18 results
Spontaneous locomotion of a symmetric squirmer
- Richard Cobos, Aditya S. Khair, Ory Schnitzer
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- Journal:
- Journal of Fluid Mechanics / Volume 983 / 25 March 2024
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- 18 March 2024, R3
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The squirmer is a popular model to analyse the fluid mechanics of a self-propelled object, such as a micro-organism. We demonstrate that some fore–aft symmetric squirmers can spontaneously self-propel above a critical Reynolds number. Specifically, we numerically study the effects of inertia on spherical squirmers characterised by an axially and fore–aft symmetric ‘quadrupolar’ distribution of surface-slip velocity; under creeping-flow conditions, such squirmers generate a pure stresslet flow, the stresslet sign classifying the squirmer as either a ‘pusher’ or ‘puller’. Assuming axial symmetry, and over the examined range of the Reynolds number $Re$ (defined based upon the magnitude of the quadrupolar squirming), we find that spontaneous symmetry breaking occurs in the puller case above $Re \approx 14.3$, with steady swimming emerging from that threshold consistently with a supercritical pitchfork bifurcation and with the swimming speed growing monotonically with $Re$.
Shape of sessile drops in the large-Bond-number ‘pancake’ limit
- Ehud Yariv, Ory Schnitzer
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- Journal:
- Journal of Fluid Mechanics / Volume 961 / 25 April 2023
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- 18 April 2023, A13
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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.
Isotropically active colloids under uniform force fields: from forced to spontaneous motion
- Saikat Saha, Ehud Yariv, Ory Schnitzer
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- Journal:
- Journal of Fluid Mechanics / Volume 916 / 10 June 2021
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- 14 April 2021, A47
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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 $\alpha$ associated with the chemical velocity scale. We find that the predicted velocity may become singular at $\alpha =4$, and that this happens under the same conditions on the surface parameters for which the associated unforced problem is known to exhibit, for $\alpha >4$, 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 $\alpha =4$, 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.
Rolling of non-wetting droplets down a gently inclined plane
- Ory Schnitzer, Anthony M. J. Davis, Ehud Yariv
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- Journal:
- Journal of Fluid Mechanics / Volume 903 / 25 November 2020
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- 28 September 2020, A25
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We analyse the near-rolling motion of non-wetting droplets down a gently inclined plane. Inspired by the scaling analysis of Mahadevan & Pomeau (Phys. Fluids, vol. 11, 1999, pp. 2449–2453), we focus upon the limit of small Bond numbers, where the drop shape is nearly spherical and the internal flow is approximately a rigid-body rotation except close to the flat 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 dissipation associated with that flow we obtain a closed-form approximation for the drop speed. This approximation reveals that the missing prefactor in the Mahadevan–Pomeau scaling law is $(3{\rm \pi} /16)\sqrt {3/2}\approx 0.72$ – in good agreement with experiments. An unexpected feature of the flow field is that it happens to satisfy the no-slip and shear-free conditions simultaneously 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 lies primarily in the plane locally normal to the contact line; it is analogous there to the local solution in the comparable problem of a two-dimensional rolling drop. This analogy breaks down near the two points where the contact line propagates parallel to itself, the local flow being there genuinely three dimensional. These observations illuminate a unique ‘peeling’ mechanism by which a rolling droplet avoids the familiar non-integrable stress singularity at a moving contact line.
Acoustic impedance of a cylindrical orifice
- Rodolfo Brandão, Ory Schnitzer
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- Journal:
- Journal of Fluid Mechanics / Volume 892 / 10 June 2020
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- 01 April 2020, A7
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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.
Stokes resistance of a solid cylinder near a superhydrophobic surface. Part 1. Grooves perpendicular to cylinder axis
- Ory Schnitzer, Ehud Yariv
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- Journal:
- Journal of Fluid Mechanics / Volume 868 / 10 June 2019
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- 10 April 2019, pp. 212-243
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An important class of canonical problems that is employed in quantifying the slipperiness of microstructured superhydrophobic surfaces is concerned with the calculation of the hydrodynamic loads on adjacent solid bodies whose size is large relative to the microstructure period. The effect of superhydrophobicity is most pronounced when the latter period is comparable to the separation between the solid probe and the superhydrophobic surface. We address the above distinguished limit, considering a simple configuration where the superhydrophobic surface is formed by a periodically grooved array, in which air bubbles are trapped in a Cassie state, and the solid body is an infinite cylinder. In the present part, we consider the case where the grooves are aligned perpendicular to the cylinder and allow for three modes of rigid-body motion: rectilinear motion perpendicular to the surface; rectilinear motion parallel to the surface, in the groove direction; and angular rotation about the cylinder axis. In this scenario, the flow is periodic in the direction parallel to the axis. Averaging over the small-scale periodicity yields a modified lubrication description where the small-scale details are encapsulated in two auxiliary two-dimensional cell problems which respectively describe pressure- and boundary-driven longitudinal flow through an asymmetric rectangular domain, bounded by a compound surface from the bottom and a no-slip surface from the top. Once the integral flux and averaged shear stress associated with each of these cell problems are calculated as a function of the slowly varying cell geometry, the hydrodynamic loads experienced by the cylinder are provided as quadratures of nonlinear functions of the latter distributions over a continuous sequence of cells.
Small-solid-fraction approximations for the slip-length tensor of micropillared superhydrophobic surfaces
- Ory Schnitzer, Ehud Yariv
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- Journal:
- Journal of Fluid Mechanics / Volume 843 / 25 May 2018
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- 26 March 2018, pp. 637-652
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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 $\unicode[STIX]{x1D719}^{-1/2}$ relative to the periodicity, $\unicode[STIX]{x1D719}$ 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 $\unicode[STIX]{x1D719}\rightarrow 0$. The leading $O(\unicode[STIX]{x1D719}^{-1/2})$ 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 $O(1)$ 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 $O(1)$ 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.
Slip length for longitudinal shear flow over an arbitrary-protrusion-angle bubble mattress: the small-solid-fraction singularity
- Ory Schnitzer
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- Journal:
- Journal of Fluid Mechanics / Volume 820 / 10 June 2017
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- 12 May 2017, pp. 580-603
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We study the effective slip length for unidirectional flow over a superhydrophobic mattress of bubbles in the small-solid-fraction limit $\unicode[STIX]{x1D716}\ll 1$. Using scaling arguments and utilising an ideal-flow analogy we elucidate the singularity of the slip length as $\unicode[STIX]{x1D716}\rightarrow 0$: relative to the periodicity it scales as $\log (1/\unicode[STIX]{x1D716})$ for protrusion angles $0\leqslant \unicode[STIX]{x1D6FC}<\unicode[STIX]{x03C0}/2$ and as $\unicode[STIX]{x1D716}^{-1/2}$ for $0<\unicode[STIX]{x03C0}/2-\unicode[STIX]{x1D6FC}=O(\unicode[STIX]{x1D716}^{1/2})$. 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 $0\leqslant \unicode[STIX]{x1D6FC}\leqslant \unicode[STIX]{x03C0}/2$. We thereby explicitly describe the transition with increasing protrusion angle from a logarithmic to an algebraic small-solid-fraction slip-length singularity.
Streaming-potential phenomena in the thin-Debye-layer limit. Part 3. Shear-induced electroviscous repulsion
- Ory Schnitzer, Ehud Yariv
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- Journal:
- Journal of Fluid Mechanics / Volume 786 / 10 January 2016
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- 26 November 2015, pp. 84-109
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We employ the moderate-Péclet-number macroscale model developed in part 2 of this sequence (Schnitzer et al., J. Fluid Mech., vol. 704, 2012, pp. 109–136) towards the calculation of electroviscous forces on charged solid particles engendered by an imposed relative motion between these particles and the electrolyte solution in which they are suspended. In particular, we are interested in the kinematic irreversibility of these forces, stemming from the diffusio-osmotic slip which accompanies the salt-concentration polarisation induced by that imposed motion. We illustrate the electroviscous irreversibility using two prototypic problems, one involving side-by-side sedimentation of two spherical particles, and the other involving a force-free spherical particle suspended in the vicinity of a planar wall and exposed to a simple shear flow. We focus on the pertinent limit of near-contact configurations, where use of lubrication approximations provides closed-form expressions for the leading-order lateral repulsion. In this approximation scheme, the need to solve the advection–diffusion equation governing the salt-concentration polarisation is circumvented.
The Taylor–Melcher leaky dielectric model as a macroscale electrokinetic description
- Ory Schnitzer, Ehud Yariv
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- Journal of Fluid Mechanics / Volume 773 / 25 June 2015
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- 14 May 2015, pp. 1-33
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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 reinterpretation of both the Taylor–Melcher conductivity-ratio parameter and the electrical Reynolds number. The latter, expressed in terms of fundamental electrokinetic properties, becomes $O(1)$ only for intense applied fields, comparable with the transverse field within the space-charge layers; at this limit the asymptotic scheme collapses. Surface-charge advection is accordingly absent in the macroscale description. Owing to the inevitable presence of (screened) net charge on the genuine interface, the drop also undergoes electrophoretic motion. The associated flow, however, is asymptotically smaller than that corresponding to the Taylor–Melcher circulation. Our successful matching procedure contrasts the analysis of Baygents & Saville, who considered more general electrolytes and were unable to directly match the inner and outer regions. We discuss this difference in detail.
Slender-body approximations for advection–diffusion problems
- Ory Schnitzer
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- Journal of Fluid Mechanics / Volume 768 / 10 April 2015
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- 06 March 2015, R5
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We consider steady advection–diffusion about a slender body of revolution at arbitrary $O(1)$ Péclet numbers ($\mathit{Pe}$). The transported scalar attenuates at large distances and is governed by axisymmetric (either Dirichlet or Neumann) data prescribed at the body boundary. The advecting field is assumed to be an axisymmetric Stokes flow approaching a uniform stream at large distances and satisfying impermeability at the boundary; otherwise, the interfacial distribution of tangential velocity is assumed to be arbitrary, irrotational and no-slip Stokes flows being particular cases. Employing the method of matched asymptotic expansions, we develop a systematic scheme for the calculation of the scalar concentration in increasing powers of $\ln ^{-1}(1/{\it\epsilon})$, ${\it\epsilon}\ll 1$ being the body’s characteristic thickness to length ratio. The leading term in the inner expansion coincides with the pure diffusion case, the second term depends nonlinearly on the magnitude of the far-field stream and higher-order terms depend on the boundary distribution of tangential velocity. In the special case of irrotational flow and Neumann boundary conditions the logarithmic expansion terminates, leaving an algebraic error in ${\it\epsilon}$. The general formulae developed can be directly applied to numerous physical scenarios. We here consider the classical problem of forced heat convection from an isothermal body, finding a two-term expansion for $\mathit{Nu}({\it\epsilon},\mathit{Pe})/\mathit{Nu}({\it\epsilon},0)$, the ratio of the Nusselt number to its value at $\mathit{Pe}=0$. This ratio is insensitive to the particle shape at the asymptotic orders considered; at moderately large $\mathit{Pe}$ ($\ll {\it\epsilon}^{-1}$) its deviation from unity is $O[\ln (\mathit{Pe})/\text{ln}\,(1/{\it\epsilon})]$, marking the poor effectiveness of advection about slender bodies. The expansion is compared with a numerical computation in the case of a prolate spheroid in both irrotational and no-slip Stokes flows.
Electrophoresis of bubbles
- Ory Schnitzer, Itzchak Frankel, Ehud Yariv
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- Journal of Fluid Mechanics / Volume 753 / 25 August 2014
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- 16 July 2014, pp. 49-79
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Smoluchowski’s celebrated electrophoresis formula is inapplicable to field-driven motion of drops and bubbles with mobile interfaces. We here analyse bubble electrophoresis in the thin-double-layer limit. To this end, we employ a systematic asymptotic procedure starting from the standard electrokinetic equations and a simple physicochemical interface model. This furnishes a coarse-grained macroscale description where the Debye-layer physics is embodied in effective boundary conditions. These conditions, in turn, represent a non-conventional driving mechanism for electrokinetic flows, where bulk concentration polarization, engendered by the interaction of the electric field and the Debye layer, results in a Marangoni-like shear stress. Remarkably, the electro-osmotic velocity jump at the macroscale level does not affect the electrophoretic velocity. Regular approximations are obtained in the respective cases of small zeta potentials, small ions, and weak applied fields. The nonlinear small-zeta-potential approximation rationalizes the paradoxical zero mobility predicted by the linearized scheme of Booth (J. Chem. Phys., vol. 19, 1951, pp. 1331–1336). For large (millimetre-size) bubbles the pertinent limit is actually that of strong fields. We have carried out a matched-asymptotic-expansion analysis of this singular limit, where salt polarization is confined to a narrow diffusive layer. This analysis establishes that the bubble velocity scales as the $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}2/3$-power of the applied-field magnitude and yields its explicit functional dependence upon a specific combination of the zeta potential and the ionic drag coefficient. The latter is provided to within an $O(1)$ numerical pre-factor which, in turn, is calculated via the solution of a universal (parameter-free) nonlinear flow problem. It is demonstrated that, with increasing field magnitude, all numerical solutions of the macroscale model indeed collapse on the analytic approximation thus obtained. Existing measurements of clean-bubble electrophoresis agree neither with present theory nor with previous models; we discuss this ongoing discrepancy.
Deformation of leaky-dielectric fluid globules under strong electric fields: boundary layers and jets at large Reynolds numbers
- Ory Schnitzer, Itzchak Frankel, Ehud Yariv
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- Journal:
- Journal of Fluid Mechanics / Volume 734 / 10 November 2013
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- 14 October 2013, R3
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In Taylor’s analysis of electrohydrodynamic drop deformation by a uniform electric field (Proc. R. Soc. Lond. A, vol. 291, 1966, pp. 159–166) inertia is neglected at the outset, resulting in fluid velocities that scale as the square of the applied-field magnitude. For large (i.e. millimetric) drops, with increasing field strength the Reynolds number predicted by this scaling may actually become large, suggesting the need for a complementary large-Reynolds-number investigation. Balancing viscous stresses and electrical shear forces in this limit reveals a different velocity scaling, with the $4/ 3$-power of the applied-field magnitude. For simplicity, we focus upon the flow about a spherical gas bubble. It is essentially confined to two boundary layers propagating from the poles to the equator, where they collide to form a radial jet. The transition occurs over a small deflection region about the equator where the flow is effectively inviscid. The deviation of the bubble shape from the original sphericity is quantified by the capillary number given by the ratio of a characteristic Maxwell stress to Laplace’s pressure. At leading order the bubble deforms owing to: (i) the surface distribution of the Maxwell stress, associated with the familiar electric-field profile; (ii) the hydrodynamic boundary-layer pressure, engendered here by centrifugal forces; and (iii) the intense pressure distribution acting over the narrow equatorial deflection zone, appearing on the bubble scale as a concentrated load. Remarkably, the unique flow topology and associated scalings allow the obtaining of a closed-form expression for the deformation through the mere application of integral mass and momentum balances. On the bubble scale, the concentrated pressure load is manifested in the appearance of a non-smooth equatorial dimple.
Electrokinetic flows about conducting drops
- Ory Schnitzer, Itzchak Frankel, Ehud Yariv
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- Journal:
- Journal of Fluid Mechanics / Volume 722 / 10 May 2013
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- 02 April 2013, pp. 394-423
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We consider electrokinetic flows about a freely suspended liquid drop, deriving a macroscale description in the thin-double-layer limit where the ratio $\delta $ between Debye width and drop size is asymptotically small. In this description, the electrokinetic transport occurring within the diffuse part of the double layer (the ‘Debye layer’) is represented by effective boundary conditions governing the pertinent fields in the electro-neutral bulk, wherein the generally non-uniform distribution of $\zeta $, the dimensionless zeta potential, is a priori unknown. We focus upon highly conducting drops. Since the tangential electric field vanishes at the drop surface, the viscous stress associated with Debye-scale shear, driven by Coulomb body forces, cannot be balanced locally by Maxwell stresses. The requirement of microscale stress continuity therefore brings about a unique velocity scaling, where the standard electrokinetic scale is amplified by a ${\delta }^{- 1} $ factor. This reflects a transition from slip-driven electro-osmotic flows to shear-induced motion. The macroscale boundary conditions display distinct features reflecting this unique scaling. The effective shear-continuity condition introduces a Lippmann-type stress jump, appearing as a product of the local charge density and electric field. This term, representing the excess Debye-layer shear, follows here from a systematic coarse-graining procedure starting from the exact microscale description, rather than from thermodynamic considerations. The Neumann condition governing the bulk electric field is inhomogeneous, representing asymptotic matching with transverse ionic fluxes emanating from the Debye layer; these fluxes, in turn, are associated with non-uniform tangential ‘surface’ currents within this layer. Their appearance at leading order is a manifestation of dominant advection associated with the large velocity scale. For weak fields, the linearized macroscale equations admit an analytic solution, yielding a closed-form expression for the electrophoretic velocity. When scaled by Smoluchowski’s speed, it reads
$${\delta }^{- 1} \frac{\sinh ( \overline{\zeta } / 2)/ \overline{\zeta } }{1+ { \textstyle\frac{3}{2} }\mu + 2\alpha {\mathop{\sinh }\nolimits }^{2} ( \overline{\zeta } / 2)} ,$$wherein $ \overline{\zeta } $, the ‘drop zeta potential’, is the uniform value of $\zeta $ in the absence of an applied field, $\mu $ the ratio of drop to electrolyte viscosities, and $\alpha $ the ionic drag coefficient. The difference from solid-particle electrophoresis is manifested in two key features: the ${\delta }^{- 1} $ scaling, and the effect of ionic advection, as represented by the appearance of $\alpha $. Remarkably, our result differs from the small-$\delta $ limit of the mobility expression predicted by the weak-field model of Ohshima, Healy & White (J. Chem. Soc. Faraday Trans. 2, vol. 80, 1984, pp. 1643–1667). This discrepancy is related to the dominance of advection on the bulk scale, even for weak fields, which feature cannot be captured by a linear theory. The order of the respective limits of thin double layers and weak applied fields is not interchangeable.
The electrophoretic mobility of rod-like particles
- Ehud Yariv, Ory Schnitzer
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- Journal of Fluid Mechanics / Volume 719 / 25 March 2013
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- 26 February 2013, R3
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At finite Dukhin numbers, where Smoluchowski’s formula is inapplicable to thin-double-layer electrophoresis, the mobility of non-spherical particles is generally anisotropic. We consider bodies of revolution of otherwise arbitrary shape, where a uniformly applied electric field results in a rectilinear motion in the plane spanned by the field direction and the particle symmetry axis, as well as (for particles lacking fore–aft symmetry) rigid-body rotation about an axis perpendicular to that plane. Focusing upon slender particles, where the ratio $\epsilon $ of cross-sectional and longitudinal scales is asymptotically small, the translational and rotational mobilities are obtained as quadratures which depend upon the lengthwise distribution of the scaled cross-sectional width and the force densities associated with rigid-body motion. These mobility expressions approach finite limits as $\epsilon \rightarrow 0$, yielding closed-form expressions for specific particle geometries.
Streaming-potential phenomena in the thin-Debye-layer limit. Part 2. Moderate Péclet numbers
- Ory Schnitzer, Itzchak Frankel, Ehud Yariv
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- Journal of Fluid Mechanics / Volume 704 / 10 August 2012
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- 03 July 2012, pp. 109-136
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Macroscale description of streaming-potential phenomena in the thin-double-layer limit, and in particular the associated electro-viscous forces, has been a matter of long-standing controversy. In part 1 of this work (Yariv, Schnitzer & Frankel, J. Fluid Mech., vol. 685, 2011, pp. 306–334) we identified that the product of the Hartmann () and Péclet () numbers is , being the dimensionless Debye thickness. This scaling relationship defines a one-family class of limit processes appropriate to the consistent analysis of this singular problem. In that earlier contribution we focused on the generic problems associated with moderate and large , where the streaming-potential magnitude is comparable to the thermal voltage. Here we consider the companion generic limit of moderate Péclet numbers and large Hartmann numbers, deriving the appropriate macroscale model wherein the Debye-layer physics is represented by effective boundary conditions. Since the induced electric field is asymptotically smaller, calculation of these conditions requires higher asymptotic orders in analysing the Debye-scale transport. Nonetheless, the leading-order electro-viscous forces are of the same relative magnitude as those previously obtained in the large- limit. The structure of these forces is different, however, first because the small Maxwell stresses do not contribute at leading order, and second because salt polarization results in a dominant diffuso-osmotic slip. Since the salt distribution is governed by an advection–diffusion equation, this slip gives rise to electro-viscous forces which are nonlinear in the driving flow. The resulting scheme is illustrated by the calculation of the electro-viscous excess drag in the prototype problem of a translating sphere.
Strong-field electrophoresis
- Ory Schnitzer, Ehud Yariv
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- Journal of Fluid Mechanics / Volume 701 / 25 June 2012
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- 11 May 2012, pp. 333-351
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We analyse particle electrophoresis in the thin-double-layer limit for asymptotically large applied electric fields. Specifically, we consider fields scaling as , being the dimensionless Debye thickness. The dominant advection associated with the intense flow mandates a uniform salt concentration in the electro-neutral bulk. The large tangential fields in the diffuse part of the double layer give rise to a novel ‘surface conduction’ mechanism at moderate zeta potentials, where the Dukhin number is vanishingly small. The ensuing electric current emerging from the double layer modifies the bulk electric field; the comparable transverse salt flux, on the other hand, is incompatible with the nil diffusive fluxes at the homogeneous bulk. This contradiction is resolved by identifying the emergence of a diffusive boundary layer of thickness, resembling thermal boundary layers at large-Reynolds-number flows. The modified electric field within the bulk gives rise to an irrotational flow, resembling those in moderate-field electrophoresis. At leading order, the particle electrophoretic velocity is provided by Smoluchowski’s formula, describing linear variation with applied field.
Streaming-potential phenomena in the thin-Debye-layer limit. Part 1. General theory
- Ehud Yariv, Ory Schnitzer, Itzchak Frankel
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- Journal of Fluid Mechanics / Volume 685 / 25 October 2011
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- 19 September 2011, pp. 306-334
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Electrokinetic streaming-potential phenomena are driven by imposed relative motion between liquid electrolytes and charged solids. Owing to non-uniform convective ‘surface’ current within the Debye layer Ohmic currents from the electro-neutral bulk are required to ensure charge conservation thereby inducing a bulk electric field. This, in turn, results in electro-viscous drag enhancement. The appropriate modelling of these phenomena in the limit of thin Debye layers ( denoting the dimensionless Debye thickness) has been a matter of ongoing controversy apparently settled by Cox’s seminal analysis (J. Fluid Mech., vol. 338, 1997, p. 1). This analysis predicts electro-viscous forces that scale as resulting from the perturbation of the original Stokes flow with the Maxwell-stress contribution only appearing at higher orders. Using scaling analysis we clarify the distinction between the normalizations pertinent to field- and motion-driven electrokinetic phenomena, respectively. In the latter class we demonstrate that the product of the Hartmann & Péclet numbers is contrary to Cox (1997) where both parameters are assumed . We focus on the case where motion-induced fields are comparable to the thermal scale and accordingly present a singular-perturbation analysis for the limit where the Hartmann number is and the Péclet number is . Electric-current matching between the Debye layer and the electro-neutral bulk provides an inhomogeneous Neumann condition governing the electric field in the latter. This field, in turn, results in a velocity perturbation generated by a Smoluchowski-type slip condition. Owing to the dominant convection, the present analysis yields an asymptotic structure considerably simpler than that of Cox (1997): the electro-viscous effect now already appears at and is contributed by both Maxwell and viscous stresses. The present paradigm is illustrated for the prototypic problem of a sphere sedimenting in an unbounded fluid domain with the resulting drag correction differing from that calculated by Cox (1997). Independently of current matching, salt-flux matching between the Debye layer and the bulk domain needs also to be satisfied. This subtle point has apparently gone unnoticed in the literature, perhaps because it is trivially satisfied in field-driven problems. In the present limit this requirement seems incompatible with the uniform salt distribution in the convection-dominated bulk domain. This paradox is resolved by identifying the dual singularity associated with the limit in motion-driven problems resulting in a diffusive layer of thickness beyond the familiar -wide Debye layer.