Equilibrium shapes and stability of annular fluid menisci held together by surface tension are analysed by applying asymptotic and computer-aided techniques from bifurcation theory. The shapes and locations of the menisci are governed by the Young–Laplace equation. These shapes are grouped together into families of like symmetry that branch from the basic family of annular shapes at specific values of the aspect ratio, α. Multiple equilibrium shapes exist over certain values of α. The inner, outer or both the inner and outer interfaces may possess either a cylindrical or sinusoidal equilibrium shape. Changes in applied pressure, fluid volume, or gravitational Bond number break families of the same symmetry which now develop limit points. Numerical calculations rely on a finite-element representation of the interfaces and the results compare very well with asymptotic analysis which is valid for small deformations. The results are important for the blow moulding process and are invaluable in understanding its dynamics. These dynamics are expected to be considerably different from the dynamics of a liquid jet first analysed by Rayleigh.

The stability of Couette flow in HeII is considered by an analysis of the HVBK equations. These equations are based on the Landau two-fluid model of HeII and include mutual friction between the normal and superfluid components, and the vortex tension due to the presence of superfluid vortices. We find that the vortex tension strongly affects the nature of the Taylor instability at temperatures below ≈ 2.05 K. The effect of the vortex tension is to make non-axisymmetric modes the most unstable, and to make the critical axial wavelength very long.

We compare our results with experiments.

A vortex pair, impulsively generated from a planar nozzle, is shown to have a degree of vorticity concentration in good agreement with inviscid theory, providing well-posed initial conditions for interaction with basic types of bodies (cylinders and plates). The scale of these bodies ranges from the same order as, to over an order of magnitude smaller than, the scale (distance between centres) of the incident vortex pair.

The fundamental case of a (primary) vortex pair symmetrically incident upon a very small cylinder shows rapid growth of a secondary vortex pair. These secondary vortices quickly attain a circulation of the same order as that of the corresponding primary vortices within a distance smaller than the lengthscale of the primary vortex pair. At this location, the temporal variation of integrated vorticity of primary and secondary vortices attains a maximum simultaneously. This zero phase shift between arrival of vorticity maxima provides the basis for formation of counter-rotating, primary–secondary vortex pairs, where both the primary and secondary vortices move at the same phase speed.

Visualization shows that the mode of secondary vortex formation is highly sensitive to the degree of symmetry of the initial encounter of the incident vortex pair with the body. The symmetrical mode of (in-phase) secondary vortex formation shows very rapid growth of large-scale secondary vortices; their development is relatively independent of the particulars of body shape and scale. On the other hand, the antisymmetrical mode takes two basic forms: large-scale secondary vortex formation, with the phase shift between their formation determined by the lengthscale of the body; and small-scale, antisymmetrical shedding of secondary vortices from the body occurring for a body lengthscale an order of magnitude smaller than that of the incident vortex pair. Correspondingly, there are several types of distortion of the cores and trajectories of the primary (incident) vortices.