Nonlinear oscillations and other motions of large axially symmetric liquid drops in zero gravity are studied numerically by a boundary-integral method. The effect of small viscosity is included in the computations by retaining first-order viscous terms in the normal stress boundary condition. This is accomplished by making use of a partial solution of the boundary-layer equations which describe the weak vortical surface layer. Small viscosity is found to have a relatively large effect on resonant-mode coupling phenomena.

Undular disturbances of arbitrarily small steepness to the boundary of a circular patch of uniform vorticity are found to give way to a qualitatively different type of behaviour after a time inversely proportional to the initial wave steepness squared. Repeatedly, thin filaments are drawn out at a frequency of nearly half the vorticity jump across the interface (the intrinsic frequency of linear waves on the interface), and the interaction between the filaments and the undulating boundary generate new disturbances from which still greater numbers of filaments of increasingly complicated shape are drawn out. The vortex boundary thereby experiences an extraordinary, continual and apparently irreversible growth in complexity.

Essentially the same phenomenon occurs on all vortex-patch equilibria, e.g. the Kirchhoff elliptical vortex, with even greater complexity. The steepening of a disturbance on a non-circular vortex proceeds faster than that on a circular vortex, because of the varying mean strain and shear seen by the disturbance as it travels around the vortex boundary.

Generalizations to more than one vorticity interface, to flows on the surface of a sphere, and to sharp but not infinitely sharp vorticity gradients are also discussed. The results support the view that almost any two-dimensional, inviscid, incompressible flow with large vorticity gradients will exhibit repeated filamentation.

In an incompressible perfectly conducting fluid the Navier-Stokes equations become \[ \frac{\partial v}{\partial t} + v.\nabla v = -\nabla {\rm P} - \nu\Delta v + j\times {\boldmath H}, \] where j = curl H, div H = 0, div v = 0, and dH/dt = curl (v × H). The last equation follows from the Maxwell equation dH/dt = − curl E and the assumption of perfect conduction — that the electric field in the frame of the fluid vanishes, E′ = E + v × H = 0. In geometric terms, it says that the system evolves so that the time derivative of H is equal to minus its spatial Lie derivative: \[ \frac{{\rm d}{\boldmath H}}{{\rm d}t} = -L_{\rm v}{\boldmath H}. \]

Thus H is equivariant with respect to the evolution (or ‘frozen in the fluid’) as long as the evolution follows these equations. Since the first equation tends to dissipate magnetic energy E = ∫ ∥H∥2 the question naturally rises whether the topology of H determines lower bounds on E. We treat this question in the general context of a divergence-free vector field H on a closed Riemannian 3-manifold M. We obtain a result bounding E from below but make no assertion on the existence of extremals. Arnol'd (1986) has defined a quadratic form for any ‘null-homologous’ H

A simple phenomenological drag relation is used to characterize weak dynamic coupling of a liquid membrane to an adjacent solid substrate. With this linear velocity-dependent drag relation, the inertialess equations of motion for membrane flow are easily solved for steady translation and rotation of a disk-like particle. The resulting drag coefficients exhibit functional dependencies on the dimensionless particle size very similar to the relations obtained for particle motion in a membrane bounded by semi-infinite liquid domains (Hughes et al. 1981), although the scaling of particle size is different. Within this phenomenological approach, diffusivities of molecular probes in membranes can be used to investigate the intrinsic molecular drag at a solid-liquid membrane interface and to estimate properties of thin lubricating liquid layers between membrane and substrate.

The fluid dynamics of moving drops in the presence of a soluble surfactant and an insoluble impurity is examined in detail. The main purpose of this analysis is to establish a fairly general theory that agrees with experimental measurements. Particular attention is paid to situations involving a stagnant cap which arise when low-solubility surfactants are present. Earlier theories on stagnant caps have not satisfactorily explained the experimental results and a two-impurity model is therefore proposed. The analysis is carried out semi-analytically using a matched asymptotic analysis of the Proudman-Pearson type for weakly inertial flows. The results seem to be in good agreement with the available data at a Péclet number of about 700 for the soluble surfactant. In particular the predicted flow field within the drop is found to be consistent with the experimental measurements of Horton. The concentration profiles graphically exhibit the physical phenomena involved in the mass transport. Another new result is the analytical expression for the drag force corrected up to O(Re) for the case involving only the insoluble surfactant.

We consider the situation where a deep-water wavetrain approaching from infinity forms a circular caustic, is glancingly reflected at the caustic, and then propagates on out to infinity. At every point in the wavefield there are two wavetrains, the incident and reflected waves. Thus the wavefield can be treated as a slowly varying field of short-crested waves. This work generalizes that of Peregrine (1981) who considered a wavefield of incident waves only. The problem is formulated using the averaged-Lagrangian variational approach of Whitham (1974). Owing to the circular symmetry of the problem, the governing differential equations can be reduced to a set of algebraic equations at each radius. Results for the wave steepness and wavenumber are presented. These indicate that the nonlinear caustic occurs at a larger radius than does the linear caustic, and that the ray paths are no longer straight but curve away from the caustic. It is found that the slowly varying assumption is invalid at the caustic radius. To overcome this we derive, by the method of multiple scales, a modified nonlinear Schrödinger equation which is valid in this region. The solution of this equation, involving the second Painlevé transcendent, is then asymptotically matched to the slowly varying solution to provide a complete description of the wavefield.