The occurrence of dual solutions in curved ducts is investigated through a numerical solution of the Navier-Stokes equations in a bipolar-toroidal co-ordinate system. With the shape of duct being the region formed by the natural co-ordinate surfaces, it was possible to alter the duct geometry gradually and preserve the prevailing form of the velocity field, in a manner suggested by Benjamin (1978).

In addition to the Dean number Dn = Re/Rc½, a geometrical parameter that defines the shape of the duct was also varied systematically to study the bifurcation of a two-vortex solution into a two- and four-vortex solution. Dual solutions have been found for all geometrical shapes investigated here. Of particular interest are the shapes of a full circle and a semicircle with a curved outer wall.

The outer cylinder was kept stationary whilst the inner rotated. Weak circulatory motion (‘embryo cells’) was observed in all the fluids investigated at Taylor numbers Ta less than the critical Taylor number Tac. These cells grew with increasing Ta, then burst into vigorous Taylor vortices at Ta = Tac. In solutions of rigid elongated molecules having concentrations above a minimum concentration, an axial oscillation (overstability?) of the embryo cells was observed. The oscillation did not occur in solutions of flexible folded molecules nor in the solvent. The oscillation was observed over a range Ta0 ≤ Ta ≤ Tac. For concentrations above the minimum Ta0/Tac decreased with increasing concentration.

Other phenomena of the flow were also studied. It was found for solvent and solutions that the torque on the outer cylinder increased relatively less rapidly with the speed of rotation of the inner cylinder when \[ Ta/Ta_c > 1.08 \] than it did when \[ 1 < Ta/Ta_c\leqslant 1.08. \] In this latter range there was a torque reduction in the flow of the solutions relative to that in the flow of solvent. At sufficiently high speeds of rotation circumferential waves occurred in the fully developed Taylor cells (at Ta = Taw) and it was found that when R1/R2 = 0·95 then Taw/Tac = 1·5 for the solvent, whereas for the solutions Taw/Tac ≥ 1·5. When R1/R2 = 0·90 then Taw/Tac = 1·25, and for the solutions Taw/Tac > 1·25. No experiment was done at Ta > Taw. The normal-stress coefficients of the solution have been evaluated from the results.

We consider the behaviour of an interface between two immiscible inviscid incompressible fluids of different density moving under the action of gravity, inertial and interfacial tension forces. A vortex-sheet model of the exact nonlinear two-dimensional motion of this interface is formulated which includes expressions for an appropriate set of integral invariants. A numerical method for solving the vortex-sheet initial-value equations is developed, and is used to study the nonlinear growth of finite-amplitude normal modes for both Kelvin-Helmholtz and Rayleigh-Taylor instability. In the absence of an interfacial or surface-tension term in the integral-differential equation that describes the evolution of the circulation distribution on the vortex sheet, it is found that chaotic motion of, or the appearance of curvature singularities in, the discretized interface profiles prevent the simulations from proceeding to the late-time highly nonlinear phase of the motion. This unphysical behaviour is interpreted as a numerical manifestation of possible ill-posedness in the initial-value equations equivalent to the infinite growth rate of infinitesimal-wavelength disturbances in the linearized stability theory. The inclusion of an interfacial tension term in the circulation equation (which stabilizes linearized short-wavelength perturbations) was found to smooth profile irregularities but only for finite times. While coherent interfacial motion could then be followed well into the nonlinear regime for both the Kelvin-Helmholtz and Rayleigh-Taylor modes, locally irregular behaviour eventually reappeared and resisted subsequent attempts at numerical smoothing or suppression. Although several numerical and/or physical mechanisms are discussed that might produce irregular behaviour of the discretized interface in the presence of an interfacial-tension term, the basic cause of this instability remains unknown. The final description of the nonlinear interface motion thus awaits further research.