This paper extends two earlier papers in which DiPrima & Stuart calculated first (1972b) the critical Taylor number to order ε2, where the eccentricity ε is proportional to the displacement of the axes of the circular cylinders, and second (1975) the torque and load to order ε associated with nonlinear effects of Taylor vortices. In the latter paper, it was shown that to order ε the torque arising from the Taylor vortices is identical with that for the concentric problem, which was first calculated, by a perturbation method, by Davey (1962). This deficiency is remedied in the present paper, where the calculation is taken to order ε2. It is found that, as ε rises, the torque associated with the Taylor vortices falls slightly when we keep constant the percentage elevation of the Taylor number above the ε-dependent critical value. This result is in accordance with experimental observations by Vohr (1967, 1968). In addition, results of calculations of the pressure field developed by the Taylor-vortex flow in association with the eccentric geometry are presented; this is larger than in the concentric case owing to a Reynolds lubrication effect. Also given are the associated components of the load on the inner cylinder, but only for Taylor numbers close to the critical value.

One additional observation by Vohr, for cylinders with a mean ratio of the gap to the inner radius of 0·099, was that the maximum Taylor-vortex strength with ε = 0·475 occurred some 50° downstream of the maximum gap for a 20% elevation of the Taylor number above the critical value. Calculations in the two earlier papers (1972b, 1975) gave 90 and 76°, respectively, for that angle. Note that in the 1975 paper a geometrical correction of order ε was included. Here, with an additional modification of order ε due to the flow, this result is improved to 49° by the extended analysis presented, although the ‘small’ parameters are somewhat outside the range for which perturbation theory is expected to be valid.

The linear stability of Poiseuille flow in an elliptic pipe which is nearly circular is examined by regarding the flow as a perturbation of Poiseuille flow in a circular pipe. We show that the temporal damping rates of non-axisymmetric infinitesimal disturbances which are concentrated near the wall of the pipe are decreased by the ellipticity. In particular we estimate that if the length of the minor axis of the cross-section of the pipe is less than about 96 ½% of that of the major axis then the flow will be unstable and a critical Reynolds number will exist. Also we calculate estimates of the ellipticities which will produce critical Reynolds numbers ranging from 1000 upwards.

Three-dimensional cellular convection concentrates magnetic flux into ropes when the magnetic Reynolds number is large. Amplification of the magnetic field is limited by the Lorentz force and the maximum field in a flux rope can be estimated. Boundary-layer analysis yields a completely self-consistent solution for a model of convection driven by imposed horizontal temperature gradients, and the transition from a kinematic to a dynamic regime can be followed in detail. The maximum value of the amplified field is proportional to the square root of the ratio of the viscous to the magnetic diffusivity.

A small drop of liquid 1 falls through a less dense liquid 2 and approaches the horizontal interface between liquid 2 and an underlying layer of liquid 1. After a short time the drop will be brought to rest (or nearly) in a hollow in the interface. Before the drop can coalesce with its bulk phase, the thin film of liquid 2 trapped between them must be squeezed out, and become sufficiently thin that rupture can occur. This is the film drainage problem. Early calculations, based on simple lubrication theory, fail to take proper account of two effects which are investigated here and shown to be decisive. They are the circulation induced in the drop and in the lower bulk fluid, which tends to speed up drainage, and the constriction in the film thickness at its periphery, which tends to slow it down. This constriction has been observed and some existing theories have attempted to model it in an ad hoc manner. We give here a physical explanation and calculate the minimum thickness explicitly. The effect of circulation in the adjacent fluids is also calculated.

An analysis of the stability of thin viscous liquid sheets, such as those emitted from industrial spraying nozzles, is presented. These sheets are in the form of a circular sector whose thickness reduces as the distance from the nozzle increases.

The Kelvin-Helmholtz type of instability usually observed causes the breakup and atomization of the sheet, as required in most industrial spraying processes. Waviness, like that of a flapping flag, is produced and increasing amplitudes finally cause breakup.

An analytical solution in the form of hypergeometric functions for the shape of the sheet and the waves is obtained. This solution includes, as special cases, analyses existing in the literature, in addition to establishing the possibility of a new type of instability dependent on the distance from the nozzle. Also, the classical type of instability, in which the waves increase with time, is examined and relations for unstable waves as a function of parameters such as the fluid viscosity, surface tension and sheet velocity are obtained. It is shown that there is no single wave that has a maximum growth rate, but that the wavenumber for maximum instability increases with the distance from the nozzle orifice.

The problem of the resonant response of a gas contained in a two-dimensional rectangular cavity to periodic (sinusoidal) velocity excitations at the walls of the cavity is investigated. It is found that in some neighbourhood of each resonant frequency there are discontinuous pressure disturbances (shock waves). The present theory is an extension of Chester's theory on resonances in closed tubes.

Large-scale atmospheric flow shares certain attributes with two-dimensional turbulence. In this paper, we study the effect of spherical geometry on two-dimensional turbulence.

Energy transfer is multi-component in spherical geometry in contrast to energy transfer among triads of wave vectors in Cartesian geometry. It follows that energy transfer is more local in spherical than in Cartesian geometry. Enstrophy transfer to higher wavenumbers in spherical geometry is less than enstrophy transfer to higher wavenumbers in Cartesian geometry. Since both energy and enstrophy are inviscid constants of motion, the back transfer of energy is also less in spherical than in Cartesian geometry. Therefore, with a finite viscosity, enstrophy decays more slowly in spherical geometry than in Cartesian geometry. Here these conjectures are tested numerically by spectral methods. The numerical results agree well with the conjectures.

The linear hydrodynamic stability problem for plane Poiseuille flow of a dilute suspension of rigid fibres is solved numerically. The constitutive equation given by Batchelor (1970a, b, 1971) is used to model the rheological properties of the suspension. The resulting eigenvalue problem is shown to be singular. The appropriate contour in the complex plane is determined by considering an initial-value problem. It is shown that, for a fixed, but not too large, inclination of the wave front to the mean flow, the fibres cause the critical Reynolds number to increase monotonically with the product of the volume fraction of the fibres and the square of their aspect ratio. The stabilizing influence of the fibres seems to vanish for large wave inclination angles.

Von Kármán's momentum-balance method is adopted to investigate the earthquake forces on a rigid dam with an inclined upstream face of constant slope. The distribution of the hydrodynamic pressure along the upstream face is determined. It is found that the maximum hydrodynamic pressure occurs at the base of the dam for any inclination angle between 0 and 90°. Explicit analytical formulae for evaluating the total horizontal, vertical and normal loads are presented and a useful approximate rule for practical engineers is given.

The equations for the earthquake forces on a rigid dam with an inclined upstream face of constant slope are solved exactly by two-dimensional potential-flow theory. The distribution of the hydrodynamic pressure along the upstream face and the total horizontal, vertical and normal loads on the dam are computed from the integral solutions. The results obtained from the exact theory are compared with those derived from the momentum-balance method and there is reasonable agreement.

Free shear layer stability measurements with a hot wire revealed that the probe itself can trigger and sustain upstream instability modes like the slit jet-wedge edge tones. The flow fields associated with the free shear layer tones induced in axisym-metric and plane air shear layers by a hot-wire probe and by a plane wedge were then explored experimentally, and found to be different in many ways from the widely investigated jet edge tone phenomenon.

As many as four frequency stages have been identified, there being a fifth stage associated with the subharmonic attributed to vortex pairing in the free shear layer. No evidence of hysteresis could be found in the shear layer tone. In the interstage jump (i.e. bimodal) regions, the tone occurred in only one mode at a time while intermittently switching from one to the other. Frequency variations in each stage are shown to collapse on a single curve when non-dimensionalized with the initial momentum thickness θe or with the lip-wedge distance h, and plotted as a function of h/θe.

Phase average measurements locked onto the tone fundamental show that the phase velocity and wavelength of the tone-induced velocity fluctuation are essentially independent of the stage of tone generation; in each stage, both phase velocity and wavelength decrease with increasing frequency but undergo jumps at starts of new stages. The measured amplitude and phase profiles, as well as the variations of the shear tone wavenumber and phase velocity with the Strouhal number, show reasonable agreement with the predictions of the spatial stability theory. The wavelength λ bears a unique relation to h, this h, δ relation being different from the Brown-Curle equation for the jet edge tone.

Shear layer tones would be typically induced in near-field shear layer measurements involving invasive probes, and can produce misleading results. A method for determining the true free shear layer natural instability frequency is recommended.

On the basis of a stability analysis of finite amplitude, two-dimensional convection, we have determined the dimensions of boxes containing fluid-saturated porous material in which convection is necessarily unsteady or steady and three-dimensional. For certain box sizes, convective rolls are unstable at Rayleigh numbers Ra lower than 380, the value below which rolls are stable forms of convection between infinite parallel planes. For Ra = 100 and 200, it appears unlikely that there are any box dimensions for which there is not a stable (possibly multicellular) two-dimensional steady motion. At Ra = 340 and 400, boxes in which rolls are unstable have heights which range from one to five times their horizontal dimensions.