Superhydrophobic (SHPo) surfaces can capture a thin layer of air called a plastron under water to reduce skin friction. Although a ~30 % drag reduction has been recently reported with longitudinal micro-trench SHPo surfaces under a boat and in a towing tank, the results lacked the consistency to establish a clear trend. Designed based on Yu et al. (J. Fluid Mech, vol. 962, 2023, A9), this work develops and tests a series of high-performance SHPo surface coupons that can sustain a pinned plastron underneath a passenger motorboat revamped to reach 14 knots. Importantly, plastrons in a pinned state, not just their existence, are confirmed during flow experiments for the first time. All the drag-reduction data measured on different longitudinal micro-trenches are found to collapse if plotted against slip length in wall units. In comparison, aligned posts and transverse trenches show less and little drag reduction, respectively, confirming the adverse effect of the spanwise slip in turbulent flows. This report not only verifies SHPo surfaces can provide a consistent drag reduction at high speeds in open sea but also shows that one may predict the amount of drag reduction in turbulent flows using the physical slip length obtained for Stokes flows.

This paper studies the sustainability of plastrons on superhydrophobic (SHPo) surfaces made of longitudinal micro-trenches covered by nano-grass with the main interest on hydrodynamic friction drag reduction in high-speed flows of open water, which represent the operating conditions of common watercraft. After revising the shear-driven drainage model to address the air diffusion for SHPo surfaces, the existing theories are combined to reveal the trends of how the immersion depth, air saturation level and shear stress affect the maximum attainable plastron length. Deviations from the theories by the dynamic effect at the two ends of the trench, the interfacial contaminations and turbulent fluctuation are also discussed. A combinatorial series of well-defined SHPo trench surfaces (4 cm × 7 cm in size with varying trench widths, depths, lengths and roughnesses) is microfabricated and attached underneath a 4 m long motorboat on seawater in turbulent flows up to 7.2 m s−1 (shear rate ∼83 000 s−1 and friction Reynolds number ∼5500). Because the plastron can provide a substantial slip only while its air–water interfaces are pinned (or only slightly depinned) at the trench top, two underwater cameras are employed to differentiate the pinned (and slightly depinned) interfaces from the depinned (and no) interfaces. In addition to achieving pinned plastrons on 6 cm long trenches aligned to high-speed flows in open water, the experimental results corroborate the theoretical estimations, supporting the design of SHPo surfaces for field applications.

We derive equations of motion that describe the dynamics of a fluid confined within an elastic nanotube subject to periodic bending deflections. We use the principle of least action applied to a continuous open system at constant temperature. We solve the equations analytically in two limiting situations: when the tube oscillations are so small that they do not affect the fluid motion, but this one affects the tube dynamics; and when the flow magnitude is so small that it has no influence on the tube dynamics, but this one affects fluid motion. In the first case, we find out that the characteristic bending frequency spectrum of the tube depends not only on the magnitude of flow velocity, as previously stated in the literature, but also on the fluid velocity profile. This could constitute the basis of a strategy for indirect determination of the slip length in carbon nanotubes conveying flow via measurement of the buckling speed. In the second case, we find that tube vibrations can modify the dynamics of the fluid. Particularly, for a fluid subject to a constant pressure gradient, the tube motion induces an oscillatory motion in the fluid with twice the frequency of the tube. Moreover, the amplitude of the oscillatory fluid motion persists at high frequencies. This could constitute a strategy to generate high-frequency flows at nanoscales. Our results open up a panorama to control flow across nanotubes via tube vibrations, which could be complementary to chemical functionalization of nanostructures.

As far as plastron is sustained, superhydrophobic (SHPo) surfaces are expected to reduce skin-friction drag in any flow conditions including large-scale turbulent boundary-layer flows of marine vessels. However, despite many successful drag reductions reported using laboratory facilities, the plastron on SHPo surfaces was persistently lost in high-Reynolds-number flows on open water, and no reduction has been reported until a recent study using certain microtrench SHPo surfaces underneath a boat (Xu et al., Phys. Rev. Appl., vol. 13, no. 3, 2020, 034056). Since scientific studies with controlled flows are difficult with a boat on ocean water, in this paper we test similar SHPo surfaces in a high-speed towing tank, which provides well-controlled open-water flows, by developing a novel $0.7\ \textrm {m} \times 1.4\ \textrm {m}$ towing plate, which subjects a $4\ \textrm {cm} \times 7\ \textrm {cm}$ sample to the high-Reynolds-number flows of the plate. In addition to the 7 cm long microtrenches, trenches divided into two in length are also tested and reveal an improvement. The skin-friction drag ratio relative to a smooth surface is found to be decreasing with increasing Reynolds number, down to 73 % (i.e. 27 % drag reduction) at $Re_x\sim 8\times 10^6$, before starting to increase at higher speeds. For a given gas fraction, the trench width non-dimensionalized to the viscous length scale is found to govern the drag reduction, in agreement with previous numerical results.

The interaction of a thin viscous film with an elastic sheet results in coupling of pressure and deformation, which can be utilized as an actuation mechanism for surface deformations in a wide range of applications, including microfluidics, optics and soft robotics. Implementation of such configurations inherently takes place over finite domains and often requires some pre-stretching of the sheet. Under the assumptions of strong pre-stretching and small deformations of the lubricated elastic sheet, we use the linearized Reynolds and Föppl–von Kármán equations to derive closed-form analytical solutions describing the deformation in a finite domain due to external forces, accounting for both bending and tension effects. We provide a closed-form solution for the case of a square-shaped actuation region and present the effect of pre-stretching on the dynamics of the deformation. We further present the dependence of the deformation magnitude and time scale on the spatial wavenumber, as well as the transition between stretching- and bending-dominant regimes. We also demonstrate the effect of spatial discretization of the forcing (representing practical actuation elements) on the achievable resolution of the deformation. Extending the problem to an axisymmetric domain, we investigate the effects arising from nonlinearity of the Reynolds and Föppl–von Kármán equations and present the deformation behaviour as it becomes comparable to the initial film thickness and dependent on the induced tension. These results set the theoretical foundation for implementation of microfluidic soft actuators based on elastohydrodynanmics.

Sprays are a class of multiphase flows which exhibit a wide range of drop size and velocity scales spanning several orders of magnitude. The objective of the current work is to experimentally investigate the prospect of dynamical similarity in these flows. We are also motivated to identify a choice of length and time scales which could lead towards a universal description of the drop size and velocity spectra. Towards this end, we have fabricated a cohort of geometrically similar pressure swirl atomizers using micro-electromechanical systems (MEMS) as well as additive manufacturing technology. We have characterized the dynamical characteristics of the sprays as well as the drop size and velocity spectra (in terms of probability density functions, p.d.f.s) over a wide range of Reynolds ($Re$) and Weber numbers ($We$) using high-speed imaging and phase Doppler interferometry, respectively. We show that the dimensionless Sauter mean diameter ($D_{32}$) scaled to the boundary layer thickness in the liquid sheet at the nozzle exit ($\unicode[STIX]{x1D6FF}_{o}$) exhibits self-similarity in the core region of the spray, but not in the outer zone. In addition, we show that global drop size spectra in the sprays show two distinct characteristics. The spectra from varying $Re$ and $We$ collapse onto a universal p.d.f. for drops of size $x$ where $x/\unicode[STIX]{x1D6FF}_{o}>1$. For $x/\unicode[STIX]{x1D6FF}_{o}<1$, a residual effect of $Re$ and $We$ persists in the size spectra. We explain this characteristic by the fact that the physical mechanisms that cause large drops is different from that which is responsible for the small drops. Similarly, with the liquid sheet velocity at the nozzle exit ($u_{s}$) as the choice of velocity scale, we show that drops moving with a velocity $u$ such that $u/u_{s}<1$ collapse onto a universal p.d.f., while drops with $u/u_{s}>1$ exhibit a residual effect of $Re$ and $We$. From these observations, we suggest that physically accurate models for drop size and velocity spectra should rely on piecewise descriptions of the p.d.f. rather than invoking a single mathematical form for the entire distribution. Finally, we show from a dynamical modal analysis that the conical liquid sheet flapping characteristics exhibit a sharp transition in Strouhal number ($St$) at a critical $Re$.

Recent experiments have demonstrated that small-scale rotary devices installed in a microfluidic channel can be driven passively by the underlying flow alone without resorting to conventionally applied magnetic or electric fields. In this work, we conduct a theoretical and numerical study on such a flow-driven ‘watermill’ at low Reynolds number, focusing on its hydrodynamic features. We model the watermill by a collection of equally spaced rigid rods. Based on the classical resistive force (RF) theory and direct numerical simulations, we compute the watermill’s instantaneous rotational velocity as a function of its rod number $N$, position and orientation. When $N\geqslant 4$, the RF theory predicts that the watermill’s rotational velocity is independent of $N$ and its orientation, implying the full rotational symmetry (of infinite order), even though the geometrical configuration exhibits a lower-fold rotational symmetry; the numerical solutions including hydrodynamic interactions show a weak dependence on $N$ and the orientation. In addition, we adopt a dynamical system approach to identify the equilibrium positions of the watermill and analyse their stability. We further compare the theoretically and numerically derived rotational velocities, which agree with each other in general, while considerable discrepancy arises in certain configurations owing to the hydrodynamic interactions neglected by the RF theory. We confirm this conclusion by employing the RF-based asymptotic framework incorporating hydrodynamic interactions for a simpler watermill consisting of two or three rods and we show that accounting for hydrodynamic interactions can significantly enhance the accuracy of the theoretical predictions.

A size-dependent dynamical model is suggested to investigate the vibrational characteristics of an Euler-Bernoulli micro/nanoscale beam. The strain gradient type of nonlocal elasticity is employed and a small intrinsic length scale parameter is considered in the theoretical model. A partial differential equation that governs transverse motion is derived and the corresponding ordinary differential equation and its dispersion relation are determined from the governing equation by the method of separation of variables. The problem is solved for several sets of strain gradient nonlocal boundary conditions by use of the eigenvalue method. These examples show that strain gradient nonlocality affects the natural frequencies of micro/nanoscale beams significantly. The initial axial force is also proved to play an important role in the vibrational behaviors of a micro/nanoscale beam. The critical compression and critical strain are derived and they are compared with some other approaches. The material constant in nonlocal elasticity theory can be determined by comparing the nonlocal theoretical results with molecular dynamic simulation and it is consistent with the estimation in previous work. Some further comments on the mechanisms of size dependence and physical meaning of micro/nanoscale parameter are presented. Comparisons of natural frequencies by nonlocal theory, classical theory, molecular dynamic simulation and surface effects are also constructed and they indicate the validity of the model developed in the present study.

Electro-osmotic pumping of fluid through a nanopore that traverses an insulating membrane is considered. The density of surface charge on the membrane is assumed to be uniform and sufficiently low for the Poisson–Boltzmann equation to be linearized. The reciprocal theorem gives the flow rate generated by an applied weak electric field, expressed as an integral over the fluid volume. For a circular hole in a membrane of zero thickness, an analytical result is possible up to quadrature. For a membrane of arbitrary thickness, the full Poisson–Nernst–Planck–Stokes system of equations is solved numerically using a finite volume method. The numerical solution agrees with the standard analytical result for electro-osmotic flux through a long cylindrical pore when the membrane thickness is large compared to the hole diameter. When the membrane thickness is small, the flow rate agrees with that calculated using the reciprocal theorem.

Despite the confirmation of slip flows and successful drag reduction (DR) in small-scaled laminar flows, the full impact of superhydrophobic (SHPo) DR remained questionable because of the sporadic and inconsistent experimental results in turbulent flows. Here we report a systematic set of bias-free reduction data obtained by measuring the skin-friction drags on a SHPo surface and a smooth surface at the same time and location in a turbulent boundary layer (TBL) flow. Each monolithic sample consists of a SHPo surface and a smooth surface suspended by flexure springs, all carved out from a $2.7 \times 2.7 {\mathrm{mm}}^{2}$ silicon chip by photolithographic microfabrication. The flow tests allow continuous monitoring of the plastron on the SHPo surfaces, so that the DR data are genuine and consistent. A family of SHPo samples with precise profiles reveals the effects of grating parameters on turbulent DR, which was measured to be as much as ${\sim }75\, \%$.

We present a fluid-mechanical model of the coalescence of a number of elastic objects due to surface tension. We consider an array of spring–block elements separated by thin liquid films, whose dynamics are modelled using lubrication theory. With this simplified model of elastocapillary coalescence, we present the results of numerical simulations for a large number of elements, $N=O(10^4)$. A linear stability analysis shows that pairwise coalescence is always the most unstable mode of deformation. However, the numerical simulations show that the cluster sizes actually produced by coalescence from a small white-noise perturbation have a distribution that depends on the relative strength of surface tension and elasticity, as measured by an elastocapillary number $K$. Both the maximum cluster size and the mean cluster size scale like $K^{-1/2}$ for small $K$. An analytical solution for the response of the system to a localized perturbation shows that such perturbations generate propagating disturbance fronts, which leave behind ‘frozen-in’ clusters of a predictable size that also depends on $K$. A good quantitative comparison between the cluster-size statistics from noisy perturbations and this ‘frozen-in’ cluster size suggests that propagating fronts may play a crucial role in the dynamics of coalescence.

We consider the elastocapillary interaction of a liquid drop placed between two elastic beams, which are both clamped at one end to a rigid substrate. This is a simple model system relevant to the problem of surface-tension-induced collapse of flexible micro-channels that has been observed in the manufacture of microelectromechanical systems (MEMS). We determine the conditions under which the beams remain separated, touch at a point, or stick along a portion of their length. Surprisingly, we show that in many circumstances multiple equilibrium states are possible. We develop a lubrication-type model for the flow of liquid out of equilibrium and thereby investigate the stability of the multiple equilibria. We demonstrate that for given material properties two stable equilibria may exist, and show via numerical solutions of the dynamic model that it is the initial state of the system that determines which stable equilibrium is ultimately reached.

Kinetic theory provides a rigorous foundation for calculating the dynamics of gas flow at arbitrary degrees of rarefaction, with solutions of the Boltzmann equation requiring numerical methods in many cases of practical interest. Importantly, the near-continuum regime can be examined analytically using asymptotic techniques. These asymptotic analyses often assume steady flow, for which analytical slip models have been derived. Recently, developments in nanoscale fabrication have stimulated research into the study of oscillatory non-equilibrium flows, drawing into question the applicability of the steady flow assumption. In this article, we present a formal asymptotic analysis of the unsteady linearized Boltzmann–BGK equation, generalizing existing theory to the oscillatory (time-varying) case. We consider the near-continuum limit where the mean free path and oscillation frequency are small. The complete set of hydrodynamic equations and associated boundary conditions are derived for arbitrary Stokes number and to second order in the Knudsen number. The first-order steady boundary conditions for the velocity and temperature are found to be unaffected by oscillatory flow. In contrast, the second-order boundary conditions are modified relative to the steady case, except for the velocity component tangential to the solid wall. Application of this general asymptotic theory is explored for the oscillatory thermal creep problem, for which unsteady effects manifest themselves at leading order.

We use LVDSMC (low-variance deviational Monte Carlo) simulations to calculate, under linearized conditions, the second-order temperature jump coefficient for a dilute gas whose temperature is governed by the Poisson equation with a constant forcing term, as in the case of homogeneous volumetric heating. Both the hard-sphere gas and the BGK model of the Boltzmann equation, for which slip/jump coefficients are not functions of temperature, are considered. The temperature jump relation and jump coefficient determined here are closely linked to the general jump relations for time-dependent problems that have yet to be systematically treated in the literature; as a result, they are different from those corresponding to the well-known linear and steady case where the temperature is governed by the homogeneous heat conduction (Laplace) equation.

Many swimming microorganisms are able to propel themselves by the organized beating motion of numerous short flagella or cilia attached to their body surface. For their small size and the inherently viscous nature of the motion, this mechanism is very effective and they can swim several body lengths per second. The quest has been to see if artificial cilia may be developed and if the strategy of cilia propulsion can be used in microfluidic devices to transport fluids in a localized and controllable manner. Babataheri et al. (J. Fluid Mech., this issue, vol. 678, 2011, pp. 5–13) explore the response of chains of small paramagnetic beads that are elastically bonded together to form artificial cilia. The chain or fleximag is tethered to the surface and driven by external magnetic fields, responding also to both fluid and elastic forces. A key observation from their experiments and model is that for a simple planar-forcing strategy there is a hidden symmetry that limits the net transport of fluid.

We study the elasto-hydrodynamic (EHD) interaction of a sphere with a flat elastic surface in the prospect of measuring the elastic moduli of soft supported thin films, with non-contact dynamic surface forces or atomic force microscopy measurements. When the sphere is oscillated at a very small amplitude close to the surface, the linear force response undergoes a dynamic transition from a viscous-dominated behaviour at large distance to an elastic-dominated behaviour at short distance. In the limit of very thin or very thick supported layers, we show that the force response obeys simple scaling laws which allow to unambiguously determine the absolute elastic modulus of the layer. In the general case, we establish the very rich phase diagram of the EHD interaction and discuss its application for optimizing experimental parameters.

Flexible superparamagnetic filaments (‘fleximags’) are very slender elastic filaments, which can be driven by distributed magnetic torques to mimic closely the behaviour of biological flagella. Previously, fleximags have been used as a basis for artificial micro-swimmers capable of transporting small cargos Dreyfus et al. (Nature, vol. 437, 2005, p. 862). Here, we demonstrate how these filaments can be anchored to a wall to make carpets of artificial micro-magnetic cilia with tunable densities. We analyse the dynamics of an artificial cilium under both planar and three-dimensional beating patterns. We show that the dynamics are controlled by a single characteristic length scale varying with the inverse square root of the driving frequency, providing a mechanism to break the fore and aft symmetry and to generate net fluxes and forces. However, we show that an effective geometrical reciprocity in the filament dynamics creates intrinsic limitations upon the ability of the artificial flagellum to pump fluid when driven in two dimensions.

We report on the direct experimental observation of laser-induced flows in isotropic liquids that scatter light. We use a droplet microemulsion in the two-phase regime, which behaves like a binary mixture. Close to its critical consolute line, the microemulsion undergoes large refractive index fluctuations that scatter light. The radiation pressure of a laser beam is focused onto the soft interface between the two phases of the microemulsion and induces a cylindrical liquid jet that continuously emits droplets. We demonstrate that this dripping phenomenon takes place as a consequence of a steady flow induced by the transfer of linear momentum from the optical field to the liquid due to light scattering. We first show that the cylindrical jet guides light as a step-index liquid optical fiber whose core diameter is self-adapted to the light itself. Then, by modelling the light-induced flow as a low-Reynolds-number, parallel flow, we predict the dependence of the dripping flow rate on the thermophysical properties of the microemulsion and the laser beam power. Satisfying agreement is found between the model and experiments.