This paper considers the mixing of a turbulent radial wall jet with a secondary fluid introduced into the impingement area of the wall jet so as to form a steady axisymmetric state. Similarity of the concentration profiles perpendicular to the wall has been assumed and, by solving the momentum and mass flow equations, the concentration distribution through the layer and a similarity exponent giving the variation of concentration along the wall have been determined.

A shallow layer of silicone oil on a plane, circular copper plate was uniformly heated from below. The air on its surface was kept at constant temperature by a uniformly cooled glass plate close to the oil, which at the same time inhibited air motions. Motions began with concentric circular rolls which, after the centre ring had formed, broke down into a hexagonal pattern. A rather accurate determination of the wavelength of the motions was possible. The wavelength was found to be variable with the depth of the fluid layer in qualitative accordance with the theory of Nield. Supercritical motions are briefly discussed.

The evolution of disturbances after a laminar, slightly supercritical flow between parallel planes is disturbed is considered as an initial-value problem. An asymptotic solution of the disturbances for large time possesses the same characteristic features as the turbulent spots observed by Emmons (1951).

A largely qualitative experimental investigation has been made of cellular convection patterns produced by the instability of a horizontal layer of fluid heated uniformly throughout its body and cooled from above. The fluid was aqueous zinc sulphate solution, and the heating was produced by an electrolytic current. The flow was visualized and photographed using polystyrene beads which, because of a differential expansion, came out of suspension in regions where the fluid was hotter or colder than average.

The development of the cellular patterns as the Rayleigh number (a modified form) was varied was in many respects similar to that for Bénard convection. There were, however, two striking differences. First, the fluid descended in the centres of the cells and ascended at the peripheries; this was probably associated with the asymmetry between the heating and cooling. Secondly, and more surprisingly, the horizontal scale of the convection patterns was unusually large (except at the lowest Rayleigh numbers); the distance between rising and falling currents was observed to reach typically five times the depth of the layer.

This last observation may be relevant to theories of convection within the Earth's mantle, to mesoscale convection in the atmosphere, and to patterns formed in ice.

A theoretical study has been made of an experiment by Tritton & Zarraga (1967) in which eonvective motions were generated in a horizontal layer of water (cooled from above) by the application of uniform heating. The marginal stability problem for such a layer is solved, and a critical Rayleigh number of 2772 is obtained, at which patterns of wave-number 2·63 times the reciprocal depth of the layer are marginally stable.

The remainder of the paper is devoted to the finite amplitude convection which ensues when the Rayleigh number, R, exceeds 2772. The theory is approximate, the basic simplification being that, to an adequate approximation, Fourier decompositions of the convective motions in the horizontal (x, y) directions can be represented by their dominant (planform) terms alone. A discussion is given of this hypothesis, with illustrations drawn from the (better studied) Bénard situation of convection in a layer heated below, cooled from above, and containing no heat sources. The hypothesis is then used to obtain ‘mean-field equations’ for the convection. These admit solutions of at least three distinct forms: rolls, hexagons with upward flow at their centres, and hexagons with downward flow at their centres. Using the hypothesis again, the stability of these three solutions is examined. It is shown that, for all R, a (neutrally) stable form of convection exists in the form of rolls. The wave-number of this pattern increases gradually with R. This solution is, in all respects, independent of Prandtl number. It is found, numerically, that the hexagons with upward motions in their centres are unstable, but that the hexagons with downward motions at their centres are completely stable, provided R exceeds a critical value (which depends on Prandtl number, P, and which for water is about 3Rc), and provided the wave-number of the pattern lies within certain limits dependent on R and P.

The one-dimensional problem of shock-wave reflexion with relaxation is treated numerically by combining the shock-wave, characteristic, and Rayleigh-line equations. The theoretical results are compared with pressure and density measurements in CO2, and the agreement is found to be excellent.

The effect of an applied transverse magnetic field on the development of a two-dimensional jet of incompressible fluid is examined. The jet is prescribed in terms of its mass flux ρQ0 and its lateral scale d at an initial section x = 0. The three dimensionless numbers characterizing the problem are a Reynolds number R = Q0/ν, a magnetic Reynolds number Rm = μσQ0, and a magnetic interaction parameter N = σB20d2/ρQ0, where ρ represents density, σ conductivity, μ permeability and B0 applied field strength, and it is assumed that \[ R_m \ll 1,\quad R\gg 1,\quad N\ll 1. \] It is shown that when M2 = RN [Gt ] 1, an inviscid treatment is appropriate, and that the effect of the magnetic field is then to destroy the jet momentum within a distance of order N−1 in the downstream direction. A general solution for inviscid development is obtained, and it is shown that a large class of velocity profiles (though not all of them) are self-preserving.

When M2 [Lt ] 1, it is shown that the viscous similarity solution obtained by Moreau (1963a, b) is relevant. This solution is re-derived and re-interpreted; it implies that the jet momentum is destroyed within a distance of order $R^{\frac{1}{4}}N^{-\frac{3}{4}}$ in the downstream direction.

Some further aspects of the jet annihilation problem are qualitatively discussed in § 4, viz. the nature of the overall flow field, the effect of the presence of distant boundaries, the effect of increasing Rm to order unity and greater, and the effect of oblique injection. Finally the development of a jet of conducting fluid into a nonconducting environment is considered; in this case the jet is not stopped by the magnetic field unless a return path outside the fluid for the induced current is available.

The magnetohydrodynamic lubrication flow in a step-type thrust bearing has been investigated both theoretically and experimentally. The experimental configuration consisted of nine identical segments arranged on a circular plate. Flow was induced by rotating the upper plate. Pressures, voltages and torques were recorded. Provision was made for passing current radially through the segments, and it was shown that the torque could be reduced to zero and even reversed by this means. Good agreement has been obtained between the analysis and the experimental results.

Helmholtz stated and Korteweg proved that of all divergenceless velocity fields in a domain, with prescribed values on the boundary, the solution of the Stokes equation minimizes the rate of viscous energy dissipation. Hill & Power and also Kearsley proved the corresponding reciprocal maximum principle involving the stress tensor. We prove generalizations of both these principles to the flow of a liquid containing one or more solid bodies and drops of another liquid. The essential point in doing this is to take account of the motion of the solids or drops, which must be determined along with the flow. We illustrate the use of these principles by deducing several consequences from them. In particular we obtain upper and lower bounds on the effective coefficient of viscosity and a lower bound on the sedimentation velocity of suspensions of any concentration. The results involve the statistical properties of the distribution of suspended particles or drops. Graphs of the bounds are shown for special cases. For very low concentrations of spheres, both bounds on the effective viscosity coefficient are the same, and agree with the results of Einstein and Taylor.

The stability to small inviscid disturbances of a rotating flow, whose velocity components in cylindrical polars (r, 0, z) are (0, V(r), 0), is investigated when one boundary of the flow (r = b) is a free surface under the action of surface tension (γ), and the other is either at infinity, or a rigid cylinder (r = a ≠ b), or at the axis (r = 0). The free surface may be the inner or the outer boundary. A necessary and sufficient condition for stability to axisymmetric disturbances is derived, which requires that Rayleigh's criterion of increasing circulation be satisfied, and otherwise depends only on b, V(b), γ and the density of the swirling liquid. This condition may be extended to include non-axisymmetric disturbances when V ∝ 1/r and when V ∝ r although in the latter case it is no longer a necessary one. It is shown that, in the case V ∝ r, as well as V ∝ 1/r, the ‘most unstable’ disturbance on a rotating column of fluid will be non-axisymmetric if the rotation speed at the surface is sufficiently great. Several applications of the theory are suggested, and a possible experiment to test it is described.

A non-linear analysis of cellular convection driven by surface tension in a semi-infinite liquid layer heated from below has been made. The purpose is to determine whether or not one can predict the emergence of the hexagonal flow pattern from the interaction of a certain large class of important disturbances. The principal conclusion is that, compared with gravity driven convection, there is generally a much greater band of imposed temperature difference associated with hexagonal convective patterns. Partial results for the more realistic assumption of finite depth support this conclusion.

Laboratory and field measurements of the generation of gravity waves by turbulent winds imply that a theoretical model based on laminar flow may be adequate on a laboratory, but not an oceanographic, scale. This suggests that the significance of wave-induced perturbations in the turbulent Reynolds stresses for momentum transfer from wind to waves must increase with an appropriate scale parameter. A generalization of the laminar model is constructed by averaging the linearized equations of motion for a turbulent shear flow in a direction (say y) parallel to the wave crests of a particular Fourier component of the surface-wave field. It is shown that the resulting, mean momentum transfer to this component comprises: (i) a singular part, which is proportional to the product of the velocity-profile curvature and the mean square of the wave-induced vertical velocity in the critical layer, where the mean wind speed is equal to the wave speed; (ii) a vertical integral of the mean product of the vertical velocity and the vorticity ω, where ω is the wave-induced perturbation in the total vorticity along a streamline of the y-averaged motion; (iii) the perturbation in the mean turbulent shear stress at the air-water interface. The equation that governs the advection of the vorticity ω under the action of the perturbations in the turbulent Reynolds stresses is derived. Further theoretical progress appears to demand some ad hoc hypothesis for the specification of these turbulent Reynolds stresses. Two such hypotheses are discussed briefly, but it does not appear worth while, in the absence of more detailed experimental data, to carry out elaborate numerical calculations at this time.

Most of the existing methods for calculating the inviscid flow past a delta wing with leading-edge vortices are based on slender-body theory. When these vortices are represented by rolled-up vortex sheets in an otherwise irrotational flow, some of the assumptions of slender-body theory are violated near the centres of the spirals. The aim of the present report is to describe for the vortex core an alternative method in which only the assumption of a conical velocity field is made. An asymptotic solution valid near the centre of a rolled-up vortex sheet is derived for incompressible flow. Further asymptotic solutions are determined for two-dimensional flow fields with vortex sheets which vary with time in such a manner that the sheets remain similar in shape. A particular two-dimensional flow corresponds to the slender theory approximation for conical sheets.

Experiments giving the mass efflux of a Poiseuille flow over a naturally permeable block are reported. The efflux is greatly enhanced over the value it would have if the block were impermeable, indicating the presence of a boundary layer in the block. The velocity presumably changes across this layer from its (statistically average) Darcy value to some slip value immediately outside the permeable block. A simple theory based on replacing the effect of the boundary layer with a slip velocity proportional to the exterior velocity gradient is proposed and shown to be in reasonable agreement with experimental results.