While the problem governing Stokes flow about a single particle that is subject to an external force is ill posed in two dimensions (the ‘Stokes paradox’), the related problem of two mutually repellent particles is well posed. Motivated by self-assembly phenomena in thin viscous membranes, we consider this problem in the limit of remote particles. Such limits are typically handled in the literature using reflection techniques, which provide successive approximations to the mutual hydrodynamic interactions. Since their starting point is a single particle in an unbounded fluid domain, these techniques are futile in the present two-dimensional problem. We show how this apparent contradiction is resolved via use of singular perturbations. We obtain a two-term approximation for the velocity acquired by circular disks, considering both rigid and free particle surfaces. We also illustrate our perturbation scheme for elliptic disks, deriving a renormalised single-particle velocity. The utility of our asymptotic scheme is illustrated in the general problem of hydrodynamic interaction between a cluster of remote disks.

We consider heat or mass transport from a circular cylinder under a uniform crossflow at small Reynolds numbers, $\mathrm{Re}\ll 1$. This problem has been thwarted in the past by limitations inherent in the classical analyses of the singular flow problem, which have used asymptotic expansions in inverse powers of $\log \mathrm{Re}$. We here make use of the hybrid approximation of Kropinski, Ward & Keller [(1995) SIAM J. Appl. Math. 55, 1484], based upon a robust asymptotic expansion in powers of $\mathrm{Re}$. In that approximation, the “inner” streamfunction is provided by the product of a pre-factor $S$, a slowly varying function of $\mathrm{Re}$, with a $\mathrm{Re}$-independent “canonical” solution of a simple mathematical form. The pre-factor, in turn, is determined as an implicit function of $\log \mathrm{Re}$ via asymptotic matching with a numerical solution of the nonlinear single-scaled “outer” problem, where the cylinder appears as a point singularity. We exploit the hybrid approximation to analyse the transport problem in the limit of large Péclet number, $\mathrm{Pe}\gg 1$. In that limit, transport is restricted to a narrow boundary layer about the cylinder surface – a province contained within the inner region of the flow problem. With $S$ appearing as a parameter, a similarity solution is readily constructed for the boundary-layer problem. It provides the Nusselt number as $0.5799(S\,\mathrm{Pe})^{1/3}$. This asymptotic prediction is in remarkably close agreement with that of the numerical solution of the exact problem [Dennis, Hudson & Smith (1968) Phys. Fluids 11, 933] even for moderate $\mathrm{Re}$-values.

The motion of a disk in a Langmuir film bounding a liquid substrate is a classical hydrodynamic problem, dating back to Saffman (J. Fluid Mech., vol. 73, 1976, p. 593) who focused upon the singular problem of translation at large Boussinesq number, ${\textit {Bq}}\gg 1$. A semianalytic solution of the dual integral equations governing the flow at arbitrary ${\textit {Bq}}$ was devised by Hughes et al. (J. Fluid Mech., vol. 110, 1981, p. 349). When degenerated to the inviscid-film limit ${\textit {Bq}}\to 0$, it produces the value $8$ for the dimensionless translational drag, which is $50\,\%$ larger than the classical $16/3$-value corresponding to a free surface. While that enhancement has been attributed to surface incompressibility, the mathematical reasoning underlying the anomaly has never been fully elucidated. Here we address the inviscid limit ${\textit {Bq}}\to 0$ from the outset, revealing a singular mechanism where half of the drag is contributed by the surface pressure. We proceed beyond that limit, considering a nearly inviscid film. A naïve attempt to calculate the drag correction using the reciprocal theorem fails due to an edge singularity of the leading-order flow. We identify the formation of a boundary layer about the edge of the disk, where the flow is primarily in the azimuthal direction with surface and substrate stresses being asymptotically comparable. Utilising the reciprocal theorem in a fluid domain tailored to the asymptotic topology of the problem produces the drag correction $(8\,{\textit {Bq}}/{\rm \pi} ) [ \ln (2/{\textit {Bq}}) + \gamma _E+1]$, $\gamma _E$ being the Euler–Mascheroni constant.

It is theoretically known that an isotropic chemically active particle in an unbounded solution undergoes symmetry breaking when the intrinsic Péclet number ${{Pe}}$ exceeds a finite critical value (Michelin et al., Phys. Fluids, vol. 25, 2013, 061701). At that value, a transition takes place from a stationary state to spontaneous motion. In two dimensions, where no stationary state is possible in an unbounded domain, a linear stability analysis in a large bounded domain (Hu et al., Phys. Rev. Lett., vol. 123, 2019, 238004) reveals that the critical ${{Pe}}$ value slowly diminishes as the domain size increases. Motivated by these findings, we here consider an unbounded domain from the outset, addressing the two-dimensional problem of steady self-propulsion with a focus on the limit ${{Pe}}\ll 1$. This singular limit is handled using matched asymptotic expansions, conceptually decomposing the fluid domain into a particle-scale region, where the leading-order solute transport is diffusive, and a remote region, where diffusion and advection are comparable. The expansion parameter is provided by the product of ${{Pe}}$ and $U$, the unknown particle speed (normalised by the standard autophoretic scale). The problem is unconventional in that the scaling of $U$ with ${{Pe}}$ must be determined in the course of the perturbation analysis. The resulting approximation, $U=4\exp ({-2/{Pe}-\gamma _{E}-1})/{{Pe}}$ ($\gamma _{E}$ being the Euler–Mascheroni constant), is in remarkable agreement with the numerical predictions of Hu et al. in the common interval of validity.

Compound surfaces, consisting of periodic arrays of solid patches and free surfaces, exhibit hydrodynamic slipperiness which is quantified by their slip length. The limit of small solid fractions, where the slip length diverges, is of fundamental interest. This paper addresses longitudinal liquid flows over a periodic array of grooves which are partially invaded by the liquid. Assuming that the slats separating the grooves are infinitely thin, the solid fraction $\epsilon$ is set by the invasion depth. Inspired by the singular small-solid-fraction limit for non-invaded grooves (Schnitzer, Phys. Rev. Fluids, vol. 1, 2016, 052101R), we consider the idealised geometry of $90^{\circ }$ protrusion angles, where an integral force balance in the limit $\epsilon \to 0$ implies a slip length that scales as $\epsilon ^{-1}$. The problem exhibits a nested structure, where the liquid domain is conceptually decomposed into four distinct regions: a unit-cell region on the scale of the period, where the wetted portion of the slat appears as a point singularity; two regions on the scale of the wetted slat, where the flow essentially varies in one dimension; and a transition region about the tip of the slat. Analysing these regions using matched asymptotic expansions and conformal mappings yields the ratio of slip length to semi-period as $2\epsilon ^{-1} - (10/{\rm \pi} )\ln 2 + \cdots$.

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.

The shape of a sessile drop on a horizontal substrate depends upon the Bond number $Bo$ and the contact angle $\alpha$. Inspired by puddle approximations at large $Bo$ (Quéré, Rep. Prog. Phys., vol. 68, 2005, p. 2495), we address here the limit of small contact angles at fixed drop volume and arbitrary $Bo$. It readily leads to a pancake shape approximation, where the drop height and radius scale as $\alpha$ and $\alpha ^{-1/2}$, respectively, with capillary forces being appreciable only near the edge. The pancake approximation breaks down for $Bo=\textrm {ord}(\alpha ^{2/3})$. In that distinguished limit, capillary and gravitational forces are comparable throughout, and the drop height and radius scale as $\alpha ^{2/3}$ and $\alpha ^{-1/3}$, respectively. For $Bo\ll \alpha ^{2/3}$ these scalings remain, with the drop shape turning into a spherical cap. The asymptotic results are compared with a numerical solution of the exact problem.

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.

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.

As a result of field fringing, the capacitance of a parallel-plate capacitor differs from that predicted by the textbook formula. Using singular perturbations and conformal mapping techniques, we calculate the leading-order correction to the capacitance in the limit of large aspect ratio. We additionally obtain a comparable approximation for the electrostatic attraction between the plates.

The hydrodynamic quantification of superhydrophobic slipperiness has traditionally employed two canonical problems – namely, shear flow about a single surface and pressure-driven channel flow. We here advocate the use of a new class of canonical problems, defined by the motion of a superhydrophobic particle through an otherwise quiescent liquid. In these problems the superhydrophobic effect is naturally measured by the enhancement of the Stokes mobility relative to the corresponding mobility of a homogeneous particle. We focus upon what may be the simplest problem in that class – the rotation of an infinite circular cylinder whose boundary is periodically decorated by a finite number of infinite grooves – with the goal of calculating the rotational mobility (velocity-to-torque ratio). The associated two-dimensional flow problem is defined by two geometric parameters – namely, the number $N$ of grooves and the solid fraction $\unicode[STIX]{x1D719}$. Using matched asymptotic expansions we analyse the large-$N$ limit, seeking the mobility enhancement from the respective homogeneous-cylinder mobility value. We thus find the two-term approximation,

$$\begin{eqnarray}\displaystyle 1+{\displaystyle \frac{2}{N}}\ln \csc {\displaystyle \frac{\unicode[STIX]{x03C0}\unicode[STIX]{x1D719}}{2}}, & & \displaystyle \nonumber\end{eqnarray}$$

$N=1,2,4,8,\ldots$

$N$

We consider two-dimensional mass transport to a finite absorbing strip in a uniform shear flow as a model of surface-based biosensors. The quantity of interest is the Sherwood number Sh, namely the dimensionless net flux onto the strip. Considering early-time absorption, it is a function of the Péclet number Pe and the Damköhler number Da, which, respectively, represent the characteristic magnitude of advection and reaction relative to diffusion. With a view towards modelling nanoscale biosensors, we consider the limit Pe«1. This singular limit is handled using matched asymptotic expansions, with an inner region on the scale of the strip, where mass transport is diffusively dominated, and an outer region at distances that scale as Pe-1/2, where advection enters the dominant balance. At the inner region, the mass concentration possesses a point-sink behaviour at large distances, proportional to Sh. A rescaled concentration, normalised using that number, thus possesses a universal logarithmic divergence; its leading-order correction represents a uniform background concentration. At the outer region, where advection by the shear flow enters the leading-order balance, the strip appears as a point singularity. Asymptotic matching with the concentration field in that region provides the Sherwood number as

$${\rm{Sh}} = {\pi \over {\ln (2/{\rm{P}}{{\rm{e}}^{1/2}}) + 1.0559 + \beta }},$$

We consider the thermocapillary motion of a liquid layer which is bounded between two superhydrophobic surfaces, each made up of a periodic array of highly conducting solid slats, with flat bubbles trapped in the grooves between them. Following the recent analysis of the longitudinal problem (Yariv, J. Fluid Mech., vol. 855, 2018, pp. 574–594), we address here the transverse problem, where the macroscopic temperature gradient that drives the flow is applied perpendicular to the grooves, with the goal of calculating the volumetric flux between the two surfaces. We focus upon the situation where the slats separating the grooves are long relative to the groove-array period, for which case the temperature in the solid portions of the superhydrophobic plane is piecewise uniform. This scenario, which was investigated numerically by Baier et al. (Phys. Rev. E, vol. 82 (3), 2010, 037301), allows for a surprising analogy between the harmonic conjugate of the temperature field in the present problem and the unidirectional velocity in a comparable longitudinal pressure-driven flow problem over an interchanged boundary. The main body of the paper is concerned with the limit of deep channels, where the problem reduces to the calculation of the heat transport and flow about a single surface and the associated ‘slip’ velocity at large distance from that surface. Making use of Lorentz’s reciprocity, we obtain that velocity as a simple quadrature, providing the analogue to the expression obtained by Baier et al. (2010) in the comparable longitudinal problem. The rest of the paper is devoted to the diametric limit of shallow channels, which is analysed using a Hele-Shaw approximation, and the singular limit of small solid fractions, where we find a logarithmic scaling of the flux with the solid fraction. The latter two limits do not commute.

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.

A common realisation of superhydrophobic surfaces is based on a periodically grooved solid substrate, with air bubbles trapped in a Cassie state within the grooves. Following Baier, Steffes & Hardt (Phys. Rev. E, vol. 82 (3), 2010, 037301) we consider the thermocapillary flow of a liquid bounded between two such surfaces, driven by a macroscopic temperature gradient. Assuming zero protrusion angle of the free menisci, the periodic geometry is described by two parameters, namely the ratio $\unicode[STIX]{x1D6EC}$ of the groove-array period to the channel depth and the gas fraction $\unicode[STIX]{x1D6E5}$ of the surface. The flow and heat transport depend upon both these parameters, as well as the Marangoni number $Ma$, which quantifies the relative magnitudes of advection and conduction. This paper is concerned with the longitudinal problem, where the temperature gradient is applied along the grooves. The temperature within the highly conducting solid substrate varies linearly with distance and may be regarded as prescribed. For any non-zero value of $Ma$, however, advection necessitates the formation of an excess temperature profile in the liquid domain, above and beyond that linearly varying distribution. The associated Marangoni forces, in turn, imply the formation of flow in the cross-sectional plane. The nonlinearly coupled problem governing the excess temperature and cross-sectional flow is independent of the longitudinal flow. Conversely, the latter satisfies an independent problem in the Stokes limit, and accordingly possesses the standard thermocapillary scaling. A simple transformation reveals a linkage between the longitudinal velocity in the present problem and that in the comparable pressure-driven flow. In studying the coupled problems governing the excess temperature and cross-sectional velocity components, we focus upon deep channels, where $\unicode[STIX]{x1D6EC}\ll 1$. Towards this end, we employ matched asymptotic expansions, with two $O(\unicode[STIX]{x1D6EC})$-deep inner regions adjacent to the surfaces and an outer region in the remaining fluid domain. The small-$\unicode[STIX]{x1D6EC}$ limit is compatible with two possible scalings of $Ma$, the first where it is $O(1)$ and the second where it is $O(\unicode[STIX]{x1D6EC}^{-1})$. In the first scaling, the excess temperature in the outer region is driven by a balance between longitudinal advection and conduction perpendicular to the bounding surfaces. In the inner region, the excess temperature is governed by pure conduction; the cross-sectional flow it animates, which possesses the thermocapillary scaling, is linear in $Ma$. In the second scaling, the cross-sectional velocity becomes $O(\unicode[STIX]{x1D6EC}^{-2/3})$ large relative to the thermocapillary scaling; a boundary layer, of $O(\unicode[STIX]{x1D6EC}^{4/3})$ dimensionless width, is formed within the inner region. In that layer the excess temperature is governed by a dominant balance between conduction perpendicular to the surface and cross-sectional advection. The dependence upon $Ma$ is inherently nonlinear.

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.

Because of the associated far-field logarithmic divergence, the transport problem governing two-dimensional phoretic self-propulsion lacks a steady solution when the Péclet number $\mathit{Pe}$ vanishes. This indeterminacy, which has no counterpart in three dimensions, is remedied by introducing a non-zero value of $\mathit{Pe}$, however small. We consider that problem employing a first-order kinetic model of solute absorption, where the ratio of the characteristic magnitudes of reaction and diffusion is quantified by the Damköhler number $\mathit{Da}$. As $\mathit{Pe}\rightarrow 0$ the dominance of diffusion breaks down at distances that scale inversely with $\mathit{Pe}$; at these distances, the leading-order transport represents a two-dimensional point source in a uniform stream. Asymptotic matching between the latter region and the diffusion-dominated near-particle region provides the leading-order particle velocity as an implicit function of $\log \mathit{Pe}$. Another scenario involving weak advection takes place under strong reactions, where $\mathit{Pe}$ and $\mathit{Da}$ are large and comparable. In that limit, the breakdown of diffusive dominance occurs at distances that scale as $\mathit{Da}^{2}/\mathit{Pe}$.

When suspended in a liquid solution, chemically active colloids may self-propel due to an asymmetry in either particle shape or the interfacial distribution of solute absorption. We here consider a chemically homogeneous spherical particle which undergoes self-diffusiophoresis due to the presence of nearby inert wall. In particular, we focus upon the near-contact limit where it was recently observed (Yariv, Phys. Rev. Fluids, vol. 1 (3), 2016, 032101) that the solute-concentration profile within the narrow gap separating the particle and the wall cannot be uniquely determined by a gap-scale analysis. We here revisit this near-contact limit using matched asymptotic expansions, the inner region being the gap domain and the outer region being on the particle scale. Asymptotic matching with the Hankel-transform representation of the outer distribution of solute concentration serves to determine both the scaling and magnitude of the corresponding inner profile. The ensuing gap-scale pressure field, set by a lubrication mechanism, gives rise to an anomalous particle–wall interaction, scaling as an irrational power of the gap clearance.

In their bipolar-coordinate analysis of circular-cylinder electrophoresis near a dielectric wall, Keh et al. (J. Fluid Mech., vol. 231, 1991, pp. 211–228) found that, when an electric field is applied parallel to the wall, the translational and rotational electrophoretic mobilities increase monotonically as the ratio $\unicode[STIX]{x1D6FF}$ of the cylinder–wall separation to the cylinder radius decreases, eventually diverging as $\unicode[STIX]{x1D6FF}^{-1/2}$ when $\unicode[STIX]{x1D6FF}\rightarrow 0$. Considering the singular limit $\unicode[STIX]{x1D6FF}\ll 1$ from the outset, we conduct here an asymptotic analysis of that electrokinetic problem, providing insight to the manner by which the intense electric field in the narrow gap is transformed into $O(\unicode[STIX]{x1D6FF}^{-3/2})$ shear stresses; these stresses, in turn, overcome the large Stokes resistance so as to provide the large electrophoretic mobilities. In a companion problem, where the cylinder is exposed to a uniform current emanating from a nearby reactive electrode, the intense gap-scale electric field results in an $O(\unicode[STIX]{x1D6FF}^{-2})$ pressure, giving rise in turn to a large repulsive force. In that problem we find that the cylinder velocity perpendicular to the wall approaches a finite limit as $\unicode[STIX]{x1D6FF}\rightarrow 0$. We also discuss the role of ‘dielectrophoretic’ forces which are inevitable in the above semi-bounded configurations.

The mechanism of surface-charge convection, quantified by the electric Reynolds number $Re$, renders the Melcher–Taylor electrohydrodynamic model inherently nonlinear, with the electrostatic problem coupled to the flow. Because of this nonlinear coupling, the settling speed of a drop under a uniform electric field differs from that in its absence. This difference was calculated by Xu & Homsy (J. Fluid Mech., vol. 564, 2006, pp. 395–414) assuming small $Re$. We here address the same problem using a different route, considering the case where the applied electric field is weak in the sense that the magnitude of the associated electrohydrodynamic velocity is small compared with the settling velocity. As convection is determined at leading order by the well-known flow associated with pure settling, the electrostatic problem becomes linear for arbitrary value of $Re$. The electrohydrodynamic correction to the settling speed is then provided as a linear functional of the electric-stress distribution associated with that problem. Calculation of the settling speed eventually amounts to the solution of a difference equation governing the respective coefficients in a spherical harmonics expansion of the electric potential. It is shown that, despite the present weak-field assumption, our model reproduces the small-$Re$ approximation of Xu and Homsy as a particular case. For finite $Re$, inspection of the difference equation reveals a singularity at the critical $Re$-value $4S(1+R)(1+M)/(1+S)M$, wherein $R$, $S$ and $M$ respectively denote the ratios of resistivity, permittivity and viscosity values in the suspending and drop phases, as defined by Melcher & Taylor (Annu. Rev. Fluid Mech., vol. 1, 1969, pp. 111–146). Straightforward numerical solutions of this equation for electric Reynolds numbers smaller than the critical value reveal a non-monotonic dependence of the settling speed upon the electric field magnitude, including a transition from velocity enhancement to velocity decrement.