## Publications

66 results found

Yariv E, Schnitzer O, 2023, Shape of sessile drops in the large-Bond-number ‘pancake’ limit, *Journal of Fluid Mechanics*, Vol: 961, Pages: 1-19, ISSN: 0022-1120

We revisit the classical problem of calculating the pancake-like shape of a sessile drop at large Bond numbers. Starting from a formulation where drop volume and contact angle are prescribed, we develop an asymptotic scheme which systematically produces approximations to the two key pancake parameters, height and radius. The scheme is based on asymptotic matching of a ‘flat region’ where capillarity is negligible and an ‘edge region’ near the contact line. Major simplifications follow from the distinction between algebraically and exponentially small terms, together with the use of two exact integral relations. The first represents a force balance in the vertical direction. The second, which can be interpreted as a radial force balance on the drop edge (up to exponentially small terms), generalises an approximate force balance used in classical treatments. The resulting approximations for the geometric pancake parameters, which go beyond known leading-order results, are compared with numerical calculations tailored to the pancake limit. These, in turn, are facilitated by an asymptotic approximation for the exponentially small apex curvature, which we obtain using a Wentzel–Kramers–Brillouin method. We also consider the comparable two-dimensional problem, where similar integral balances explicitly determine the pancake parameters in closed form up to an exponentially small error.

Schnitzer O, 2023, Weakly nonlinear dynamics of a chemically active particle near the threshold for spontaneous motion: Adjoint method, *Physical Review Fluids*, Vol: 8, ISSN: 2469-990X

In this Series, we study the weakly nonlinear dynamics of chemically active particles near thethreshold for spontaneous motion. In this Part, we focus on steady solutions and develop an‘adjoint method’ for deriving the nonlinear amplitude equation governing the particle’s velocity,first assuming the canonical model in the literature of an isotropic chemically active particle andthen considering general perturbations about that model. As in previous works, the amplitudeequation is obtained from a solvability condition on the inhomogeneous problem at second order ofa particle-scale weakly nonlinear expansion, the formulation of that problem involving asymptoticmatching with a leading-order solution in a remote region where advection and diffusion are balanced. We develop a generalised solvability condition based on a Fredholm Alternative argument,which entails identifying the adjoint linear operator at the threshold and calculating its kernel.This circumvents the apparent need in earlier theories to solve the second-order inhomogeneousproblem, resulting in considerable simplification and adding insight by making it possible to treata wide range of perturbation scenarios on a common basis. To illustrate our approach, we deriveand solve amplitude equations for a number of perturbation scenarios (external force and torquefields, non-uniform surface properties, first-order surface kinetics and bulk absorption), demonstrating that sufficiently near the threshold weak perturbations can appreciably modify and enrichthe landscape of steady solutions.

Peng G, Schnitzer O, 2023, Weakly nonlinear dynamics of a chemically active particle near the threshold for spontaneous motion. II. History-dependent motion, *Physical Review Fluids*, Vol: 8, ISSN: 2469-990X

We develop a reduced model for the slow unsteady dynamics of an isotropic chemically active particle near the threshold for spontaneous motion. Building on the steady theory developed in part I of this series, we match a weakly nonlinear expansion valid on the particle scale with a leading-order approximation in a larger-scale unsteady remote region, where the particle acts as a moving point source of diffusing concentration. The resulting amplitude equation for the velocity of the particle includes a term representing the interaction of the particle with its own concentration wake in the remote region, which can be expressed as a time integral over the history of the particle motion, allowing efficient simulation and theoretical analysis of fully three-dimensional unsteady problems. To illustrate how to use the model, we study the effects of a weak force acting on the particle, including the stability of the steady states and how the velocity vector realigns toward the stable one, neither of which previous axisymmetric and steady models were able to capture. This unsteady formulation could also be applied to most of the other perturbation scenarios studied in part I as well as the dynamics of interacting active particles.

Kurzthaler C, Brandão R, Schnitzer O,
et al., 2023, Shape of a tethered filament in various low-Reynolds-number flows, *Physical Review Fluids*, Vol: 8, ISSN: 2469-990X

We consider the steady-state deformation of an elastic filament in various unidirectional, low-Reynolds-number flows, with the filament either clamped at one end, perpendicular to the flow, or tethered at its center and deforming symmetrically about a plane parallel to the flow. We employ a slender-body model [Pozrikidis, J. Fluids Struct. 26, 393 (2010)] to describe the filament shape as a function of the background flow and a nondimensional compliance η characterizing the ratio of viscous to elastic forces. For η≪1, we describe the small deformation of the filament by means of a regular perturbation expansion. For η≫1, the filament strongly bends such that it is nearly parallel to the flow except close to the tether point; we analyze this singular limit using boundary-layer theory, finding that the radius of curvature near the tether point, as well as the distance of the parallel segment from the tether point, scale like η−1/2 for flow profiles that do not vanish at the tether point, and like η−1/3 for flow profiles that vanish linearly away from the tether point. We also use a Wentzel-Kramers-Brillouin approach to derive a leading-order approximation for the exponentially small slope of the filament away from the tether point. We compare numerical solutions of the model over a wide range of η values with closed-form predictions obtained in both asymptotic limits, focusing on particular uniform, shear and parabolic flow profiles relevant to experiments.

Brandao R, Schnitzer O, 2022, Leidenfrost levitation of a spherical particle above a liquid bath: evolution of the vapour-film morphology with particle size, *European Journal of Applied Mathematics*, Vol: 33, Pages: 1117-1169, ISSN: 0956-7925

We consider a spherical particle levitating above a liquid bath owing to the Leidenfrost effect, where the vapour of either the bath or sphere forms an insulating film whose pressure supports the sphere’s weight. Starting from a reduced formulation based on a lubrication-type approximation, we use matched asymptotics to describe the morphology of the vapour film assuming that the sphere is small relative to the capillary length (small Bond number) and that the densities of the bath and sphere are comparable. We find that this regime is comprised of two formally infinite sequences of distinguished limits which meet at an accumulation point, the limits being defined by the smallness of an intrinsic evaporation number relative to the Bond number. These sequences of limits reveal a sur16 prisingly intricate evolution of the film morphology with increasing sphere size. Initially, the vapour film transitions from a featureless morphology, where the thickness profile is parabolic, to a neck-bubble morphology, which consists of a uniform-pressure bubble bounded by a narrow and much thinner annular neck. Gravity effects then become im20 portant in the bubble leading to sequential formation of increasingly smaller neck-bubble pairs near the symmetry axis. This process terminates when the pairs closest to the sym metry axis become indistinguishable and merge. Subsequently, the inner section of that merger transitions into a uniform-thickness film that expands radially, gradually squish24 ing larger and larger neck-bubble pairs into a region of localised oscillations sandwiched between the uniform film and what remains of the bubble whose radial extent is presently comparable to the uniform film; the neck-bubble pairs farther from the axis remain es sentially intact. Ultimately, the uniform film gobbles up the largest outermost bubble, whereby the morphology simplifies to a uniform film bounded by localised oscillations. Overall, the asymptotic analysis describes the cont

Schnitzer O, Porter R, 2022, Acoustics of a partially partitioned narrow slit connected to a half-plane: case study for exponential quasi-bound states in the continuum and their resonant excitation, *SIAM Journal on Applied Mathematics*, Vol: 82, ISSN: 0036-1399

Localised wave oscillations in an open system that do not decay or grow in time, despite their frequency lying within a continuous spectrum of radiation modes carrying energy to or from infinity, are known as bound states in the continuum (BIC). Small perturbations from the typically delicate conditions for BIC almost always result in the waves weakly coupling with the radiation modes, leading to leaky states called quasi-BIC that have a large quality factor. We study the asymptotic nature of this weak coupling in the case of acoustic waves interacting with a rigid substrate featuring a partially partitioned slit — a setup that supports quasi-BIC that exponentiallyapproach BIC as the slit is made increasingly narrow. In that limit, we use the method of matched asymptotic expansions in conjunction with reciprocal relations to study those quasi-BIC and their resonant excitation. In particular, we derive a leading approximation for the exponentially small imaginary part of each wavenumber eigenvalue (inversely proportional to quality factor), which is beyond all orders of the expansion for the wavenumber eigenvalue itself. Furthermore, we derive a leading approximation for the exponentially large amplitudes of the states in the case where they are resonantly excited by a plane wave at oblique incidence. These resonances occur in exponentially narrow wavenumber intervals and are physically manifested in cylindrical-dipolar waves emanating from the slit aperture and exponentially large field enhancements inside the slit. The asymptotic approximations are validated against numerical calculations.

Schnitzer O, Brandao R, 2022, Absorption characteristics of large acoustic metasurfaces, *Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences*, Vol: 380, ISSN: 1364-503X

Metasurfaces formed of arrays of subwavelength resonators are often tuned to ‘critically couple’ with incident radiation, so that at resonance dissipativeand radiative damping are balanced and absorption is maximised. Such design criteria are typically derived assuming an infinite metasurface, whereas the absorption characteristics of finite metasurfaces, even very large ones, can be markedly different in certain frequency intervals. This is due to the excitation of surface waves, intrinsic to resonant metasurfaces, and especially meta-resonances, namely collective resonances where the surface waves form standing-wave patterns over the planar metasurfacedomain. We illustrate this issue using a detailed model of a Helmholtz-type acoustic metasurface formed of cavity-neck pairs embedded into a rigid substrate, with geometric and dissipation effects included from first principles (R. Brandão and O. Schnitzer, Wave Motion, 97 102583, 2020).

Ruiz M, Schnitzer O, 2022, Plasmonic resonances of slender nanometallic rings, *Physical Review B*, Vol: 105, ISSN: 2469-9950

We develop an approximate quasistatic theory describing the low-frequency plasmonic resonances of slender nanometallic rings and configurations thereof. First, we use asymptotic arguments to reduce the plasmonic eigenvalue problem governing the geometric (material- and frequency-independent) modes of a given ring structure to a one-dimensional periodic integrodifferential problem in which the eigenfunctions are represented by azimuthal voltage and polarization-charge profiles associated with each ring. Second, we obtain closed-form solutions to the reduced eigenvalue problem for azimuthally invariant rings (including torus-shaped rings but also allowing for noncircular cross-sectional shapes), as well as coaxial dimers and chains of such rings. For more general geometries, involving azimuthally nonuniform rings and noncoaxial structures, we solve the reduced eigenvalue problem using a semianalytical scheme based on Fourier expansions of the reduced eigenfunctions. Third, we used the asymptotically approximated modes, in conjunction with the quasistatic spectral theory of plasmonic resonance, to study and interpret the frequency response of a wide range of nanometallic slender-ring structures under plane-wave illumination.

Saha S, Yariv E, Schnitzer O, 2021, Isotropically active colloids under uniform force fields: from forced to spontaneous motion, *Journal of Fluid Mechanics*, Vol: 916, Pages: 1-19, ISSN: 0022-1120

We consider the inertia-free motion of an isotropic chemically active particle which is exposed to a weak uniform force field. This problem is characterised by two velocity scales, a ‘chemical’ scale associated with diffusio-osmosis and a ‘mechanical’ scale associated with the external force. The motion animated by the force deforms the originally spherically symmetric solute cloud surrounding the particle, thus resulting in a concomitant diffusio-osmotic flow which, in turn, modifies the particle speed. A weak-force linearisation furnishes a closed-form expression for the particle velocity as a function of the intrinsic Péclet number α associated with the chemical velocity scale. We find that the predicted velocity may become singular at α=4, and that this happens under the same conditions on the surface parameters for which the associated unforced problem is known to exhibit, for α>4, a symmetry-breaking instability giving rise to steady spontaneous motion (Michelin, Lauga & Bartolo, Phys. Fluids, vol. 25, 2013, 061701). Here, a local analysis in a distinguished region near α=4, wherein the velocity scaling is amplified, yields a closed-form description of the imperfect bifurcation which bridges between a perturbed stationary state and a perturbed spontaneous motion. Remarkably, while the direction of spontaneous motion in the absence of an external force is random, in the perturbed case that motion is rendered steady solely in the directions parallel or antiparallel to the external force.

Schnitzer O, Davis AMJ, Yariv E, 2020, Rolling of non wetting droplets down a gently inclined plane, *Journal of Fluid Mechanics*, Vol: 903, Pages: A25-1-A25-29, ISSN: 0022-1120

We analyse the near-rolling motion of nonwetting droplets down a gently inclined plane.Inspired by the scaling analysis of Mahadevan & Pomeau (Phys. Fluids, vol. 11, 1999,pp. 2449), we focus upon the limit of small Bond numbers, where the drop shape is nearlyspherical and the internal flow is approximately a rigid-body rotation except close to theflat spot at the base of the drop. In that region, where the fluid interface appears flat,we obtain an analytical approximation for the flow field. By evaluating the dissipationassociated with that flow we obtain a closed-form approximation for the drop speed.This approximation reveals that the missing prefactor in the Mahadevan–Pomeau scalinglaw is (3π/16)p3/2 ≈ 0.72 — in good agreement with experiments. An unexpectedfeature of the flow field is that it happens to satisfy the no-slip and shear-free conditionssimultaneously over both the solid flat spot and the mobile fluid interface in its vicinity.Furthermore, we show that close to the near-circular contact line the velocity field liesprimarily in the plane locally normal to the contact line; it is analogous there to thelocal solution in the comparable problem of a two-dimensional rolling drop. This analogybreaks down near the two points where the contact line propagates parallel to itself,the local flow being there genuinely three dimensional. These observations illuminate aunique ‘peeling’ mechanism by which a rolling droplet avoids the familiar non-integrablestress singularity at a moving contact line.

Brandão R, Holley J, Schnitzer O, 2020, Boundary-layer effects on electromagnetic and acoustic extraordinary transmission through narrow slits, *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, Vol: 476, ISSN: 1364-5021

We study the problem of resonant extraordinary transmission of electromagnetic and acoustic waves through subwavelength slits in an infinite plate, whose thickness is close to a half-multiple of the wavelength. We build on the matched-asymptotics analysis of Holley & Schnitzer (2019 Wave Motion91, 102381 (doi:10.1016/j.wavemoti.2019.102381)), who considered a single-slit system assuming an idealized formulation where dissipation is neglected and the electromagnetic and acoustic problems are analogous. We here extend that theory to include thin dissipative boundary layers associated with finite conductivity of the plate in the electromagnetic problem and viscous and thermal effects in the acoustic problem, considering both single-slit and slit-array configurations. By considering a distinguished boundary-layer scaling where dissipative and diffractive effects are comparable, we develop accurate analytical approximations that are generally valid near resonance; the electromagnetic–acoustic analogy is preserved up to a single parameter that is provided explicitly for both scenarios. The theory is shown to be in excellent agreement with GHz-microwave and kHz-acoustic experiments in the literature.

Brandão R, Schnitzer O, 2020, Spontaneous dynamics of two-dimensional Leidenfrost wheels, *Physical Review Fluids*, Vol: 5, Pages: 091601-1-091601-10, ISSN: 2469-990X

Recent experiments have shown that liquid Leidenfrost drops levitated by their vapor above a flat hot surface can exhibit symmetry-breaking spontaneous dynamics (A. Bouillantet al.,Nature Physics,141188–1192, 2018). Motivated by these experiments, we theoretically investigate the translational and rotational dynamics of Leidenfrost drops on the basis of a simplified two-dimensional model, focusing on near-circular drops small in comparison to the capillary length.The model couples the equations of motion of the drop, which flows as a rigid wheel, and a thin-film model governing the vapor flow, the profile of the deformable vapor-liquid interface and thus the hydrodynamic forces and torques on the drop. In contrast to previous analytical models of Leidenfrost drops, which predict only symmetric solutions, we find that the symmetric Leidenfrost state is unstable above a critical drop radius: R1 for a free drop and R2> R1 for an immobilized drop. In these respective cases, symmetry breaking is manifested in supercritical-pitchfork bifurcations into steady states of pure rolling and constant angular velocity. In further qualitative agreement with the experiments, when a symmetry-broken immobilized drop is suddenly released it initially moves at constant accelerationαg, whereαis an angle characterizing the slope of the liquid-vapor profile and g is the gravitational acceleration; moreover,αexhibits a maximum with respect to the drop radius, at a radius increasing with the temperature difference between the surface and the drop.

Brandao R, Schnitzer O, 2020, Asymptotic modeling of Helmholtz resonators including thermoviscous effects, *Wave Motion*, Vol: 97, Pages: 1-25, ISSN: 0165-2125

We systematically employ the method of matched asymptotic expansions to model Helmholtz resonators, with thermoviscous effects incorporated starting from first principles and with the lumped parameters characterizing the neck and cavity geometries precisely defined and provided explicitly for a wide range of geometries. With an eye towards modeling acoustic metasurfaces, we consider resonators embedded in a rigid surface, each resonator consisting of an arbitrarily shaped cavity connected to the external half-space by a small cylindrical neck. The bulk of the analysis is devoted to the problem where a single resonator is subjected to a normally incident plane wave; the model is then extended using “Foldy’s method” to the case of multiple resonators subjected to an arbitrary incident field. As an illustration, we derive critical-coupling conditions for optimal and perfect absorption by a single resonator and a model metasurface, respectively.

Brandao R, Schnitzer O, 2020, Acoustic impedance of a cylindrical orifice, *Journal of Fluid Mechanics*, Vol: 892, ISSN: 0022-1120

We use matched asymptotics to derive analytical formulae for the acoustic impedance of a subwavelength orifice consisting of a cylindrical perforation in a rigid plate. In the inviscid case, an end correction to the length of the orifice due to Rayleigh is shown to constitute an exponentially accurate approximation in the limit where the aspect ratio of the orifice is large; in the opposite limit, we derive an algebraically accurate correction, depending upon the logarithm of the aspect ratio, to the impedance of a circular aperture in a zero-thickness screen. Viscous effects are considered in the limit of thin Stokes boundary layers, where a boundary-layer analysis in conjunction with a reciprocity argument provides the perturbation to the impedance as a quadrature of the basic inviscid flow. We show that for large aspect ratios the latter perturbation can be captured with exponential accuracy by introducing a second end correction whose value is calculated to be in between two guesses commonly used in the literature; we also derive an algebraically accurate approximation in the small-aspect-ratio limit. The viscous theory reveals that the resistance exhibits a minimum as a function of aspect ratio, with the orifice radius held fixed. It is evident that the resistance grows in the long-aspect-ratio limit; in the opposite limit, resistance is amplified owing to the large velocities close to the sharp edge of the orifice. The latter amplification arrests only when the plate is as thin as the Stokes boundary layer. The analytical approximations derived in this paper could be used to improve circuit modelling of resonating acoustic devices.

Schnitzer O, 2020, Asymptotic approximations for the plasmon resonances of nearly touching spheres, *European Journal of Applied Mathematics*, Vol: 31, Pages: 246-276, ISSN: 0956-7925

Excitation of surface-plasmon resonances of closely spaced nanometallic structures is a key technique used in nanoplasmonics to control light on subwavelength scales and generate highly confined electric-field hotspots. In this paper, we develop asymptotic approximations in the near-contact limit for the entire set of surface-plasmon modes associated with the prototypical sphere dimer geometry. Starting from the quasi-static plasmonic eigenvalue problem, we employ the method of matched asymptotic expansions between a gap region, where the boundaries are approximately paraboloidal, pole regions within the spheres and close to the gap, and a particle-scale region where the spheres appear to touch at leading order. For those modes that are strongly localised to the gap, relating the gap and pole regions gives a set of effective eigenvalue problems formulated over a half space representing one of the poles. We solve these problems using integral transforms, finding asymptotic approximations, singular in the dimensionless gap width, for the eigenvalues and eigenfunctions. In the special case of modes that are both axisymmetric and odd about the plane bisecting the gap, where matching with the outer region introduces a logarithmic dependence upon the dimensionless gap width, our analysis follows Schnitzer [Singular perturbations approach to localized surface-plasmon resonance: nearly touching metal nanospheres. Phys. Rev. B92(23), 235428 (2015)]. We also analyse the so-called anomalous family of even modes, characterised by field distributions excluded from the gap. We demonstrate excellent agreement between our asymptotic formulae and exact calculations.

Holley J, Schnitzer O, 2019, Extraordinary transmission through a narrow slit, *Wave Motion*, Vol: 91, Pages: 1-9, ISSN: 0165-2125

We revisit the problem of extraordinary transmission of acoustic (electromagnetic) waves through a slit in a rigid (perfectly conducting) wall. We use matched asymptotic expansions to study the pertinent limit where the slit width is small compared to the wall thickness, the latter being commensurate with the wavelength. Our analysis focuses on near-resonance frequencies, furnishing elementary formulae for the field enhancement, transmission efficiency, and deviations of the resonances from the Fabry–Pérot frequencies of the slit. We find that the apertures’ near fields play a dominant role, in contrast with the prevalent approximate theory of Takakura (2001). Our theory agrees remarkably well with numerical solutions and electromagnetic experiments (Suckling et al., 2004), thus providing a paradigm for analysing a wide range of wave propagation problems involving small holes and slits.

Ruiz M, Schnitzer O, 2019, Slender-body theory for plasmonic resonance, *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, Vol: 475, ISSN: 1364-5021

We develop a slender-body theory for plasmonic resonance of slender metallic nanoparticles, focusing on a general class of axisymmetric geometries with locally paraboloidal tips. We adopt a modal approach where one first solves the plasmonic eigenvalue problem, a geometric spectral problem which governs the surface-plasmon modes of the particle; then, the latter modes are used, in conjunction with spectral-decomposition, to analyse localized-surface-plasmon resonance in the quasi-static limit. We show that the permittivity eigenvalues of the axisymmetric modes are strongly singular in the slenderness parameter, implying widely tunable, high-quality-factor, resonances in the near-infrared regime. For that family of modes, we use matched asymptotics to derive an effective eigenvalue problem, a singular non-local Sturm–Liouville problem, where the lumped one-dimensional eigenfunctions represent axial voltage profiles (or charge line densities). We solve the effective eigenvalue problem in closed form for a prolate spheroid and numerically, by expanding the eigenfunctions in Legendre polynomials, for arbitrarily shaped particles. We apply the theory to plane-wave illumination in order to elucidate the excitation of multiple resonances in the case of non-spheroidal particles.

Yariv E, Schnitzer O, 2019, Speed of rolling droplets, *Physical Review Fluids*, Vol: 4, ISSN: 2469-990X

We analyze the near-rolling motion of two-dimensional nonwetting drops down a gently inclined plane. Inspired by the scaling analysis of Mahadevan & Pomeau [Phys. Fluids 11, 2449 (1999)], we focus upon the limit of small Bond numbers, B≪1, where the drop shape is nearly circular and the internal flow is approximately a rigid-body rotation except close to the flat spot at the base of the drop. Our analysis reveals that the leading-order dissipation is contributed by both the flow in the flat-spot region and the correction to rigid-body rotation in the remaining liquid domain. The resulting leading-order approximation for the drop velocity U is given by We analyze the near-rolling motion of two-dimensional nonwetting drops down a gently inclined plane. Inspired by the scaling analysis of Mahadevan & Pomeau [Phys. Fluids 11, 2449 (1999)], we focus upon the limit of small Bond numbers, B≪1, where the drop shape is nearly circular and the internal flow is approximately a rigid-body rotation except close to the flat spot at the base of the drop. Our analysis reveals that the leading-order dissipation is contributed by both the flow in the flat-spot region and the correction to rigid-body rotation in the remaining liquid domain. The resulting leading-order approximation for the drop velocity U is given by We analyze the near-rolling motion of two-dimensional nonwetting drops down a gently inclined plane. Inspired by the scaling analysis of Mahadevan & Pomeau [Phys. Fluids 11, 2449 (1999)], we focus upon the limit of small Bond numbers, B≪1, where the drop shape is nearly circular and the internal flow is approximately a rigid-body rotation except close to the flat spot at the base of the drop. Our analysis reveals that the leading-order dissipation is contributed by both the flow in the flat-spot region and the correction to rigid-body rotation in the remaining liquid domain. The resulting leading-order approximation for the drop velocity U is given by μU&ga

Schnitzer O, 2019, Geometric quantization of localized surface plasmons, *IMA Journal of Applied Mathematics*, Vol: 84, Pages: 813-832, ISSN: 0272-4960

We consider the quasi-static problem governing the localized surface plasmon modes and permittivityeigenvaluesεof smooth, arbitrarily shaped, axisymmetric inclusions. We develop an asymptotic theoryfor the dense part of the spectrum, i.e., close to the accumulation valueε=−1 at which a flat interfacesupports surface plasmons; in this regime, the field oscillates rapidly along the surface and decays expo-nentially away from it on a comparable scale. Withτ=−(ε+1)as the small parameter, we developa surface-ray description of the eigenfunctions in a narrow boundary layer about the interface; the fastphase variation, as well as the slowly varying amplitude and geometric phase, along the rays are deter-mined as functions of the local geometry. We focus on modes varying at most moderately in the azimuthaldirection, in which case the surface rays are meridian arcs that focus at the two poles. Asymptoticallymatching the diverging ray solutions with expansions valid in inner regions in the vicinities of the polesyields the quantization rule1τ∼πnΘ+12(πΘ−1)+o(1),wheren 1 is an integer andΘa geometric parameter given by the product of the inclusion length andthe reciprocal average of its cross-sectional radius along its symmetry axis. For a sphere,Θ=π, wherebythe formula returns the exact eigenvaluesε=−1−1/n. We also demonstrate good agreement with exactsolutions in the case of prolate spheroids

Schnitzer O, Yariv E, 2019, Stokes resistance of a solid cylinder near a superhydrophobic surface. Part 1. Grooves perpendicular to cylinder axis, *Journal of Fluid Mechanics*, Vol: 868, Pages: 212-243, ISSN: 0022-1120

An important class of canonical problems which is employed in quantifying the slip-periness of microstructured superhydrophobic surfaces is concerned with the calculationof the hydrodynamic loads on adjacent solid bodies whose size is large relative to themicrostructure period. The effect of superhydophobicity is most pronounced when thelatter period is comparable with the separation between the solid probe and the su-perhydrophobic surface. We address the above distinguished limit, considering a simpleconfiguration where the superhydrophobic surface is formed by a periodically groovedarray, in which air bubbles are trapped in a Cassie state, and the solid body is an in-finite cylinder. In the present Part, we consider the case where the grooves are alignedperpendicular to the cylinder and allow for three modes of rigid-body motion: rectilinearmotion perpendicular to the surface; rectilinear motion parallel to the surface, in thegrooves direction; and angular rotation about the cylinder axis. In this scenario, the flowis periodic in the direction parallel to the axis. Averaging over the small-scale periodicityyields a modified lubrication description where the small-scale details are encapsulatedin two auxiliary two-dimensional cell problems which respectively describe pressure- andboundary-driven longitudinal flow through an asymmetric rectangular domain, boundedby a compound surface from the bottom and a no-slip surface from the top. Once theintegral flux and averaged shear stress associated with each of these cell problems arecalculated as a function of the slowly varying cell geometry, the hydrodynamic loadsexperienced by the cylinder are provided as quadratures of nonlinear functions of thelatter distributions over a continuous sequence of cells.

Schnitzer O, Brandao R, Yariv E, 2019, Acoustics of bubbles trapped in micro-grooves: from isolated subwavelength resonators to superhydrophobic metasurfaces, *Physical review B: Condensed matter and materials physics*, Vol: 99, ISSN: 1098-0121

We study the acoustic response of flat-meniscus bubbles trapped in the grooves of a microstructured hydrophobic substrate immersed in water. In the first part of the paper, we consider a single bubble subjected to a normally incident plane wave. We use the method of matched asymptotic expansions, based on the smallness of the gas-to-liquid density ratio, to describe the near field of the groove, where the compressibility of the liquid can be neglected, and an acoustic region, on the scale of the wavelength, which is much larger than the groove opening in the resonance regime of interest. We find that bubbles trapped in grooves support multiple subwavelength resonances, which are damped—radiatively—even in the absence of dissipation. Beyond the fundamental resonance, at which the pinned meniscus is approximately parabolic, we find a sequence of higher-order antiresonance and resonance pairs; at the antiresonances (whose frequencies are independent of the gas properties and groove size), the gas is idle and the scattering vanishes, while the liquid pressure is in balance with capillary forces.In the second part of the paper, we develop a multiple-scattering theory for dilute arrays of trapped bubbles, where the frequency response of a single bubble enters via a scattering coefficient. For an infinite array and subwavelength spacing between the bubbles, the resonances are suppressed by an interference effect associated with the strong logarithmic interactions between quasistatic line sources; the antiresonances are robust. In contrast, for finite arrays, however large, we find strong and highly oscillatory deviations from the frequency response of an infinite array in a sequence of intervals about the resonance frequencies of a single bubble; these deviations are shown to be associated with edge excitation, in the finite case, of surface “spoof plasmon” waves, which exist in the infinite case precisely in the said frequency intervals; the resona

Vanel AL, Craster RV, Schnitzer O, 2019, Asymptotic modelling of phononic box crystals, *SIAM Journal on Applied Mathematics*, Vol: 79, Pages: 506-524, ISSN: 0036-1399

We introduce phononic box crystals, namely arrays of adjoined perforated boxes, as a three-dimensional prototype for an unusual class of subwavelength metamaterials based on directly coupling resonating elements. In this case, when the holes coupling the boxes are small, we create networks of Helmholtz resonators with nearest-neighbour interactions. We use matched asymptotic expansions, in the small hole limit, to derive simple, yet asymptotically accurate, discrete wave equations governing the pressure field. These network equations readily furnish analytical dispersion relations for box arrays, slabs and crystals, that agree favourably with finite-element simulations of the physical problem. Our results reveal that the entire acoustic branch is uniformly squeezed into a subwavelength regime; consequently, phononic box crystals exhibit nonlinear-dispersion effects (such as dynamic anisotropy) in a relatively wide band, as well as a high effective refractive index in the long-wavelength limit. We also study the sound field produced by sources placed within one of the boxes by comparing and contrasting monopole- with dipole-type forcing; for the former the pressure field is asymptotically enhanced whilst for the latter there is no asymptotic enhancement and the translation from the microscale to the discrete description entails evaluating singular limits, using a regularized and efficient scheme, of the Neumann's Green's function for a cube. We conclude with an example of using our asymptotic framework to calculate localized modes trapped within a defected box array.

Yariv E, Schnitzer O, 2018, Pressure-driven plug flows between superhydrophobic surfaces of closely spaced circular bubbles, *Journal of Engineering Mathematics*, Vol: 111, Pages: 15-22, ISSN: 0022-0833

Shear-driven flows over superhydrophobic surfaces formed of closely spaced circular bubbles are characterized by giant longitudinal slip lengths, viz., large compared with the periodicity (Schnitzer, Phys Rev Fluids 1(5):052101, 2016). This hints towards a strong superhydrophobic effect in the concomitant scenario of pressure-driven flow between two such surfaces, particularly for non-wide channels where bubble-to-bubble pitch and bubble radius are commensurate with channel width. We show here that such pressure-driven flows can be analyzed asymptotically and in closed form based on the smallness of the gaps separating the bubbles relative to the channel width (and bubble radius). We find that the flow adopts an unconventional plug profile away from the inter-bubble gaps, with the uniform velocity being asymptotically larger than the corresponding Poiseuille scale. For a given solid fraction and channel width, the net volumetric flux is maximized when the length of each semi-circular bubble-liquid interface is equal to the channel width. The plug flow identified herein cannot be obtained via a naive implementation of a Navier condition, which is indeed inapplicable for non-wide channels.

Schnitzer O, Yariv E, 2018, Small-solid-fraction approximations for the slip-length tensor of micropillared superhydrophobic surfaces, *Journal of Fluid Mechanics*, Vol: 843, Pages: 637-652, ISSN: 0022-1120

Fakir-like superhydrophobic surfaces, formed by doubly periodic arrays of thin pillars that sustain a lubricating gas layer, exhibit giant liquid-slip lengths that scale as relative to the periodicity, being the solid fraction (Ybert et al., Phys. Fluids, vol. 19, 2007, 123601). Considering arbitrarily shaped pillars distributed over an arbitrary Bravais lattice, we employ matched asymptotic expansions to calculate the slip-length tensor in the limit . The leading slip length is determined from a local analysis of an ‘inner’ region close to a single pillar, in conjunction with a global force balance. This leading term, which is independent of the lattice geometry, is related to the drag due to pure translation of a flattened disk shaped like the pillar cross-section; its calculation is illustrated for the case of elliptical pillars. The slip length is associated with the excess velocity induced about a given pillar by all the others. Since the field induced by each pillar corresponds on the ‘outer’ lattice scale to a Stokeslet whose strength is fixed by the shear rate, the slip length depends upon the lattice geometry but is independent of the cross-sectional shape. Its calculation entails asymptotic evaluation of singular lattice sums. Our approximations are in excellent agreement with existing numerical computations for both circular and square pillars.

Schnitzer O, Yariv E, 2018, Resistive-force theory for mesh-like superhydrophobic surfaces, *Physical Review Fluids*, Vol: 3, ISSN: 2469-990X

A common realization of superhydrophobic surfaces makes use of a mesh-like geometry, where pockets of air are trapped in a periodic array of holes in a no-slip solid substrate. We consider the small-solid-fraction limit where the ribs of the mesh are narrow. In this limit, we obtain a simple leading-order approximation for the slip-length tensor of an arbitrary mesh geometry. This approximation scales as the solid-fraction logarithm, as anticipated by Ybert et al. [Phys. Fluids19, 123601 (2007)]; in the special case of a square mesh it agrees with the analytical results obtained by Davis & Lauga [Phys. Fluids21, 113101 (2009)].

Schnitzer O, Craster RV, 2017, Bloch waves in an arbitrary two-dimensional lattice of subwavelength Dirichlet scatterers, *SIAM Journal on Applied Mathematics*, Vol: 77, Pages: 2119-2135, ISSN: 0036-1399

We study waves governed by the planar Helmholtz equation, propagating in aninfinite lattice of subwavelength Dirichlet scatterers, the periodicity beingcomparable to the wavelength. Applying the method of matched asymptoticexpansions, the scatterers are effectively replaced by asymptotic pointconstraints. The resulting coarse-grained Bloch-wave dispersion problem issolved by a generalised Fourier series, whose singular asymptotics in thevicinities of scatterers yield the dispersion relation governing modes that arestrongly perturbed from plane-wave solutions existing in the absence of thescatterers; there are also empty-lattice waves that are only weakly perturbed.Characterising the latter is useful in interpreting and potentially designingthe dispersion diagrams of such lattices. The method presented, that simplifiesand expands on Krynkin & McIver [Waves Random Complex, 19 347 2009], could beapplied in the future to study more sophisticated designs entailing resonantsubwavelength elements distributed over a lattice with periodicity on the orderof the operating wavelength.

Vanel AL, Schnitzer O, Craster RV, 2017, Asymptotic network models of subwavelength metamaterials formed by closely packed photonic and phononic crystals, *Europhysics Letters: a letters journal exploring the frontiers of physics*, Vol: 119, ISSN: 1286-4854

We demonstrate that photonic and phononic crystals consisting of closely spaced inclusions constitute a versatile class of subwavelength metamaterials. Intuitively, the voids and narrow gaps that characterise the crystal form an interconnected network of Helmholtz-like resonators. We use this intuition to argue that these continuous photonic (phononic) crystals are in fact asymptotically equivalent, at low frequencies, to discrete capacitor-inductor (mass-spring) networks whose lumped parameters we derive explicitly. The crystals are tantamount to metamaterials as their entire acoustic branch, or branches when the discrete analogue is polyatomic, is squeezed into a subwavelength regime where the ratio of wavelength to period scales like the ratio of period to gap width raised to the power $1/4$ ; at yet larger wavelengths we accordingly find a comparably large effective refractive index. The fully analytical dispersion relations predicted by the discrete models yield dispersion curves that agree with those from finite-element simulations of the continuous crystals. The insight gained from the network approach is used to show that, surprisingly, the continuum created by a closely packed hexagonal lattice of cylinders is represented by a discrete honeycomb lattice. The analogy is utilised to show that the hexagonal continuum lattice has a Dirac-point degeneracy that is lifted in a controlled manner by specifying the area of a symmetry-breaking defect.

Schnitzer O, 2017, Spoof surface plasmons guided by narrow grooves, *Physical Review B*, Vol: 96, ISSN: 1550-235X

An approximate description of surface waves propagating along periodically grooved surfaces is intuitively developed in the limit where the grooves are narrow relative to the period. Considering acoustic and electromagnetic waves guided by rigid and perfectly conducting gratings, respectively, the wave field is obtained by interrelating elementary approximations obtained in three overlapping spatial domains. Specifically, above the grating and on the scale of the period the grooves are effectively reduced to point resonators characterized by their dimensions as well as the geometry of their apertures. Along with this descriptive physical picture emerges an analytical dispersion relation, which agrees remarkably well with exact calculations and improves on preceding approximations. Scalings and explicit formulas are obtained by simplifying the theory in three distinguished propagation regimes, namely where the Bloch wave number is respectively smaller than, close to, or larger than that corresponding to a groove resonance. Of particular interest is the latter regime where the field within the grooves is resonantly enhanced and the field above the grating is maximally localized, attenuating on a length scale comparable with the period.

Schnitzer O, Yariv E, 2017, Longitudinal pressure-driven flows between superhydrophobic grooved surfaces: large effective slip in the narrow-channel limit, *Physical Review Fluids*, Vol: 2, ISSN: 2469-990X

The gross amplification of the fluid velocity in pressure-driven flows due to the introduction of superhydrophobic walls is commonly quantified by an effective slip length. The canonical duct-flow geometry involves a periodic structure of longitudinal shear-free stripes at either one or both of the bounding walls, corresponding to flat-meniscus gas bubbles trapped within a periodic array of grooves. This grating configuration is characterized by two geometric parameters, namely the ratio κ of channel width to microstructure period and the areal fraction Δ of the shear-free stripes. For wide channels, κ≫1, this geometry is known to possess an approximate solution where the dimensionless slip length λ, normalized by the duct semiwidth, is small, indicating a weak superhydrophobic effect. We here address the other extreme of narrow channels, κ≪1, identifying large O(κ−2) values of λ for the symmetric configuration, where both bounding walls are superhydrophobic. This velocity enhancement is associated with an unconventional Poiseuille-like flow profile where the parabolic velocity variation takes place in a direction parallel (rather than perpendicular) to the boundaries. Use of matched asymptotic expansions and conformal-mapping techniques provides λ up to O(κ−1), establishing the approximation λ∼κ−2Δ33+κ−1Δ2πln4+⋯, which is in excellent agreement with a semianalytic solution of the dual equations governing the respective coefficients of a Fourier-series representation of the fluid velocity. No similar singularity occurs in the corresponding asymmetric configuration, involving a single superhydrophobic wall; in that geometry, a Hele-Shaw approximation shows that λ=O(1).

Schnitzer O, 2017, Slip length for longitudinal shear flow over an arbitrary-protrusion-angle bubble mattress: The small-solid-fraction singularity, *Journal of Fluid Mechanics*, Vol: 820, Pages: 580-603, ISSN: 1469-7645

We study the effective slip length for unidirectional flow over a superhydrophobic mattress of bubbles in the small-solid-fraction limit . Using scaling arguments and utilising an ideal-flow analogy we elucidate the singularity of the slip length as : relative to the periodicity it scales as for protrusion angles and as for . We continue with a detailed asymptotic analysis using the method of matched asymptotic expansions, where ‘inner’ solutions valid close to the solid segments are matched with ‘outer’ solutions valid on the scale of the periodicity, where the bubbles protruding from the solid grooves appear to touch. The analysis yields asymptotic expansions for the effective slip length in each of the protrusion-angle regimes. These expansions overlap for intermediate protrusion angles, which allows us to form a uniformly valid approximation for arbitrary protrusion angles . We thereby explicitly describe the transition with increasing protrusion angle from a logarithmic to an algebraic small-solid-fraction slip-length singularity.

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