In this paper we examine the effect that physiological non-polar lipids, residing on the surface of an aqueous tear film, have on the film evolution. In our model we track the evolution of the thickness of the non-polar lipid layer, the thickness of the aqueous layer and the concentration of polar lipids which reside at the interface between the two. We also utilise a force balance in the non-polar lipid layer in order to determine its velocity. We show how to obtain previous models in the literature from our model by making particular choices of the parameters. We see the formation of boundary layers in some of these submodels, across which the concentration of polar lipid and the non-polar lipid velocity and film thickness vary. We solve our model numerically for physically realistic parameter values, and we find that the evolution of the aqueous layer and the polar lipid layer are similar to that described by previous authors. However, there are interesting dynamics for the non-polar lipid layer. The effects of altering the key parameters are highlighted and discussed. In particular, we see that the Marangoni number plays a key role in determining how far over the eye the non-polar lipid spreads.

A mechanistic model for the low-Reynolds-, high-Strouhal-number behaviour of a system consisting of a spherical particle attached to an inelastic tether under uniform sinusoidal cross-flow is presented. Unsteady history drag and virtual mass effects are considered for both the sphere and the tether. The mechanics of the problem is such that the resulting coupled fractional differential equations are linear and solvable analytically. The stationary solutions obtained in this work show that there are limiting dimensions for the length and thickness of the tether when compared to the radius of the particle that allow for the motion of the particle–tether system to simulate the motion of a free particle. These conditions exist for the range of small oscillation amplitudes that are required for keeping the particle Reynolds number smaller than unity while oscillating the particle–tether system at high frequencies (Strouhal numbers larger than unity). The fractional order model for the particle–tether system is compared against detailed experimental results for tethered particles for a wide range of experimental frequencies, including the low-frequency range where tether effects are measurable.

A combination of cross-wire probes with an array of flush-mounted skin-friction sensors are used to study the three-dimensional conditional organisation of large-scale structures in a high-Reynolds-number turbulent boundary layer. Previous studies have documented the amplitude modulation of small-scale motions in response to conditionally averaged large-scale events, but the data are largely restricted to the streamwise component of velocity alone. Here, we report results based on all three components of velocity and find that the small-scale spanwise and wall-normal fluctuations ($v$ and $w$) and the instantaneous Reynolds shear stress ($-{uw}$) are modulated in a very similar manner to that previously noted for the streamwise fluctuations ($u$). The envelope of the small scale fluctuations for all velocity components is well described by the large-scale component of the $u$ fluctuation. These results also confirm the conditional existence of roll modes associated with the very large-scale or ‘superstructure’ motions.

We prove that there are no three-dimensional bounded travelling gravity waves with constant non-zero vorticity on water of finite depth. The result also holds for gravity–capillary waves under a certain condition on the pressure at the surface, which is satisfied by sufficiently small waves. The proof relies on unique continuation arguments and Liouville-type results for elliptic equations.

A recent hydrodynamic theory of liquid slippage on a solid substrate (Kirkinis & Davis, Phys. Rev. Lett., vol. 110, 2013, 234503) gives rise to a sequence of eddies (Moffatt vortices) that co-move with a moving contact line (CL) in a liquid wedge. The presence of these vortices is established through secular equations that depend on the dynamic contact angle $\alpha $ and capillary number Ca. The limiting case $\alpha \rightarrow 0$ is associated with the appearance of such vortices in a channel. The vortices are generated by the relative motion of the interfaces, which in turn is due to the motion of the CL. This effect has yet to be observed in experiment.

The fall velocity of a dense large ball in a suspension of neutrally buoyant non-Brownian particles subjected to horizontal oscillatory shear is studied. As the strain amplitude is increased, the velocity increases up to a maximum value before decreasing to the value that it would have in a resting suspension. The higher the frequency is, the stronger the effect is. The falling ball velocity can be largely increased in the presence of the oscillatory cross-shear flow. For instance, for a particle volume fraction of $\varPhi =0.47$ it reaches four times the value it has in the unsheared suspension. At small strain amplitudes, it turns out that the velocity of the falling ball is determined by a balance between the steady drag flow, which drives the apparent suspension viscosity toward a high value, and the oscillatory cross-shear, which lessens it. A simple model is proposed to explain the experimental observations at small strain amplitude. The velocity decrease observed at larger amplitude is not completely understood yet.