## Abstract

An initially uncharged ideally polarizable particle is freely suspended in an electrolyte solution in the vicinity of an uncharged dielectric wall. A uniform electric field is externally applied parallel to the wall, inflicting particle drift perpendicular to it. Assuming a thin Debye thickness, the electrokinetic flow is analysed for large particle–wall separations using reflection methods, thereby yielding an asymptotic approximation for the particle velocity. The leading-order correction term in that approximations stems from wall polarization.

## 1. Introduction

In the classical view of electrokinetics (Saville 1977), surface charge (and whence zeta potential) is considered a fixed physicochemical property of the solid–electrolyte interface, unaffected by the applied field. Implicit in this view is the assumption of a *non-polarizable* surface (‘a perfect dielectric’). The resulting mathematical model in the thin-Debye-layer limit is described by Keh & Anderson (1985).

This model is inapplicable to conducting solid surfaces, which are effectively *infinitely polarizable*. The analysis of electrokinetic flows about conducting particles began with Levich (1962) and was followed by other researchers in the former USSR (Simonov & Dukhin 1973; Shilov & Simonova 1981; Dukhin & Murtsovkin 1986; Gamayunov *et al*. 1986). On conducting walls the surface charge is mobile, and it arranges itself to ensure zero interior electric field. This polarization mechanism affects the zeta potential, which can no longer be considered a fixed quantity. A similar polarization mechanism also appears at electrokinetic flows about dielectric surfaces (Murtsovkin 1996; Nadal *et al*. 2002) which possess a finite polarizability (represented by their dielectric constant). Indeed, surface polarization was shown to be responsible to observed vortices around sharp corners in micro-channels (Thamida & Chang 2002; Yossifon *et al*. 2006). A thin-Debye-layer macroscale formulation for flows about polarizable surfaces was developed by Yossifon *et al*. (2007).

Squires & Bazant (2004) coined the term ‘induced-charge’ flows to describe the entire host of electrokinetic processes in which surface polarization affects the zeta potential. The archetypical configuration of induced-charge electro-osmosis consists of an initially uncharged ideally polarizable spherical particle that is suspended in an unbounded electrolyte and is exposed to an otherwise uniform faradaic current. It is common to assume that the particle boundary is chemically inert; thus, the dipolar charge distribution that is induced on it corresponds to zero net charge. The steady-state flow in this configuration was studied by Gamayunov *et al*. (1986). In view of the resulting flow symmetry, such a particle does not experience any hydrodynamic force, and would therefore remain stationary.

When the preceding symmetry is violated, it is possible to affect particle motion despite the zero net charge (Bazant & Squires 2004). The theoretical possibility of animating induced-charge electrophoresis has led to a series of theoretical investigations of non-spherically symmetric configurations. Using general symmetry arguments, Yariv (2005) discussed flows about arbitrary particle shapes. Squires & Bazant (2006) employed regular perturbations to analyse near spheres and near cylinders. These authors also considered other modes of asymmetry for spherical particles; the electrophoretic motion in one of these, Janus-type particles, was experimentally observed by Gangwal *et al*. (2008). Spheroids were studied by Saintillan *et al*. (2006*a*) in the slender limit and by Yossifon *et al*. (2007) in general. Spheroids exhibit many features that are absent in spherical geometries; having both fore–aft and axial symmetries, however, they do not experience electrophoresis when exposed to a uniformly applied field (Yariv 2005). This limitation motivated the recent investigation of arbitrarily shaped slender particles (Yariv 2008*b*); when lacking fore–aft symmetry, such particles do experience electrophoretic motion. Asymmetry can also be animated by the presence of neighbouring particles. Interactions between spherical particles were investigated using both analytic approximations (Dukhin & Murtsovkin 1986; Gamayunov *et al*. 1986) and numerical methods (Saintillan 2008). Saintillan *et al*. (2006*a*) used slender-body approximations to calculate the interactions between elongated spheroids; these were employed in the subsequent analyses of rod-like particle suspensions (Saintillan *et al*. 2006*b*; Rose *et al*. 2007).

Another category of asymmetric geometries comprises bounded configurations. This category is of special importance since all practical devices are bounded in one or more dimensions; a dielectric wall, for example, can represent the boundary of a microfluidic channel. The simplest scenario entails a dielectric plane wall, the applied current directed parallel to it. Even in the absence of electrokinetic flow, and despite its zero net charge, the particle experiences a net electric force that tends to repel it from the wall (Yariv 2006). It is plausible that the induced-charge electro-osmosis will result in an additional force along that direction; it was already speculated by Gangwal *et al*. (2008) that such a force may explain a recent theory–experiment discrepancy in the motion of Janus-type particles. The goal of this paper is to investigate this induced-charge phenomenon.

Wall effects were analysed by Zhao & Bau (2007) for a cylindrical particle. In the thin-Debye-layer limit, the electrostatic and flow problems were solved using eigenfunction expansions in bipolar coordinates. In principle, this procedure can be adapted to a spherical particle via an appropriate use of bi-spherical coordinates (see, e.g. Keh & Chen 1989). Since these eigenfunction expansions do not provide direct mathematical insight, we here adopt a different approach, following Keh & Anderson (1985). Rather than considering arbitrary particle–wall separations, we focus from the start upon the remote wall scenario. This allows to obtain closed-form analytic approximations, which, in turn, can be used in modelling of more complicated bounded systems. Our approach is motivated by the existing approximations for remote particle–particle interactions (Dukhin & Murtsovkin 1986; Gamayunov *et al*. 1986) which were recently improved by Saintillan (2008).

Towards this end, we consider the simplest particle–wall configuration, consisting of an initially uncharged ideally polarizable (i.e. perfectly conducting) spherical particle (radius *a*) that is suspended in a symmetric (valency , ionic strength ^{2}*n*_{∞}) electrolyte solution (viscosity *μ*, electrical permittivity *ϵ*) in the vicinity of an uncharged non-polarizable plane wall. At time zero, a uniform faradaic current is externally driven through the solution in a direction parallel to the wall.

Our interest lies in the motion of the particle following the transient period (Squires & Bazant 2004; Chu & Bazant 2006; Yossifon *et al*. in press) during which the induced Debye layer about it is formed. At temperature *T*, this layer is characterized by the Debye–Hückel parameter *κ*, defined by (*k* being Boltzmann's constant and *e* the elementary charge)(1.1)Throughout our investigation, we will assume that the Debye thickness 1/*κ* is small compared with particle size(1.2)

Following Keh & Anderson (1985), we introduce an iterative scheme that naturally handles the particle–wall geometry. When focusing upon the remote wall limit, it is necessary to calculate only several terms in that scheme. When limiting our attention to the leading-order term and to its leading-order correction, we find that the force experienced by the particle is not affected by Maxwell stresses. When focus lies at these asymptotic orders, it is possible to employ the Robin condition of Yossifon *et al*. (2007) so as to generalize the analysis to polarizable walls. It is found that a finite wall polarizability affects the leading-order correction to the force.

The paper is arranged as follows: In §2, we formulate the dimensionless electrokinetic problem. The iterative scheme is delineated in §3. The remote wall approximation is obtained in §4. In §5, we derive a generalization for polarizable walls. Conclusions appear in §6.

## 2. Problem formulation

The system that we consider is described in figure 1. It comprises an electrolyte solution that is bounded by a non-polarizable planar wall and a spherical conducting particle of radius *a*. The particle centre *O* is instantaneously positioned at distance *a*/*λ* (*λ*<1) from the wall. The system is exposed to a uniform and constant external electric field *E*_{∞}=*E*_{∞}** Ê** (

**being a unit vector in the field direction), which is applied parallel to the wall.**

*Ê*We employ a dimensionless notation, using *a*, *E*_{∞}, *aE*_{∞} and as the respective units of length, electric field, electric potential and stress; velocities are accordingly normalized by . It is convenient to employ a Cartesian coordinate system centred about *O*, with the *z*-axis lying perpendicular to wall (which is then given by *z*=−1/*λ*) and the *x*-axis lying in the applied field direction (** Ê**=

*ê*_{x}). In addition, we also employ spherical polar coordinates, the radial coordinate

*r*measured from

*O*and the polar angle

*θ*measured from the

*x*-axis. We analyse the electrokinetic flow using the thin-Debye-layer limit (1.2), where it is understood that the description in the preceding coordinates is a ‘coarse-grained’ one. Accordingly, the no-flux boundary condition and the Smoluchowski slip condition apply at both the sphere boundary

*r*=1 and the wall

*z*=−1/

*λ*.

The electric potential is governed by (i) Laplace's equation in the fluid domain(2.1)(ii) the no-flux condition on both the particle boundary(2.2)and the wall(2.3)and (iii) the far-field condition(2.4)The above Neumann-type boundary-value problem uniquely defines *φ* up to a physically meaningless integration constant. It is readily verified that the electric potential can be made an odd function of *x* by a proper choice of that constant.

The induced zeta potential on the particle is(2.5)in which *Φ* is the uniform particle potential.1 The value of *Φ* is determined from an integral constraint representing the zero net charge of the particle (Yariv 2005). In view of the oddness of *φ* and the odd dependence of the Debye-layer capacitance upon *ζ* (Yariv 2008*a*), that constraint is trivially satisfied by choosing *Φ*=0. We calculate the electrokinetic flow assuming a stationary particle. Once the loads on such a particle are calculated, the velocities of a comparable freely suspended particle are readily obtained using the known mobility relations of the sphere–wall configuration (Happel & Brenner 1965). The velocity field ** v** and the pressure field

*p*are calculated in an inertial reference system attached to the wall. Thus, the hydrodynamics are described by (i) the Stokes equations,(2.6)(ii) Smoluchowski's slip condition on the particle, which, upon using (2.5), appears as(2.7)(iii) the no-slip condition on the wall (representing the presumed zero zeta potential there)(2.8)and (iv) the condition of velocity decay at large distances from the particle.

Once the electric and velocity fields are evaluated, it is possible to calculate the force and torque (respectively normalized with and ) exerted on the stationary particle. These loads consist of (i) hydrodynamic contributions(2.9)which result from the tractions caused by the Newtonian stresses ( being the idem factor and ^{†} denoting transposition)(2.10)and from (ii) electric contributions(2.11)which result from the tractions caused by the Maxwell stresses(2.12)Our interest lies in the total force and torque, and .

Even without solving the governing equations, it is possible to use symmetry arguments so as to predict the force and torque directions. Since the electrical problem is linear and homogeneous in the constant vector ** Ê**, the electric potential must be linear in it. In view of the quadratic slip structure (2.7) and the linearity of the flow problem, it becomes clear that all the flow variables are quadratic in

**and then so must also be the hydrodynamic loads (2.9). These loads can therefore be represented in the invariant tensorial notation(2.13)in which is a third-order tensor and a third-order pseudo-tensor. These dimensionless coefficients can only depend upon the instantaneous configuration of the particle–wall system. This configuration introduces only a single constant vector: , a unit normal to the wall, which points into the fluid (). Thus, the only candidates for are as well as the three permutations of , while the only candidates for are the alternating pseudo-tensor**

*Ê***as well as and . In general, all of these candidates are multiplied by functions of**

*ϵ**λ*, the single scalar parameter in the problem. In view of the contraction with

**, it becomes evident that the sphere does not experience a hydrodynamic torque and that the hydrodynamic force is of the form(2.14)Identical arguments imply that vanishes2 and that is of the form . Lastly, the preceding tensorial arguments can be repeated for a freely suspended particle, showing that it must acquire a rectilinear velocity of the form(2.15)and no angular velocity.**

*ÊÊ*## 3. Iterative reflections

Following Keh & Anderson (1985), we employ an iterative reflection scheme. The electric potential is provided by the following series:(3.1)where we define(3.2)The potential *φ*^{(0)} is the solution in the absence of a wall. Specifically, (which automatically satisfies the no-flux condition (2.3) on the wall) corresponds to the applied field and is the dipole(3.3)required to satisfy the boundary condition (2.2) on the particle.

The harmonic corrections and for *n*>0 represent successive reflections that alternately satisfy the no-flux conditions on the two surfaces: The ‘wall correction’ decays at large distances from the wall and restores the no-flux condition (2.3) violated by (3.4)similarly, the ‘particle correction’ decays at large distances from the particle and restores the no-flux condition (2.2) violated by (3.5)

We also introduce the iterative expansion for the velocity field:(3.6)with a similar series for the pressure *p*. Each corresponding pair in these expansions separately satisfies the Stokes equations (2.6); then, due to the linearity of the hydrodynamic stress in ** v** and

*p*(see (2.10)), a similar series is automatically induced for

**.**

*σ*The first term in (3.6), , represents the flow in the absence of a wall. It decays at large distances from the particle, and is driven by the slip condition (cf. (2.7))(3.7)This field was calculated by Gamayunov *et al*. (1986) who obtained the quadrupolar profile(3.8)

The field (*n*≥1) decays at large distances from the wall and satisfies the boundary condition(3.9)which restores the null value (violated by ) of ** v** on the wall.

The field (*n*≥1) is split into two sub-fields (with a similar decomposition being applied to both and )(3.10)Both sub-fields satisfy the Stokes equations and decay at large distances from the particle; The sub-field , triggered by the distribution of on the particle, satisfies the boundary condition(3.11)the sub-field , triggered by the additional electrokinetic slip animated by *φ*^{(n)}, satisfies the boundary condition (cf. (3.7))(3.12)The decomposition (3.10) also applies for *n*=0 when it is understood that is null (i.e. ).

The iterative decomposition (3.6) directly induces a comparable decomposition for the hydrodynamic force,(3.13)in which we naturally define(3.14)Since the wall reflections are regular for *z*>−1/*λ*, vanishes inside the particle; thus, these reflections do not contribute to the hydrodynamic force.

In view of the boundary condition (3.11), the contribution is simply provided by Faxén's laws (Happel & Brenner 1965) applied upon the wall reflection that ‘triggered’ the field (3.15)In what follows, we will refer to as the force ‘provoked’ by .

In view of the quadratic dependence of the Maxwell stresses (2.12) upon the electric field, we define(3.16)

The contribution of to the electric force is(3.17)

## 4. Remote wall approximation

We focus upon the remote wall limit, *λ*≪1. While consecutive terms in the iterative representations are not asymptotically ordered; they eventually generate separate asymptotic expansions in two asymptotic regions. The first, characterized by the ‘particle scale,’ lies at the *O*(1) neighbourhood of the particle; the second, characterized by the ‘gap scale,’ lies at *O*(1/*λ*) distances from the particle. Following Ho & Leal (1974), the gap region is treated using the stretched coordinates(4.1)In these coordinates the wall is described by the plane *Z*=−1, while the particle boundary is the sphere *R*=*λ*.

In the particle scale, the leading-order approximation is provided by the solution of Gamayunov *et al*. (1986) for a particle in an unbounded fluid domain. This solution is highly symmetric and does not result in either a hydrodynamic or an electric force .3

Owing to the *r*^{–2} type decay of the (see (3.8)), it transforms from *O*(1) in the particle scale to *O*(*λ*^{2}) in the gap scale. In view of (3.9), it becomes evident that is also *O*(*λ*^{2}), and then, following Faxén's law (3.15), so must be the hydrodynamic force provoked by it. We will therefore focus upon obtaining a leading-order *O*(*λ*^{2}) approximation for *F* together with an *O*(*λ*^{3}) correction term.

Evaluating requires first expressingin terms of the gap-scale variables(4.2)The *O*(*λ*^{4}) error in the above expressions stems from terms that decay at an *r*^{−4} rate in the particle-scale description (3.8) of .

Since satisfies the Stokes equations, its Cartesian components can be expressed as Fourier transforms (Happel & Brenner 1965). Following Ho & Leal (1974) we express the Cartesian components ofin the form(4.3)Here, denotes a two-dimensional Fourier transform, defined generically by(4.4)*k*=(*ξ*^{2}+*η*^{2})^{1/2}; and *g*_{1}, *g*_{2} and *g*_{3} are arbitrary functions of *ξ* and *η*. These functions are determined by imposing (3.9) for *n*=1.

It is therefore necessary to express the various terms in (4.2) as Fourier transforms (evaluated at *Z*=−1). Manipulating the identity (Happel & Brenner 1965)(4.5)yields the following relations at *Z*=−1 (see Yariv & Miloh in press)(4.6)For future reference, we also find that(4.7)

Applying the boundary condition (3.9) for *n*=1 yields *g*_{1}, *g*_{2} and *g*_{3}; straightforward integration over the (*ξ*, *η*)-plane yields(4.8)In view of (4.1), the Laplacian of is *O*(*λ*^{4}); thus, Faxén's law (3.15) gives(4.9)

Consider now the contribution of , whose evaluation requires the calculation of and at the particle region. The first wall reflection represents a mirror dipole to (3.3), positioned at *z*=−2/*λ* (Keh & Anderson 1985). In the gap-scale variables(4.10)Expanding (4.10) into a Taylor series about *O* (Keh & Anderson 1985) yields at the particle region(4.11)To leading order, this expression represents a uniform electric field in the *x*-direction of magnitude *λ*^{3}/16. The following terms in the Taylor expansion are of progressively smaller asymptotic magnitude. At *O*(*λ*^{3}), the evaluation of is similar to that of ; thus, the leading-order term in the particle-scale expansion of is a dipole in the *x*-direction, identical to (see (3.3)) with a *λ*^{3}/16 multiplicative factor.

We therefore conclude that(4.12)Recall that the field , triggered by the quadratic interaction (3.7) in the leading-order potential *φ*^{(0)}, does not result in a force. It is therefore evident from (4.12) that the slip-driven field , triggered by quadratic interactions (3.12) in *φ*^{(0)} and *φ*^{(1)}, produces a force that is *O*(*λ*^{4}) at most. This must also be the order of magnitude of the electrical force , which also results from quadratic interactions in *φ*^{(0)} and *φ*^{(1)} (see (3.16) for *n*=1).4

Consider now the field , induced by the two components of : and . In the particle region, is *O*(*λ*^{3}); in view of its *r*^{−2} decay, it is *O*(*λ*^{5}) at the gap region; this is then the order of magnitude of the reaction to it in . Accordingly, we need only consider the effect of .

Expanding to a Taylor series about *O* yields(4.13)in which the leading-order term is *O*(*λ*^{2}). We are interested in the reaction of to that term—namely the disturbance caused by a sphere that is positioned within an *O*(*λ*^{2}) uniform stream in the *z*-direction. This reaction consists of two parts (Happel & Brenner 1965), both *O*(*λ*^{2}) in the particle region: the first is a Stokeslet that decays at an *r*^{−1} rate; the second is a dipole that decays like *r*^{−3}. At the gap region, these terms are *O*(*λ*^{3}) and *O*(*λ*^{5}). The leading-order *O*(*λ*^{3}) reaction in to is therefore triggered by the *O*(*λ*^{2}) Stokeslet of .

Thus, to obtain to *O*(*λ*^{3}), we only need to consider the Stokeslet of that is triggered by the leading-order term in (4.13), and then the reaction in to that Stokeslet. As a matter of fact, no calculations are required: to leading order, the ratio of the force provoked by the reaction in to the Stokeslet and that provoked by the leading-order uniform-stream term in (4.13) is 9*λ*/8: this is the well-known (Happel & Brenner 1965) leading-order wall-effect appearing in the classical drag problem of a sphere translating away from a wall (cf. (6.1)).5

Since the contribution of all the other reflections is *o*(*λ*^{3}), we conclude that(4.14)

## 5. Generalization to a polarizable wall

Implicit in the no-slip condition (2.8) on the dielectric wall is the assumption of zero zeta potential. This assumption is tantamount to that of an ideally non-polarizable wall. In reality, the dielectric wall material possesses a finite polarizability and a zeta potential can be induced at the wall–fluid interface (Squires & Bazant 2004). Here, we analyse the effect of the wall polarization upon the hydrodynamic force exerted on a stationary particle.

Consistently with the thin-Debye-layer limit, the zeta potential on the wall is simply(5.1)wherein is the electric potential within the wall. Since the interior of the dielectric wall is charge free, is harmonic. This potential needs to match the electric potential inside the induced Debye layer surrounding the wall. Assuming small zeta potentials, it was shown by Yossifon *et al*. (2007) that the requisite matching is equivalent to the macroscale Robin-type condition(5.2)Here, , where is the dielectric permittivity of the wall and 1/*κ* is the Debye thickness, see (1.1).

The common model of an ideally non-polarizable wall corresponds to vanishingly small *α*, whereby and the zeta potential vanishes. Within the thin-Debye layer regime (1.2) it seems plausible to assume small *α* even for polarizable walls, see Yossifon *et al*. (2006).6 When considering, however, the entire range of dielectric constants that appear in specific applications, we find that moderate *α*-values can appear as well. (For certain ceramic materials is quite large, see Rodriguez & Markx (2006).) In what follows, we follow Yossifon *et al*. (2007) and present a general analysis for arbitrary *α*-values.

It is natural to evaluate using the gap-scale variables, whereby condition (5.2) appears as(5.3)Then, in view of condition (5.3), the zeta potential is(5.4)The no-slip condition (2.8) is therefore modified to(5.5)

Just like *φ*, is an odd function of *X*. We postulate the iterative solution(5.6)where . The harmonic corrections (*n*≥1) are driven by corresponding reflections in *φ* through the Robin condition (5.3), that is(5.7)in addition, they are required to decay at large distances(5.8)The iterative expansion (5.6) affects a comparable expansion for ζ_{W} through (5.4).

Consider now the limit *λ*≪1. The gap-scale electric fields associated with and are both *O*(*λ*^{3}); it is therefore clear that , and then the slip on the wall, begin at this asymptotic order. Consequently, it is sufficient to calculate . Substitution of (3.3) and (4.10) into (5.7) for *n*=1 gives(5.9)To leading order, this is simply a Dirichlet condition at *Z*=−1. Solving the boundary-value problem to that order using Sine transforms yields:(5.10)This is a dipole centred about *O*; aside from having twice the magnitude, it is identical to (see (3.3)). Using (5.4), we then find(5.11)The small *O*(*λ*^{3}) zeta potential *a posteriori* justifies the use of condition (5.2).

In view of (5.5), the leading-order *O*(*λ*^{3}) wall slip results from interaction between the *O*(*λ*^{3}) wall zeta potential and the *O*(1) leading-order electric field in the bulk fluid, :(5.12)The velocity field generated by this slip condition is calculated using a Fourier representation, similar to that of (see (4.3)). The Fourier transform of the slip condition (5.12) is obtained from (4.7). Evaluation at *O* yields(5.13)The force provoked by is obtained using Faxén's formula (3.15); it is of magnitude(5.14)and it is directed parallel to the *z*-axis.

Up to *O*(*λ*^{3}), the force induced by the dielectric wall is simply provided by combining (4.14) and (5.14), the case of an ideally non-polarizable wall corresponding to *α*→0.

## 6. Concluding remarks

We have calculated the wall-induced force acting on a stationary particle. When the particle is freely suspended in the electrolyte, this force imparts it with the velocity required to keep it force free (see (2.15)). Multiplying the mobility (normalized with 1/*aμ*) of a spherical particle in a direction normal to solid wall (Happel & Brenner 1965)(6.1)by the sum of (4.14) and (5.14) yields(6.2)The leading-order term in this expression was independently found by Saintillan (in preparation). It is different from that calculated for a pair of spherical particles whose line of centres lies perpendicular to the applied field (Saintillan 2008); indeed, these two problems are not physically equivalent. Note that the *O*(*λ*) wall effect in (6.1) cancels out the contribution provoked by . This is to be expected, since that contribution represents a Stokeslet associated with a fixed particle; this Stokeslet must disappear when a force-free particle is considered.

In principle, it is possible to improve the approximation (6.2). It should be noted that once *O*(*λ*^{4}) terms are retained, this velocity becomes affected by the electric force (Yariv 2006), and does not formally qualify as ‘electrophoretic’.

The present investigation of a sphere–wall system was motivated by the bipolar calculation of Zhao & Bau (2007) for a two-dimensional cylinder–wall system. It may appear that the present iterative scheme could be applied to the comparable two-dimensional problem as well, thereby providing asymptotic formulae that can supplement the numerical results of Zhao & Bau (2007). Recall, however, that as *λ*→0 the iterative reflection method represents a limit process at which the particle–wall distance approaches infinity. In view of the Stokes paradox, no such limit exists in the two-dimensional problem: specifically, the two-dimensional equivalents of do not exist. This, of course, is implicit in the absence of a two-dimensional counterpart of Faxén's law.

## Footnotes

↵We here assume for simplicity that all the potential drop in the double layer occurs in its diffuse part, thereby neglecting the Stern layer voltage.

↵This is also evident from (2.2) and (2.12), which together imply that the Maxwell tractions are radial.

↵This is also evident from tensorial arguments: in the absence of a wall the system is istropic, whence no candidates exist for in (2.13).

↵The calculation of to this order was carried out by Yariv (2006).

↵The two problems are not completely analogous owing to the differences in particle motion. The mere effect of this motion on the velocity field, however, is the additional of a dipole term; this term decays as

*r*^{−3}and does not affect the*O*(*λ*) leading wall effect.↵Physically, the smallness of

*α*represents the intensive electric displacement within the thin Debye layer, as compared with the moderate displacement in the wall. The dominance of the former in Gauss's electrostatic boundary condition decouples the bulk electrostatics from the wall polarization.- Received August 3, 2008.
- Accepted October 20, 2008.

- © 2008 The Royal Society