## Publications

58 results found

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.

Maling B, Schnitzer O, Craster RV, 2017, Radiation from structured-ring resonators, *SIAM Journal on Applied Mathematics*, ISSN: 0036-1399

We investigate the scalar-wave resonances of systems composed of identicalNeumann-type inclusions arranged periodically around a circular ring. Drawingon natural similarities with the undamped Rayleigh-Bloch waves supported byinfinite linear arrays, we deduce asymptotically the exponentially smallradiative damping in the limit where the ring radius is large relative to theperiodicity. In our asymptotic approach, locally linear Rayleigh-Bloch wavesthat attenuate exponentially away from the ring are matched to a ring-scaleWKB-type wave field. The latter provides a descriptive physical picture of howthe mode energy is transferred via tunnelling to a circularevanescent-to-propagating transition region a finite distance away from thering, from where radiative grazing rays emanate to the far field. Excluding thezeroth-order standing-wave modes, the position of the transition circlebifurcates with respect to clockwise and anti-clockwise contributions,resulting in striking spiral wavefronts.

Schnitzer O, 2017, Waves in Slowly Varying Band-Gap Media, *SIAM Journal on Applied Mathematics*, Vol: 77, Pages: 1516-1535, ISSN: 0036-1399

Schnitzer O, 2016, Singular effective slip length for longitudinal flow over a dense bubble mattress, *Physical Review Fluids*, Vol: 1, ISSN: 2469-990X

We consider the effective hydrophobicity of a periodically grooved surface immersed in liquid,with trapped shear-free bubbles protruding between the no-slip ridges at a π/2 contact angle.Specifically, we carry out a singular-perturbation analysis in the limit ǫ ≪ 1 where the bubbles areclosely spaced, finding the effective slip length (normalised by the bubble radius) for longitudinalflow along the the ridges as π/√2ǫ − (12/π) ln 2 + (13π/24)√2ǫ + o(√ǫ), the small parameter ǫbeing the planform solid fraction. The square-root divergence highlights the strong hydrophobiccharacter of this configuration; this leading singular term (along with the third term) follows froma local lubrication-like analysis of the gap regions between the bubbles, together with generalmatching considerations and a global conservation relation. The O(1) constant term is found bymatching with a leading-order solution in the “outer” region, where the bubbles appear to betouching. We find excellent agreement between our slip-length formula and a numerical schemerecently derived using a “unified-transform” method (D. Crowdy, IMA J. Appl. Math., 80 1902,2015). The comparison demonstrates that our asymptotic formula, together with the diametric“dilute-limit” approximation (D. Crowdy, J. Fluid Mech., 791 R7, 2016), provides an elementaryanalytical description for essentially arbitrary no-slip fractions.

Schnitzer O, Giannini V, Maier SA,
et al., 2016, Surface-plasmon resonances of arbitrarily shaped nanometallic structures in the small-screening-length limit, *Proceedings of the Royal Society A: Mathematical, Physical & Engineering Sciences*, Vol: 472, ISSN: 1364-5021

According to the hydrodynamic Drude model,surface-plasmon resonances of metallic nanostructuresblueshift owing to the nonlocal response of the metal’selectron gas. The screening length characterisingthe nonlocal effect is often small relative to theoverall dimensions of the metallic structure, whichenables us to derive a coarse-grained nonlocaldescription using matched asymptotic expansions; aperturbation theory for the blueshifts of arbitraryshaped nanometallic structures is then developed.The effect of nonlocality is not always a perturbationand we present a detailed analysis of the “bonding”modes of a dimer of nearly touching nanowires wherethe leading-order eigenfrequencies and eigenmodedistributions are shown to be a renormalisation ofthose predicted assuming a local metal permittivity.

Schnitzer O, Giannini V, Craster RV,
et al., 2016, Asymptotics of surface-plasmon redshift saturation at subnanometric separations, *Physical Review B*, Vol: 93, ISSN: 2469-9950

Many promising nanophotonics endeavors hinge upon the unique plasmonic properties of nanometallic structures with narrow nonmetallic gaps, which support superconcentrated bonding modes that singularly redshift with decreasing separations. In this Rapid Communication, we present a descriptive physical picture, complemented by elementary asymptotic formulas, of a nonlocal mechanism for plasmon redshift saturation at subnanometric gap widths. Thus, by considering the electron-charge and field distributions in the close vicinity of the metal-vacuum interface, we show that nonlocality is asymptotically manifested as an effective potential discontinuity. For bonding modes in the near-contact limit, the latter discontinuity is shown to be effectively equivalent to a widening of the gap. As a consequence, the resonance-frequency near-contact asymptotics are a renormalization of the corresponding local ones. Specifically, the renormalization furnishes an asymptotic plasmon-frequency lower bound that scales with the 1/4 power of the Fermi wavelength. We demonstrate these remarkable features in the prototypical cases of nanowire and nanosphere dimers, showing agreement between our elementary expressions and previously reported numerical computations.

Schnitzer O, Yariv E, 2016, Streaming-potential phenomena in the thin-Debye-layer limit. Part 3. Shear-induced electroviscous repulsion, *Journal of Fluid Mechanics*, Vol: 786, Pages: 84-109, ISSN: 1469-7645

We employ the moderate-P´eclet-number macroscale model developed in part 2 of this sequence(Schnitzer, Frankel & Yariv, J. Fluid Mech., vol. 704, 2012, pp. 109–136) towardsthe calculation of electroviscous forces on charged solid particles, engendered by an imposedrelative motion between these particles and the electrolyte solution in which theyare suspended. In particular, we are interested in the kinematic irreversibility of theseforces, stemming from the diffusio-osmotic slip which accompanies the salt-concentrationpolarisation induced by that imposed motion. We illustrate the electroviscous irreversibilityusing two prototypic problems, one involving side-by-side sedimentation of two sphericalparticles, and the other involving a force-free spherical particle suspended in the vicinityof a planar wall and exposed to a simple shear flow. We focus upon the pertinent limitof near-contact configurations, where use of lubrication approximations provides closedformexpressions for the leading-order lateral repulsion. In this approximation schemethe need to solve the advection–diffusion equation governing the salt-concentration polarisationis circumvented.

Schnitzer O, 2015, Singular perturbations approach to localized surface-plasmon resonance: Nearly touching metal nanospheres, *Physical Review. B, Condensed Matter*, Vol: 92, ISSN: 0163-1829

Metallic nano-structures characterised by multiple geometric length scales support low-frequencysurface-plasmon modes, which enable strong light localisation and field enhancement. We suggestto study such configurations using singular perturbation methods, and demonstrate the efficacyof this approach by considering, in the quasi-static limit, a pair of nearly touching metallic nanospheressubjected to an incident electromagnetic wave polarised with the electric field along the lineof sphere centres. Rather than attempting an exact analytical solution, we construct the pertinent(longitudinal) eigen-modes by matching relatively simple asymptotic expansions valid in overlappingspatial domains. We thereby arrive at an effective boundary eigenvalue problem in a half-spacerepresenting the metal region in the vicinity of the gap. Coupling with the gap field gives rise to amixed-type boundary condition with varying coefficients, whereas coupling with the particle-scalefield enters through an integral eigenvalue selection rule involving the electrostatic capacitance ofthe configuration. By solving the reduced problem we obtain accurate closed-form expressions forthe resonance values of the metal dielectric function. Furthermore, together with an energy-likeintegral relation, the latter eigen-solutions yield also closed-form approximations for the induceddipolemoment and gap-field enhancement under resonance. We demonstrate agreement between theasymptotic formulae and a semi-numerical computation. The analysis, underpinned by asymptoticscaling arguments, elucidates how metal polarisation together with geometrical confinement enablesa strong plasmon-frequency redshift and amplified near-field at resonance.

Schnitzer O, Yariv EY, 2015, The Taylor–Melcher leaky dielectric model as a macroscale electrokinetic description, *Journal of Fluid Mechanics*, Vol: 773, Pages: 1-33, ISSN: 0022-1120

While the Taylor–Melcher electrohydrodynamic model entails ionic charge carriers, it addresses neither ionic transport within the liquids nor the formation of diffuse space-charge layers about their common interface. Moreover, as this model is hinged upon the presence of non-zero interfacial-charge density, it appears to be in contradiction with the aggregate electro-neutrality implied by ionic screening. Following a brief synopsis published by Baygents & Saville (Third International Colloquium on Drops and Bubbles, AIP Conference Proceedings, vol. 7, 1989, American Institute of Physics, pp. 7–17) we systematically derive here the macroscale description appropriate for leaky dielectric liquids, starting from the primitive electrokinetic equations and addressing the double limit of thin space-charge layers and strong fields. This derivation is accomplished through the use of matched asymptotic expansions between the narrow space-charge layers adjacent to the interface and the electro-neutral bulk domains, which are homogenized by the strong ionic advection. Electrokinetic transport within the electrical ‘triple layer’ comprising the genuine interface and the adjacent space-charge layers is embodied in effective boundary conditions; these conditions, together with the simplified transport within the bulk domains, constitute the requisite macroscale description. This description essentially coincides with the familiar equations of Melcher & Taylor (Annu. Rev. Fluid Mech., vol. 1, 1969, pp. 111–146). A key quantity in our macroscale description is the ‘apparent’ surface-charge density, provided by the transversely integrated triple-layer microscale charge. At leading order, this density vanishes due to the expected Debye-layer screening; its asymptotic correction provides the ‘interfacial’ surface-charge density appearing in the Taylor–Melcher model. Our unified electrohydrodynamic treatment provides a

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