We report experimental, theoretical and numerical results on the shapes of liquid menisci connecting a planar boundary and the surface of a horizontal cylinder placed above. The gradient of confinement traps the wetting drops in the most confined regions, which promotes their elongation along the line of smallest gap between the walls. The experimental shapes of these stretched capillary bridges are shown to be in good quantitative agreement with the numerical solution of the equation describing their contour. In particular, we show that the measured shapes are better described when taking into account the correction resulting from the coupling of in-plane and transverse interfacial curvatures calculated by Park & Homsy (J. Fluid Mech., vol. 139, 1984, pp. 291–308) over thirty years ago.

An overlooked conservation law for near-inertial waves (NIWs) propagating in a steady background flow provides a new perspective on the concentration of these waves in regions of anticyclonic vorticity. The conservation law implies that this concentration is a direct consequence of the decrease in spatial scales experienced by an initially homogeneous wave field. Scaling arguments and numerical simulations of a reduced-gravity model of mixed-layer NIWs confirm this interpretation and elucidate the influence of the strength of the background flow relative to the dispersion.

When two spherical bubbles touch, a hole is formed in the fluid sheet between them, and capillary pressure acting on its tightly curved edge drives an outward radial flow which widens the hole joining the bubbles. Recent images of the early stages of this process (Paulsen et al., Nat. Commun., vol. 5, 2014) show that the radius of the hole $r_{\!E}$ at time $t$ grows proportional to $t^{1/2}$, and that the rate is dependent on the fluid viscosity. Here, we explain this behaviour in terms of similarity solutions to a third-order system of radial extensional-flow equations for the thickness and velocity of the sheet of fluid between the bubbles, and determine the growth rate as a function of the Ohnesorge number $\mathit{Oh}$. The initially quadratic sheet profile allows the ratio of viscous and inertial effects to be independent of time. We show that the sheet is slender for $r_{\!E}\ll a$ if $\mathit{Oh}\gg 1$, where $a$ is the bubble radius, but only slender for $r_{\!E}\ll \mathit{Oh}^{2}a$ if $\mathit{Oh}\ll 1$ due to a compressional boundary layer of length $L\propto \mathit{Oh}\,r_{\!E}$, after which there is a change in the structure but not the speed of the retracting sheet. For $\mathit{Oh}\ll 1$, the detailed analysis justifies a simple momentum-balance argument, which gives the analytic prediction $r_{\!E}\sim (32a{\it\gamma}/3{\it\rho})^{1/4}t^{1/2}$, where ${\it\gamma}$ is the surface tension and ${\it\rho}$ is the density.

The mechanism of turbulent drag reduction (DR) with super-hydrophobic (SH) surfaces is investigated by direct numerical simulation (DNS) and analysis of the governing equations in channel flow. The DNS studies were performed using lattice Boltzmann methods in channels with ‘idealized’ SH surfaces on both walls, comprised of longitudinal micro-grooves (MG), transverse MG, or micro-posts. DRs of $5\,\%$ to $83\,\%$, $-4\,\%$ to $20\,\%$, and $14\,\%$ to $81\,\%$ were realized in DNS with longitudinal MG, transverse MG, and micro-posts, respectively. By mathematical analysis of the governing equations, it is shown that, in SH channel flows with any periodic SH micro-pattern on the walls, the magnitude of DR can be expressed as $DR=U_{slip}/U_{bulk}+O({\it\varepsilon})$, where the first term represents the DR resulting from the effective slip on the walls, and the second term represents the DR or drag increase (DI) resulting from modifications to the turbulence dynamics and any secondary mean flows established in the SH channel compared to a channel flow with no-slip walls at the same bulk Reynolds number as the SH channel. Comparison of this expression to DNS results shows that, with all SH surface micro-patterns studied, between 80 % and 100 % of the DR in turbulent flow arises from the effective slip on the walls. Modifications to the turbulence dynamics contribute no more than 20 % of the total DR with longitudinal MG or micro-posts of high shear-free fraction (SFF), and a DI with transverse MG or micro-posts of moderate SFF. The effect of the SH surface on the normalized dynamics of turbulence is found to be small in all cases, and confined to additional production of turbulence kinetic energy (TKE) within a thin ‘surface layer’ of thickness of the order of the width of surface micro-indentations. Outside of this ‘surface layer’, the normalized dynamics of turbulence proceeds as in a turbulent channel flow with no-slip walls at the friction Reynolds number of the SH channel flow.