This theoretical and experimental investigation inquires into the various steady and unsteady motions that are possible when a highly viscous liquid partly fills a closed circular cylinder rotated about its horizontal axis at constant angular velocity. Fillings leaving an air bubble in the range roughly 10-20% by volume provide the most lively variety of observable phenomena.

The full hydrodynamic problem is too complicated to be amenable to quantitative theoretical treatment, except by numerical analysis which is not yet available; but the abstract qualitative theory developed in § 2 appears to capture all the essentials of experimentally found behaviour. An analogous finite-dimensional system, such as would be presented by a close finite-element approximation, is used to illuminate principles governing the order of multiple solutions and their stability. Then the connection between the full problem and the analogue is demonstrated. Finally a simple argument is outlined confirming the observed stability of the motion at small rates of rotation.

The experiments are described in § 3 and their results presented in § 4. For various values of the cylinder's aspect ratio, estimated singularities of the time-independent solution set are recorded as several-branched graphs of ωv/gR versus volume fraction filled by liquid (ω is the angular velocity of the container, R its radius and v the kinematic viscosity of the liquid). The experimental observations are discussed in § 5.

A horizontal canal of infinite length and of constant width and depth contains inviscid fluid under gravity. The fluid is bounded internally by a submerged horizontal cylinder which extends right across the canal and has its generators normal to the sidewalls. Suppose that the fluid is set in motion by a surface pressure varying across the canal, then some of the energy is radiated to infinity while some of the energy is trapped in characteristic modes (bound states) near the cylinder. The existence of trapping modes in special cases was shown by Stokes (1846) and Ursell (1951); a general treatment, given by Jones (1953), is based on the theory of elliptic partial differential equations in unbounded domains. In the present paper a much simpler treatment is given which uses only the theory of bounded symmetric linear operators together with Kelvin's minimum-energy theorem of classical hydrodynamics.

We consider the radiation of internal gravity waves from a spherical body oscillating vertically in a stratified incompressible fluid. A near-field solution (under the Boussinesq approximation) is obtained by separation of variables in an elliptic problem, followed by analytic continuation to the frequencies ω < N of internal wave radiation. Matched expansions are used to relate this solution to a far-field solution in which non-Boussinesq terms are retained. In the outer near field there are parallel conical wavefronts between characteristic cones tangent to the body, but with a wavelength found to be shorter than that for oscillations of a circular cylinder. It is also found that there are caustic pressure singularities above and below the body where the characteristics intersect. Far from the source, non-Boussinesq effects cause a diffraction of energy out of the cones. The far-field wave-fronts are hyperboloidal, with horizontal axes. The case of horizontal oscillations of the sphere is also examined and is shown to give rise to the same basic wave structure.

The related problem of a pulsating sphere is then considered, and it is concluded that certain features of the wave pattern, including the caustic singularities near the source, are common to a more general class of oscillating sources.

The evolution equation governing wavemaker-generated cross-waves near a cutoff frequency in an infinitely deep, infinitely long channel is shown to be the nonlinear Schrödinger equation with a homogeneous boundary condition at the wavemaker. With the inclusion of an empirically determined damping coefficient, numerical results for growth rate, slow modulation period, and wave amplitude show good agreement with previous experiments. The results also describe observations of trapped and propagating solutions.

The theory of homogeneous barotropic beta-plane turbulence is here extended to include effects arising from spatial inhomogeneity in the form of a zonal shear flow. Attention is restricted to the geophysically important case of zonal flows that are barotropically stable and are of larger scale than the resulting transient eddy field.

Because of the presumed scale separation, the disturbance enstrophy is approximately conserved in a fully nonlinear sense, and the (nonlinear) wave-mean-flow interaction may be characterized as a shear-induced spectral transfer of disturbance enstrophy along lines of constant zonal wavenumber k. In this transfer the disturbance energy is generally not conserved. The nonlinear interactions between different disturbance components are turbulent for scales smaller than the inverse of Rhines's cascade-arrest scale κβ≡ (β0/2urms)½ and in this regime their leading-order effect may be characterized as a tendency to spread the enstrophy (and energy) along contours of constant total wavenumber κ ≡ (k2 + l2)½. Insofar as this process of turbulent isotropization involves spectral transfer of disturbance enstrophy across lines of constant zonal wavenumber k, it can be readily distinguished from the shear-induced transfer which proceeds along them. However, an analysis in terms of total wavenumber K alone, which would be justified if the flow were homogeneous, would tend to mask the differences.

The foregoing theoretical ideas are tested by performing direct numerical simulation experiments. It is found that the picture of classical beta-plane turbulence is altered, through the effect of the large-scale zonal flow, in the following ways: (i) while the turbulence is still confined to KKβ, the disturbance field penetrates to the largest scales of motion; (ii) the larger disturbance scales K < Kβ exhibit a tendency to meridional rather than zonal anisotropy, namely towards v2 > u2 rather than vice versa; (iii) the initial spectral transfer rate away from an isotropic intermediate-scale source is significantly enhanced by the shear-induced transfer associated with straining by the zonal flow. This last effect occurs even when the large-scale shear appears weak to the energy-containing eddies, in the sense that dU/dy [Lt ] κ for typical eddy length and velocity scales.

Generalized Taylor dispersion theory, incorporating so-called coupling effects, is used to calculate the transport properties of a single deformable ‘chain’ composed of hydrodynamically interacting rigid Brownian particles bound together by internal potentials and moving through an unbounded quiescent viscous fluid. The individual rigid particles comprising the flexible chain or cluster may each be of arbitrary shape, size and density, and are supposed ‘joined’ together to form the chain by a configuration-dependent internal potential V. Each particle separately undergoes translational and rotational Brownian motions; together, their relative motions give rise to a conformational or vibrational Brownian motion of the chain (in addition to a translational motion of the chain as a whole). Sufficient time is allowed for all accessible chain configurations to be sampled many times in consequence of this internal Brownian motion. As a result, an internal equilibrium Boltzmann probabilistic distribution of conformations derived from V effectively obtains.

In contrast with prior analyses of such chain transport phenomena, no ad hoc preaveraging hypotheses are invoked to effect the averaging of the input conformation-specific hydrodynamic mobility data. Rather, the calculation is effected rigorously within the usual (quasi-static) context of configuration-specific Stokes-Einstein equations.

Explicit numerical calculations serving to illustrate the general scheme are performed only for the simplest case, namely dumb-bells composed of identically sized spheres connected by a slack tether. In this context it is pointed out that prior calculations of flexible-body transport phenomena have failed to explicitly recognize the existence of a Taylor dispersion contribution to the long-time diffusivity of sedimenting deformable bodies. This fluctuation phenomenon is compounded of shape-sedimentation dispersion (arising as a consequence of the intrinsic geometrical anisotropy of the object) and size-sedimentation dispersion (arising from fluctuations in the instantaneous ‘size’ of the object). Whereas shape dispersion exists even for rigid objects, size dispersion is manifested only by flexible bodies. These two Taylor dispersion mechanisms are relevant to interpreting the non-equilibrium sedimentation-diffusion properties of monodisperse polymer molecules in solutions or suspensions.

Surface waves in a rectangular container subjected to vertical oscillations are studied. Effects of energy dissipation along the lines of Miles (1967) and the effect of surface tension are included. Sufficient conditions, for two modes to dominate the motion, are given. The analysis is along the lines of Miles (1984a) and Holmes (1986). A complete bifurcation analysis is performed, and the modal amplitudes and phases are shown to have chaotic behaviour. This result is obtained under assumptions different from those of Holmes (1986). The conclusions regarding chaotic motions are based on a theorem of šilnikov (1970).