface-tension effects in the plane of the cell are negligible. Most of the solutions given produce fingers, both in channel flow and on a growing air bubble. Possible behaviour of these fingers is described, and a qualitative comparison with published experimental results is made.

The drag on a sphere in a random array of fixed spheres of volume concentration c ([Lt ] 1) at Reynolds number R ([Lt ] 1) is discussed. In the case when c and R2 are of the same order of magnitude, the drag is determined up to terms of O(R).

The ordinary differential equation \[ {\textstyle\frac{1}{3}}\kappa^2(g^{\prime\prime}+g) - \lambda g - {\textstyle\frac{3}{2}}g^2 + \frac{2}{\pi} \cos t = -\frac{3}{2}\int_{-\pi}^{\pi}g^2\,{\rm d}t, \] which represents forced water waves on shallow water near resonance, is considered when the dispersion κ is small. Asymptotic methods are used to show that there are multiple solutions with period 2π for a given value of the detuning parameter λ. The effects of dissipation are also considered.

We Present a mathematical model for the region of dendritic or cellular growth which often forms during the solidification of alloys. The model treats the region of mixed phase (solid and liquid) as a continuum whose properties vary with the local volume fraction of solid. It is assumed that transports of heat and of solute are by diffusion alone, and the model is closed by a condition of marginal equilibrium. Results are obtained for the unidirectional solidification of an alloy from a plane wall. The spatial variations of solid fraction are highly suggestive of the types of morphology that can occur, and a wealth of different structures are found as the physical parameters are varied. Although the model ignores gravity entirely, the results can be applied to the solidification from below of an alloy which is initially less dense than its eutectic. Predictions for the growth rate of the mixed-phase region agree well with existing experimental measurements of ice growing from aqueous salt solutions.

We describe a series of laboratory experiments to study convective structures in rotating fluids (distilled water) in ranges of Rayleigh flux number Raf from 106 to 2 × 1011 and of Taylor number Ta from 106 to 1012. An intermediate quasi-stationary ring pattern of convection was found to arise from the interaction of the onset of convection with the fluid spin-up, for which we determined the times of origin and destruction, the distances between the rings, and the diameter of the central ring in terms of Raf and Ta. The ring structure evolves into a vortex grid which can be regular or irregular. In terms of Raf and Ta the regular grid exists in the linear regime, when the number of vortices N is in accord with the linear theory, when $N \propto h^{-2} Ta^{\frac{1}{3}} \propto \Omega^{\frac{2}{3}}$, or in the nonlinear regime when N ∝ h−2Ta½Raf−⅙ ∝ where Ω is the angular velocity and h is the fluid depth. In the irregular regime we always have N ∝ Ω. The transition from the regular regime to the irregular one is rather gradual and is determined by the value of the ordinary Rayleigh number, which we found to be greater than the first critical number Ra ∝ Ta2/3 by a factor about 25–40. In the transition region vortex interactions are observed, which start with rotation of two adjacent vortices around a common axis, then the vortices come closer and rotation accelerates, following which the vortices form a double helix and then coalesce into one stronger vortex.

Some other qualitative experiments show that if the rotating vessel with the convective fluid is inclined to the horizontal, the vortex grid is formed along the rotation axis in accordance with the Proudman–Taylor theorem.