An analysis is made of the Stokes flow between parallel planes due to a three-dimensional rotlet whose axis is parallel to the boundary planes. The separation in the plane of symmetry of this flow is compared with that in its two-dimensional analogue, the Stokes flow between parallel planes due to a two-dimensional rotlet. It is found that when the rotlets are midway between the planar walls, both flows exhibit an infinite set of Moffatt eddies. However, when the rotlets are not midway between the walls, the two-dimensional flow has an infinite set of Moffatt eddies, while the three-dimensional flow has at most a finite number of eddies and behaves, far from the rotlet, like the flow due to a two-dimensional source–sink doublet in each of the planes parallel to the boundary planes. An eigenfunction expansion describing a class of asymmetric Stokes flows between parallel planes is also derived and used to show that the far-field behaviour of flows in this class generally resembles the aforementioned flow due to a two-dimensional source–sink doublet in the planes parallel to the walls.

The inviscid interaction of two-dimensional vortex flows with a free surface is studied umerically using a combined vortex/boundary integral technique. The vorticity is modelled as point vortices, vortex sheets and finite area vortex regions. Two problems are studied in considerable detail, the head-on collision of a vortex pair with a free surface and the large-amplitude Kelvin–Helmholtz instability of a submerged shear layer. The interaction is controlled by a Froude number and by the geometric parameters describing the initial vortex configuration. In the large-Froude-number limit, the surface motion follows the vortical flow, but depends only weakly on the actual value of the Froude number. For low Froude numbers, the free surface remains almost flat, and the disturbances caused by the vortical flow decrease rapidly with Froude number.

A non-inertial (zero Taylor number) viscoelastic instability is discovered for Taylor–Couette flow of dilute polymer solutions. A linear stability analysis of the inertialess flow of an Oldroyd-B fluid (using both approximate Galerkin analysis and numerical solution of the relevant small-gap eigenvalue problem) show the growth of an overstable (oscillating) mode when the Deborah number exceeds f(S) ε−½, where ε is the ratio of the gap to the inner cylinder radius, and f(S) is a function of the ratio of solvent to polymer contributions to the solution viscosity. Experiments with a solution of 1000 p.p.m. high-molecular-weight polyisobutylene in a viscous solvent show an onset of secondary toroidal cells when the Deborah number De reaches 20, for ε of 0.14, and a Taylor number of 10−6, in excellent agreement with the theoretical value of 21. The critical De was observed to increase as ε decreases, in agreement with the theory. At long times after onset of the instability, the cells become small in wavelength compared to those that occur in the inertial instability, again in agreement with our linear analysis. For this fluid, a similar instability occurs in cone-and-plate flow, as reported earlier. The driving force for these instabilities is the interaction between a velocity fluctuation and the first normal stress difference in the base state. Instabilities of the kind that we report here are likely to occur in many rotational shearing flows of viscoelastic fluids.

The surface-tension-driven breakup of viscous jets is observed for a range of Weber and Ohnesorge numbers. The breakup is enhanced with a sinusoidal modulation or pulsation of the jet's exit velocity; the velocity modulation amplitudes and wavenumbers are larger than previous values reported in the literature. The combinations of modulation amplitude and wavenumber that produce uniform droplets are identified for each pair of Weber and Ohnesorge numbers. Satellite droplets are eliminated at values of the Ohnesorge number greater than 1.6. Droplet pairing and merging occurs at high wavenumbers; droplet merging has not been reported in the jet breakup literature. The timescale for breakup is predicted within the data scatter by the thin filament equation of Bousfield et al. (1986) with no fitted parameters. An upper bound on satellite droplet size is predicted by the thin-filament equation and the average satellite droplet volume is qualitatively predicted. An algebraic expression is derived to predict the breakup time of viscous jets with large velocity modulation amplitudes.

Mean drift currents due to damped, progressive, capillary-gravity waves at an air/water interface are investigated theoretically. The analysis is based on a Lagrangian description of motion. Both media are assumed to be semi-infinite, viscous, homogeneous fluids. The system rotates about the vertical axis with a constant angular velocity ½f, where f is the Coriolis parameter. Owing to viscous effects, the wave field attenuates in time or space. Linear analysis verifies the temporal decay rate reported by Dore (1978a). The nonlinear drift velocities are obtained by a series expansion of the solutions to second order in a parameter ε, which essentially is proportional to the wave steepness. The effect of the air on the drift current in the water is shown to depend on the values of the frequency ω, wavenumber k, density ρ, and kinematic viscosity ν through one single dimensionless parameter Q defined by \[ Q = \frac{\rho_1}{\rho_2}\bigg[\frac{\omega\nu_1}{2k^2\nu^2_2}\bigg]^{\frac{1}{2}}, \] where subscripts 1 and 2 refer to the air and the water, respectively. Dynamically, the increased shear near the interface due to the presence of the air leads to a higher value of the virtual wave stress (Longuet-Higgins 1969). This yields a tendency towards a higher (Eulerian) mean drift velocity as compared to the free-surface case. For temporally damped waves, this stress, due to increased damping, effectively acts over a shorter period of time. Accordingly, the mean current associated with such waves tends to be larger for short times and smaller for large times than that obtained with a vacuum above the water. For spatially damped waves, the virtual wave stress becomes independent of time. The Coriolis force is then needed to balance the wave stress in order to avoid infinitely large drift velocities as t → ∞. Furthermore, assessment of realistic values for the turbulent eddy viscosities in the air and the ocean is shown to bring the results closer to those obtained for a vacuum/water system.

The instability of a cylindrical liquid jet encapsulated by a viscous gas in a pipe is analysed in a parameter space spanned by the Reynolds number, the Froude number, the Weber number, the density ratio, the viscosity ratio, and the diameter ratio. A convergent solution of the problem is constructed by a Galerkin projection with two orthogonal sets of functions. Two distinctively different modes of instability are obtained. The first is the Rayleigh mode which tends to break up the jet into drops of diameter comparable with the jet diameter. The amplification rate of the disturbance belonging to this mode depends weakly on all parameters except the Weber number which represents the ratio of the surface tension force to the inertia force at the interface. The mechanism of the instability remains that of capillary pinching even in the presence of a viscous gas and gravity. However, the surface tension is stabilizing in the other mode termed the Taylor mode. The Taylor mode instability is due to the pressure and shear fluctuations at the interface. This mode tends to produce droplets of diameters much smaller than that of the jet. It is shown that the former mode appears when the Weber number is much larger than the gas to liquid density ratio. When this ratio is of order one, the instability can be due to either modes depending on the values of the rest of the parameters. When the density ratio is much larger than the Weber number, Taylor's atomization mode replaces the Rayleigh mode.