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Drops on a vibrating substrate can experience a variety of motion regimes, including directional motion and climbing. The key ingredient to elicit these regimes is simultaneously activating the in-plane and out-of-plane degrees of freedom of the substrate with the proper phase difference. This is typically achieved by imposing a prescribed rigid-body motion of the entire substrate. However, this framework is unable to establish different motion conditions in different regions of the substrate, thus lacking the precious spatial selectivity necessary to elicit complex drop control patterns. Challenging this paradigm, we leverage the inherent elasticity of the substrate to provide the required in-plane and out-of-plane modal characteristics and spatial diversity. To this end, we design architected substrates exhibiting a rich landscape of deformation modes, and we exploit their multimodal response to switch between drop motion regimes and select desired spatial patterns, using the excitation frequency as our tuning parameter. 

Viscous fingering, a classic hydrodynamic instability, is governed by the the competition between destabilising viscosity ratios and stabilising surface tension or thermal diffusion. We show that the channel confinement can induce ‘diffusion’-like stabilising effects on viscous fingering even in the absence of interfacial tension and thermal diffusion, when a clear oil invades the mixture of the same oil and non-colloidal particles. The key lies in the generation of long-range dipolar disturbance flows by highly confined particles that form a monolayer inside a Hele-Shaw cell. We develop a coarse-grained model whose results correctly predict universal fingering dynamics that is independent of particle concentrations. This new mechanism offers insights into manipulating and harnessing collective motion in non-equilibrium systems. 

Particle-laden filaments form on the fluid–fluid interface when a suspension drains from a vertical Hele-Shaw cell. This instability is driven by the competition between viscous stresses on highly confined particles and stabilizing capillarity. 

Motion control of droplets has generated much attention for its application to microfluidics, where precisely controlling small fluid volumes is an imperative requirement. Mechanical vibrations can induce controllable depinning and activation of a variety of drop-motion regimes. However, existing vibration-based strategies establish homogeneous rigid-body dynamics on the entire substrate, thus lacking any form of spatial heterogeneity and tuning. Addressing this limitation, elastic metamaterials provide an ideal platform to achieve spectrally and spatially selective drop-motion control. This capability results from the intrinsic ability of metamaterials to attenuate vibrations in selected frequency bands and regions of an elastic domain. In this work, we experimentally demonstrate a variety of droplet motion capabilities on the surface of a vibrating metaplate endowed with locally resonant stubs. The experiments leverage the design reconfigurability of a LEGO^®component-enabled prototyping platform, which allows us to switch in an agile manner between different configurations of resonators. We use laser vibrometry measurements with high spatial resolution to capture the spatial variability of the metaplate response. Beyond the discipline-specific boundaries, this work begins to illustrate a broader employment of elastic metamaterials in applications where their signature wave control capability is not the end goal, but rather an enabling tool for other more complex multiphysical effects. 

Free surface flows driven by boundary undulations are observed in many biological phenomena, including the feeding and locomotion of water snails. To simulate the feeding strategy of apple snails, we develop a centimetric robotic undulator that drives a thin viscous film of liquid with the wave speed$$V_w$$. Our experimental results demonstrate that the behaviour of the net fluid flux$$Q$$strongly depends on the Reynolds number$$Re$$. Specifically, in the limit of vanishing$$Re$$, we observe that$$Q$$varies non-monotonically with$$V_w$$, which has been successfully rationalised by Pandeyet al.(Nat. Commun., vol. 14, no. 1, 2023, p. 7735) with the lubrication model. By contrast, in the regime of finite inertia ($${Re} \sim O(1)$$), the fluid flux continues to increase with$$V_w$$and completely deviates from the prediction of lubrication theory. To explain the inertia-enhanced pumping rate, we build a thin-film, two-dimensional model via the asymptotic expansion in which we linearise the effects of inertia. Our model results match the experimental data with no fitting parameters and also show the connection to the corresponding free surface shapes$$h_2$$. Going beyond the experimental data, we derive analytical expressions of$$Q$$and$$h_2$$, which allow us to decouple the effects of inertia, gravity, viscosity and surface tension on free surface pumping over a wide range of parameter space. 

Abstract Examples of fluid flows driven by undulating boundaries are found in nature across many different length scales. Even though different driving mechanisms have evolved in distinct environments, they perform essentially the same function: directional transport of liquid. Nature-inspired strategies have been adopted in engineered devices to manipulate and direct flow. Here, we demonstrate how an undulating boundary generates large-scale pumping of a thin liquid near the liquid-air interface. Two dimensional traveling waves on the undulator, a canonical strategy to transport fluid at low Reynolds numbers, surprisingly lead to flow rates that depend non-monotonically on the wave speed. Through an asymptotic analysis of the thin-film equations that account for gravity and surface tension, we predict the observed optimal speed that maximizes pumping. Our findings reveal how proximity to free surfaces, which ensure lower energy dissipation, can be leveraged to achieve directional transport of liquids. 

Partially wetting droplets under an airflow can exhibit complex behaviours that arise from the coupling of surface tension, inertia of the external flow and contact-line dynamics. Recent experiments by Hooshanginejad et al. ( J. Fluid Mech. , vol. 901, 2020) revealed that a millimetric partially wetting water droplet under an impinging jet can oscillate in place, split or depin away from the jet, depending on the magnitude (i.e. $$5\unicode{x2013}20\ {\rm m}\ {\rm s}^{-1}$$ ) and position of the jet. To rationalise the experimental observations, we develop a two-dimensional lubrication model of the droplet that incorporates the external pressure of the impinging high-Reynolds-number jet, in addition to the capillary and hydrostatic pressures of the droplet. Distinct from the previous model by Hooshanginejad et al. ( J. Fluid Mech. , vol. 901, 2020), we simulate the motion of the contact line using precursor film and disjoining pressure, which allows us to capture a wider range of droplet behaviours, including the droplet dislodging to one side. Our simulations exhibit a comparable time-scale of droplet deformations and similar outcomes as the experimental observations. We also obtain the analytical steady-state solutions of the droplet shapes and construct the minimum criteria for splitting and depinning.