A thin vertical plate makes small, simple harmonic rolling oscillations beneath the surface of an incompressible, irrotational liquid. The plate is assumed to be so wide that the resulting equations may be regarded as two-dimensional. In addition, a train of plane waves of frequency equal to the frequency of oscillation of the plate, is normally incident on the plate. The resulting linearized boundary-value problem is solved in closed form for the velocity potential everywhere in the fluid and on the plate. Expressions are derived for the first- and second-order forces and moments on the plate, and for the wave amplitudes at a large distance either side of the plate. Numerical results are obtained for the case of the plate held fixed in an incident wave-train. It is shown how these results, in the special case when the plate intersects the free surface, agree, with one exception, with results obtained by Ursell (1947) and Haskind (1959) for this problem.

The title flow has been studied by measuring the drag force on, and by observing the flow field around, a sphere rising through a large, rotating tank of water. Long, almost stagnant, regions are formed up- and downstream within the shadow of the sphere and are surrounded by a thin annular region within which the velocity is larger than the mean velocity of the approach flow. Several regions are found within which vortex-jump phenomena occur and it is concluded that such features exert a controlling influence over the dynamics of the observed flow field.

This paper describes theoretical and experimental studies for induction velometry in a shock wave. The induced potential gradient profiles across the shock is obtained with an assumption that the velocity of the charged particles is given by Mott-Smith's result. The calculated result of the induced potential gradient shows that it is not proportional to the velocity of the charged particles, since eddy currents are also induced by the existence of non-uniform velocity through the shock. The induced potential gradient is measured across the shock produced by a shock holder. This result is compared with the theoretical result. The experimental result shows that the charged particles follow the Rankine-Hugoniot relation in the velocity.

The capillary instability of vertical liquid jets of different viscosities have been examined by imposing audio-frequency disturbances. Real time sequences of photographs allow a direct measurement of growth rates of disturbances of various wavelengths. Results show that in general non-linear effects dominate the growth processes. This is in agreement with Yuen's analysis. The growth rate of the difference between the neck and the swell, however, agrees well with the linearized analysis of Rayleigh and Chandrasekhar. The non-linear effect causes a liquid jet to disintegrate into drops with ligaments in between. The sizes of the ligaments decrease with increasing wave-number. The subsequent roll up of the ligament into droplet, the eventual coalescing of the droplet with the main drop and drop oscillation have also been studied.

The main thrust of this work is to treat the initial convective phase of a fluid heated from below as a statistical initial value problem. The advantage of the approach is that it allows a continuous bandwidth of modes to be represented in the initial spectrum. We show that if the initial disturbance field is small and has a sufficiently smooth spectrum, then a natural statistical selection process chooses from the initial disorder a perfectly ordered field of single rolls. The scale of this roll is the scale corresponding to the most critical wave-number obtained from the linear stability problem. We relate this solution to the optimal solution which would be obtained by the upper bound procedures of Howard, Malkus and Busse. Moreover, we show in addition, that if the initial disturbance field is weighted in favour of a particular single roll whose scale is close to critical, the final solution reflects the initial condition providing a certain stability criterion is met. In the two-dimensional case we analyze, this turns out to be the Eckhaus stability condition previously obtained by a discrete multimodal analysis.

Flow of an incompressible inviscid fluid past a sphere is considered, where the flow upstream consists of a slight shear flow superimposed on a uniform stream. Secondary vorticity, produced by deformation of vortex elements as they are carried past the sphere, is determined by a method due to Lighthill (1956b). Components of vorticity are calculated from a drift function for which expressions were previously available only in part of the flow field. For the region in which no expansion is valid an exact integral expression is obtained to replace the rough numerical approximation used by Lighthill (1956b). The velocity distribution over the upstream part of the sphere is determined numerically using a Biot–-Savart law. These results are required for the calibration of certain forms of Pitot tubes.

Results of linear stability calculations and post-stability experimental observations are reported for horizontal fluid layers with upward heat flux bounded below by a stably stratified fluid. Stability calculations were done for several families of continuous and discontinuous temperature distributions, and it was found that as a rule the flow originating in the unstable layer penetrates into the stably stratified region, resulting in increased critical cell size and correspondingly decreased critical Rayleigh number. A notable exception to this occurs for an unstable layer with a linear temperature distribution adjacent to a stable layer of very high stable density gradient. In this case energy pumped from the unstable to the stable region is sufficient to raise the critical Rayleigh number above that of a solid boundary. It is also found that, for density distributions with a more gradual transition between the stable and the unstable regions, the effect of increased cell size upon the critical Rayleigh number is sometimes masked by effects of curvature in the density profile of the unstable region, which tends to increase the critical Rayleigh number. The inadequacy of the usual definition of Rayleigh number to characterize the stability of such complex systems is discussed. Experimentally, such a temperature distribution was produced by radiant energy from above as it was absorbed by the top few centimetres of the fluid. Within an uncertainty of ± 20%, it was found that the critical experimental Rayleigh number agreed with neutral stability calculations. The supercritical convective motion consisted of vertical jets of cool surface fluid which plunged downward into the interior of the fluid. The jets were not arranged in an orderly lattice but were in a constant state of change, each jet having a tendency to merge with a close neighbour. The net loss of jets due to merging was balanced by new jets spontaneously appearing. As Rayleigh number was increased, the mean number of jets and the intermittancy increased proportionally. Temperature scans taken with a movable probe showed that cool surface fluid plunging downward in the jets was confined to a fairly restricted region, the surrounding fluid being quite isothermal.

A theory is presented for two-dimensional incompressible potential flow external to a symmetrical bluff body and its wake. The desired flow-separation points are made the critical points of a conformal transformation to a complex plane in which surface sources in the wake create stagnation conditions at the critical points. The stagnation streamlines then transform to tangential separation streamlines in the physical plane, with separation at the desired pressure. The position and strength of the sources are determined by the requirements of separation position and pressure coefficient. The flow inside the separation streamlines is ignored and base pressure is assumed constant at the separation value. Features of the theoretical model include a finite wake width, a pressure distribution on the separation streamlines decreasing asymptotically towards the free stream value at infinity and a simple analytic expression for the pressure distribution on the body. Comparisons of the theory with experimental data and with other theories are presented for the normal plate, the circular cylinder, the 90° wedge, and the elliptical cylinder. Although simpler to apply than the other theories, the present theory produces at least as good agreement with the experimental data.

The dynamical behaviour of a system of parallel line vortices in an inviscid fluid is studied numerically. The initial configuration of the system is assumed to be such that the points of intersection of the line vortices with a plane normal to the vorticity form a regular polygon. The numerical experiments show that the vortex polygon is rearranged due to non-linear interactions among the line vortices in such a way as to produce a more or less uniform distribution of vortices inside the fluid with an approximately constant mean separation. The average angular velocity of the rotation of the vortex lines about the instantaneous centroid of the vortex system remains approximately constant. These results agree with the conjecture of Raja Gopal (1964). The results may prove to be of some value in a macroscopic model of liquid helium based on hydrodynamical principles.

A study is described of the forced inertial oscillations appearing in an axially rotating completely filled circular cylinder with plane ends. Excitation is provided by causing the top end to rotate about an axis inclined slightly to the rotation axis. Experiments demonstrate the presence of numerous low mode resonances in a densely spaced range of ratios of net cylinder height to radius in close conformance with linear inviscid theory. Where geometry permits simple corner reflexion, characteristic surfaces are revealed which confirm in part the theoretical predictions concerning their scale and form.

Detailed measurements are given of the amplitude at one point within the cylinder for the condition in which the disturbance frequency equals the rotation frequency. Amplitude column height spectra are compared with theoretical estimates, and the evolution of amplitude for the simplest mode of resonant oscillation is studied. A non-linear theory based on the integral energy of large amplitude oscillation is derived whose broad features are in fair quantitative and qualitative agreement with these observations.

Some investigation is made of the phenomenon of resonant collapse, in which larger amplitude resonant oscillations, after persisting in an apparently laminar form, degenerate abruptly into a state of agitation and disorder from which they do not recover. It is found that the time for emergence of this collapse after the introduction of the forcing disturbance has a close correspondence with the theoretical period of one ‘evolutionary’ cycle of momentum exchange between the main motion and the secondary oscillation.

A Symposium on Electrohydrodynamics sponsored by the International Unions of Theoretical and Applied Mechanics and of Pure and Applied Physics was convened at the Massachusetts Institute of Technology, Cambridge, Massachusetts, from 31 March to 1–2 April 1969. Authors from eight countries presented 44 papers to an invited audience of 112. The presentations illustrated the spectrum of scientific and engineering interests in the interactions of electric fields and moving fluids. Papers were grouped in seven sessions with the following titles: Steady flows; Bulk stability and dynamics; Interfacial dynamics: Static and steady-flow equilibria; Interfacial dynamics and stability: Drops; Basic formulation; Generation and other applications; Aerosol charging and dynamics. It is expected that the majority of authors will publish their papers in the formal literature. This brief review gives only an over-view of the Symposium as seen by the author.