By introducing small controlled perturbations prior to the onset of motion, two-dimensional convection cells of arbitrary width-depth ratios are produced in a horizontal fluid layer heated from below. The conditions employed correspond to the finite-amplitude Rayleigh stability problem for a constant viscosity, large Prandtl number, Boussinesq fluid with rigid, conducting boundaries. It was found that two-dimensional cells with width-depth ratios close to unity are stable at all Rayleigh numbers investigated (Rc [Lt ] R [Lt ] 2·5Rc). Cells whose widthdepth ratios are moderately too large or too small tend to undergo size adjustments toward a preferred value of about 1·1. If the width-depth ratios are much too large or too small, they tend to develop three-dimensional instabilities in the form of cell boundary distortions and transverse secondary cells, respectively. Eventually the flow settles into a new family of rolls with a more preferred widthdepth ratio. It is suggested that these observations may have implications on nonlinear interchange instability problems and geophysical flows.

It is known experimentally that laminar circular Couette flow between two concentric circular cylinders, the outer of which is fixed, becomes unstable when the speed of the inner cylinder is high enough. The flow is then replaced by a new circumferential flow with superimposed toroidal (or Taylor) vortices spaced periodically along the axis. At a higher speed still the new flow develops another instability, and is replaced by a flow in which the axially periodic vortices are simultaneously periodic travelling waves in the azimuth.

In the present paper an attack is made on the problem of instability of the Taylor-vortex flow against perturbations which are periodic both in the axial and azimuthal co-ordinates and, moreover, travel with some phase velocity in the latter. Subject to a number of assumptions and approximations, which are detailed in the paper, it is found that the Taylor-vortex flow is stable against perturbations with the same axial wavelength and phase, but unstable against perturbations differing in phase by ½π. After instability the new flow no longer has planes separating neighbouring vortices, but has wavy surfaces travelling in the azimuth. This feature is in accord with much (though not all) of the experimental evidence.

The critical Taylor number (proportional to the square of the speed) at which the Taylor vortices become unstable is found theoretically to be about 8% above the value for which Taylor vortices first appear. This must be compared with a value in the range 5-20% for the experiments which our work models most closely. The azimuthal wave-number given a slight preference by theory is 1, in agreement with those experiments.

In the paper, on the basis of the three-moment equations, simple formulae for properties of a laminar boundary layer with suction are obtained. An equation relating the transition point critical Reynolds number to initial turbulence and surface roughness is presented. An approximate method for prediction of optimal suction of a boundary layer, including the main effects controlling transition of a laminar boundary layer into a turbulent one, is developed.

The transformation proposed by Coles for correlating the compressible turbulent boundary layer on a solid surface with a given incompressible layer has been extended to the case of a porous surface with air injection. However, unlike Coles's work, the new transformation does not use the empirical concept of the sublayer to define one of the transformation parameters. Instead this parameter can be defined in terms of the stream function at the wall, and so is directly related to the injection rate. The present paper concerns the transformation for the boundary layer with constant pressure and zero heat transfer, but the possibility of extending the transformation to flows with pressure gradients and heat transfer is mentioned briefly.

Comparison with experiments shows that the new transformation successfully relates cf, θ and the fully turbulent part of the velocity profile with their corresponding incompressible values for a wide range of Mach numbers and injection rates.

The analysis and experiments in this paper are restricted to the flow between two coaxial, infinite disks, one rotating and one stationary. The results of numerical calculations show that many solutions can exist for a given Reynolds number Ωl2/v (Ω is the angular velocity of the rotating disk and I is the spacing between the two disks). Out of a greater number of possible solutions, three solution branches have been identified; the branches correspond to one-, two- and three-flow cells in the meridional plane.

The one-cell branch has been accorded detailed treatment. Within this branch there are two subbranches. The first, now well documented in the literature, includes solutions from zero to infinite Reynolds number. The latter limiting case is characterized by an inward-flowing boundary layer on the stationary disk and an outward-flowing boundary layer on the rotating disk. In between is a core flow rotating with a constant angular velocity. The second sub-branch of the single-cell flows, apparently unknown heretofore, begins with an infinite Reynolds number, decreases to a minimum and then increases to an infinite Reynolds number again. The first infinite Reynolds number limit again corresponds to two boundary-layer flows separated by a core flow with constantangular velocity opposite in direction to the angular velocity of the rotating disk. The second limiting case of infinite Reynolds number is the free-disk solution of von Kármán (1921). Asymptotic solutions have been obtained which more fully describe the nature of this flow as the Reynolds number increases.

The second part of the paper presents experimental measurements corresponding to the Reynolds number range 0–100. Profiles were measured with a hot-wire anemometer. The measurements are in agreement with the first, one-cell branch of solutions. A semi-quantitative evaluation of edge effects is obtained.

Linear stability theory of Bénard convection in a rotating fluid (Chandrasekhar 1961) has shown that fluids with large ([ges ] 1) Prandtl number, σ, exhibit behaviour markedly different from that of fluids with σ [les ] 1. This difference in behaviour extends also into the finite-amplitude range (Veronis 1959, 1966I). In this paper we report a numerical study of two-dimensional Be´nard convection in a rotating fluid confined between free boundaries, with σ = 6·8 and σ = 0·2 for the range of Taylor number 0 [les ] [Fscr ]2 [Lt ] 105 and for Rayleigh numbers, R, extending an order of magnitude from the critical value of linear stability theory. The behaviour of water (σ = 6·8) is dominated by the rotational constraint even for relatively moderate values (∼ 103) of [Fscr ]2. A study of the resultant velocity and temperature fields shows how rotation controls the system, with the principal behaviour reflected by the thermal wind balance; i.e. the horizontal temperature gradient is largely balanced by the vertical shear of the velocity component normal to the temperature gradient. A fluid with a small Prandtl number (σ = 0·2) becomes unstable to finite-amplitude disturbances at values of the Rayleigh number significantly below the critical value of linear stability theory. The subsequent steady vorticity and temperature fields exhibit a structure which is quite different from that of fluids with large σ. The rotational constraint is balanced primarily by non-linear processes in a limited range of Taylor number ([Fscr ]2 [les ] 103·6). For larger values of [Fscr ]2 the system first becomes unstable to infinitesimal oscillatory disturbances but a steady, finite-amplitude flow is established at supercritical values of R which are none the less smaller than the values that one would expect from linear theory. The ranges of Taylor number in which the above phenomena occur are different from those which were estimated on the basis of an earlier study (Veronis 1966 I) which made use of a minimal representation of the finite-amplitude velocity and temperature fields. No subcritical, finite-amplitude oscillatory motions were found in the present study. Comparison with some of the experimental features observed and reported by Rossby (1966) is also discussed and it is pointed out that some of the differences between theory and experiment may be traced to the restrictive conditions (two-dimensionality and free boundaries) of the present study.

A uniform liquid film on a horizontal flat plate may be unstable to small disturbances when an air stream flows over the liquid surface. The stability of such films is examined for cases where the film is contaminated by an insoluble surface-active agent.

Two approximate analyses are given, which are applicable to liquid films at moderately large Reynolds numbers and at fairly small Reynolds numbers, respectively. These supplement previous work on uncontaminated films by Miles (1960), Cohen & Hanratty (1965) and Craik (1966).

At large liquid Reynolds numbers, the presence of surface contamination enhances stability due to increased dissipation in the viscous layer just within the liquid surface; but, at small liquid Reynolds numbers, there exists a class of disturbances for which surface elasticity may be destabilizing.

An investigation of the interaction of a jet of air impinging normally on a liquid surface has been carried out. The regimes of stable, oscillating, and dispersing indentions were established as a function of velocity and surface tension.

For the stable region, energy and force balances were formulated. The force balance, when integrated, described the indention profile well and will have further use in predicting the point of onset of instability.

A linear relationship between the square of the instability velocity and the surface tension and between the square of the critical dispersion velocity and the surface tension has been determined experimentally.

The paper considers the effect on the steady flow past a sphere of a uniform upstream distribution of dust particles having a small relaxation time. Using a potential solution as an upstream model of the gas flow at large Reynolds numbers R, an equation for the concentration of dust near the sphere is derived and solved numerically. It is shown that in this inviscid model there exists a dustfree layer adjacent to the sphere. A drag force is computed, and it is also shown that particles do not collide with the sphere until the Stokes number σ is greater than $\frac{1}{12}$ if we assume the gas flow unchanged by the presence of dust particles, which is in agreement with previous work of Langmuir & Blodgett (1946). The paper concludes with a discussion of the effect of a viscous boundary layer in which it is suggested that the dust-free layer is preserved when σR½ [Gt ] 1, but is prevented from forming by the viscous boundary layer when σR½ [Lt ] 1.

In a uniform rotating liquid of uniform depth, Kelvin waves may be propagated in one direction along a straight boundary of the liquid. The Wiener-Hopf technique is used to obtain the wave field due to Kelvin waves incident at a rightangle corner. An asymptotic solution is obtained for the field far from the corner. It is shown that for low frequencies the Kelvin waves are propagated round the corner without change of amplitude, but that for high frequencies cylindrical waves of the ‘Poincaré’ type are generated at the corner, so that the amplitudes of the Kelvin waves propagated round the corner are reduced.