We study the influence of a low-frequency harmonic vibration on the formation of the two-dimensional rolling solitary waves in vertically co-flowing two-layer liquid films. The system consists of two adjacent layers of immiscible fluids with the first layer being sandwiched between a vertical solid plate and the second fluid layer. The solid plate oscillates harmonically in the horizontal direction inducing Faraday waves at the liquid–liquid and liquid–air interfaces. We use a reduced hydrodynamic model derived from the Navier–Stokes equations in the long-wave approximation. Linear stability of the base flow in a flat two-layer film is determined semi-analytically using Floquet theory. We consider sub-millimetre-thick films and focus on the competition between the long-wavelength gravity-driven and finite wavelength Faraday instabilities. In the linear regime, the range of unstable wave vectors associated with the gravity-driven instability broadens at low and shrinks at high vibration frequencies. In nonlinear regimes, we find multiple metastable states characterized by solitary-like travelling waves and short pulsating waves. In particular, we find the range of the vibration parameters at which the system is multistable. In this regime, depending on the initial conditions, the long-time dynamics is dominated either by the fully developed solitary-like waves or by the shorter pulsating Faraday waves.

We present the results of a combined experimental and theoretical investigation of sheet evolution, expansion and retraction, under unsteady fragmentation upon drop impact on a surface of comparable size to that of the drop. We quantify and model the effect of the continuous time-varying – unsteady – shedding of droplets from the sheet via its bounding rim. We present and validate especially developed advanced image processing algorithms that quantify, with high accuracy, the key quantities involved in such unsteady fragmentation, from sheet, to rim, to ligaments, to droplet properties. With these high precision measurements, we show the important effect of continuous unsteady droplet shedding on the sheet dynamics. We combine experiments and theory to derive and validate governing equations of the sheet that incorporate such continuous shedding – associated with continuous loss of momentum and mass – from unsteady fragmentation. Combining this theory with the universal unsteady rim dynamics discovered in Wang et al. (Phys. Rev. Lett., vol. 120, 2018, 204503), we show that the governing equation of the sheet can be reduced to a continuous-shedding, non-Galilean Taylor–Culick law, from which we deduce new analytical expressions for the time evolution of the sheet radius. We show the robustness of the predictions to changes of fluid properties, including surface tension and moderate fluid viscosity and elasticity, including use of physiological mucosalivary fluid. We also reconcile prior literature's inconsistent experimental results on the sheet dynamics upon drop impact.

We consider fully three-dimensional time-dependent outflow from a source into a surrounding fluid of different density. The source is distributed over a sphere of finite radius. The nonlinear problem is formulated using a spectral approach in which two streamfunctions and the density are represented as a Fourier-type series with time-dependent coefficients that must be calculated. Linearized theories are also discussed and an approximate stability condition for early stages in the outflow is derived. Nonlinear solutions are presented and different outflow shapes adopted by the fluid interface are investigated.

We elucidate the physics underlying the birth, evolution and breakup of ligaments on a rim bounding an unsteady liquid sheet. This rim destabilizes into corrugations that can grow into ligaments, which in turn, break into secondary droplets via end-pinching. Combining experiments and theory, we show that not all corrugations can grow into ligaments. The number of corrugations is captured by linear instability coupled with nonlinear rim thickness self-adjustment ($\text {Bond number} = 1 \text { criterion}$, Wang et al. (Phys. Rev. Lett., vol. 120, 2018, 204503)) and scales as $N_c \sim We^{3/4}$ with Weber number, $We$. The number of ligaments scales as $N_{\ell } \sim We^{3/8}$. The growth of a ligament is governed by the competition between the constraint imposed by the geometry of the local rim–ligament junction; the local force balance including the fictitious force from the continuously decelerating rim; and the global rim mass conservation constraint. The temporal evolution of the average width of ligaments is predicted. Key to understanding the ligament population, a minimum distance between two corrugations is required to enable their actual transition into ligaments. By predicting this minimal distance, we derive the evolution of the number of ligaments. We show that droplets are shed, one at a time, following a chaotic dripping end-pinching regime independent of $We$. Finally, the number of droplets shed per unit of time decreases over time and scales as $S_d \sim We^{3/4}$; while the volume shed per unit of time increases over time and is independent of $We$. Theoretical predictions are validated without fitting parameters.

Lipid membranes are examples of fluid deformable surfaces, which can be viewed as two-dimensional viscous fluids with bending elasticity. With this solid–fluid duality any shape change contributes to tangential flow and vice versa any tangential flow on a curved surface induces shape deformations. This tight coupling between shape and flow makes curvature a natural element of the governing equations. The modelling and numerical tools outlined in Torres-Sánchez et al. (J. Fluid Mech., vol. 872, 2019, pp. 218–271) open a new field of study by enabling the exploration of the role of curvature in this context.

This paper reports new analytical formulae for the longitudinal slip lengths for simple shear over a superhydrophobic surface, or bubble mattress, comprising a periodic array of unidirectional circular menisci, or bubbles, protruding into, or out of, the fluid. The accuracy of the formulae is tested against results from full numerical simulations; they are found to give small relative errors even at large no-shear fractions. In the dilute limit the formulae reduce to an earlier result by Crowdy (Phys. Fluids, vol. 22, 2010, 121703). They also extend analytical results of Sbragaglia & Prosperetti (Phys. Fluids, vol. 19, 2007, 043603) beyond the limit of a small protrusion angle.

Explicit analytical expressions for the drag and diffusion coefficients of a spherical particle attached to the flat interface between two immiscible fluids are constructed for the case of a vanishing viscosity ratio between the fluid phases. The model is designed to account explicitly for the dependence on the contact angle between the two fluids and the solid surface. The Lorentz reciprocal theorem is applied in the context of geometric perturbations from the limiting cases of $90^{\circ }$ and $180^{\circ }$ contact angles. The model agrees well with the experimental and numerical data from the literature. Also, an advantage of the method utilized is that the drag and diffusion coefficients can be calculated up to one order higher in the perturbation parameter than the known velocity and pressure fields. Extensions to other particle shapes with known velocity and pressure fields are straightforward.

Water drops falling on a deep pool can either coalesce to form a vortex ring or splash, depending on the impact conditions. The transition between coalescence and splashing proceeds via a number of intermediate steps, such as thick and thin jet formation and gas-bubble entrapment. We perform simulations to determine the conditions under which bubble entrapment and jet formation occur. A regime map is established for Weber numbers ranging from 50 to 300 and Froude numbers from 25 to 600. Vortex ring formation is seen for all of the regimes; it is greater for the coalescence regime and less in the case of the thin jet regime.

The collision of liquids of different densities is studied theoretically for the case of liquids having wedge-shaped configuration before the impact. Both liquids are assumed to be ideal and incompressible, and the velocity potential theory is used for the flow of each liquid. Surface tension and gravity effects are neglected. The problem is decomposed into two self-similar problems, one for each liquid. Across the interface between the liquids, continuity of the pressure and the normal component of the velocity is enforced through iteration. This determines the shape of the interface and other flow parameters. The integral hodograph method is employed to derive the solution consisting of analytical expressions for the complex-velocity potential, the complex-conjugate velocity, and the mapping function. They are all defined in the first quadrant of a parameter plane, in which the original boundary-value problem is reduced to a system of integro-differential equations in terms of the velocity magnitude and the velocity angle relative to the flow boundary. They are solved numerically using the method of successive approximations. The results are presented through streamlines, interface and free-surface shapes, the pressure and velocity distributions. Special attention is given to the structure of the splash jet rising as a result of the impact.

The shape of impacting drops onto a solid surface is investigated by probing the local flow velocity and the local thickness profile of the spreading lamella during the drop impact. First, as a model situation of no viscous coupling between the liquid and the substrate, the impact of a drop onto hot plates, above the Leidenfrost temperature, is considered. In this case, we demonstrate that the velocity and thickness profiles are in good agreement with inviscid convective flow theory. This local description allows us to revisit the modelling of well-studied global behaviour such as drop spreading. Building from this idealized situation, viscous boundary-layer effects emerging from frictional coupling on a cold surface are then captured.

This experimental study, performed under microgravity conditions, focuses on the evaporation dynamics of ethanol drops and the formation and behaviour of the hydrothermal waves that spontaneously develop on the drop surfaces. The aim of this study is to compare our results to a similar study performed under normal gravity conditions to confirm the purely thermocapillary origin of these instabilities. A scaling law predicts with good agreement the number of instabilities that form, regardless of the gravity level.

When a fluid in a tube is occluded, one finds a static configuration in which the occluding free surface of the fluid is an equilibrium capillary surface spanning the tube. We extend known criteria for existence and non-existence of such a surface, leading to an explicit mathematically rigorous occlusion criterion for cylindrical tubes in a transverse body force field, depending on the force magnitude and contact angle. For any contact angle γ ≠ π/2, we provide further an explicit design of a tube section, which will not occlude in a downward gravity field, regardless of the field strength. In addition, we derive a precise analytic occlusion criterion for liquid partially filling a circular vessel spinning about its axis.

We study the linear stability of the flow of a viscous electrically conducting capillary fluid on a planar fixed plate in the presence of gravity and a uniform magnetic field, assuming that the plate is either a perfect electrical insulator or a perfect conductor. We first confirm that the Squire transformation for magnetohydrodynamics is compatible with the stress and insulating boundary conditions at the free surface but argue that unless the flow is driven at fixed Galilei and capillary numbers, respectively parameterizing gravity and surface tension, the critical mode is not necessarily two-dimensional. We then investigate numerically how a flow-normal magnetic field and the associated Hartmann steady state affect the soft and hard instability modes of free-surface flow, working in the low-magnetic-Prandtl-number regime of conducting laboratory fluids (Pm ≤ 10−4). Because it is a critical-layer instability (moderately modified by the presence of the free surface), the hard mode exhibits similar behaviour as the even unstable mode in channel Hartmann flow, in terms of both the weak influence of Pm on its neutral-stability curve and the dependence of its critical Reynolds number Rec on the Hartmann number Ha. In contrast, the structure of the soft mode's growth-rate contours in the (Re, α) plane, where α is the wavenumber, differs markedly between problems with small, but non-zero, Pm and their counterparts in the inductionless limit, Pm ↘ 0. As derived from large-wavelength approximations and confirmed numerically, the soft mode's critical Reynolds number grows exponentially with Ha in inductionless problems. However, when Pm is non-zero the Lorentz force originating from the steady-state current leads to a modification of Rec(Ha) to either a sub-linearly increasing or a decreasing function of Ha, respectively for problems with insulating or perfectly conducting walls. In insulating-wall problems we also observe pairs of counter-propagating Alfvén waves, the upstream-propagating wave undergoing an instability driven by energy transferred from the steady-state shear to both of the velocity and magnetic degrees of freedom. Movies are available with the online version of the paper.