2.1. Equations of motion

We consider a colloid adhered to a planar interface between two immiscible Newtonian fluids of viscosities $\mu _1$ and $\mu _2$, which are quiescent in the far field and together form an unbounded domain, as illustrated in figure 1. As described in § 1, we assume the resulting three-phase contact line is pinned, that is, it cannot move relative to the surface of the colloid. For simplicity, we further assume that the interface is flat. This assumption physically requires that (i) viscous stress due to flows generated by the particle are negligible compared to surface tension $\gamma$, which determines the equilibrium shape of the interface; (ii) the weight $mg$ of the colloid is also negligible compared to surface tension; and (iii) the amplitude of the undulations in the contact line are negligibly small compared to the size of the colloid. Requirement (i) is formally satisfied when $\mathinner {Ca} = \mu U / \gamma \ll 1$, where $\mathinner {Ca}$ is the capillary number, $\mu$ is the fluid viscosity and $U$ is the characteristic velocity of the colloid. For typical colloidal systems at air–aqueous or alkane–aqueous interfaces, $\mathinner {Ca} = O(10^{-7})$ to $O(10^{-5})$. Requirement (ii) is satisfied when $\mathinner {Bo} = mg a^{2} / \gamma \ll 1$, where $\mathinner {Bo}$ is the particle Bond number and $a$ is the characteristic length scale of the colloid. In general, requirement (iii) may not be satisfied. For isolated passive particles, nanometric contact-line distortions alter the capillary energy that traps colloids on interfaces (Stamou, Duschl & Johannsmann Reference Stamou, Duschl and Johannsmann2000), and thermally activated fluctuations at the contact line are hypothesized to alter dissipation in the interface (Boniello et al. Reference Boniello, Blanc, Fedorenko, Medfai, Mbarek, In, Gross, Stocco and Nobili2015). Neither effect is included here, but the results we present may form the basis for a perturbative method to treat the problem of undulated contact lines.

Figure 1. Driven and active colloids at interfaces. Panel (a) is a colloid adhered to a fluid interface by a pinned contact line and driven into motion by external force and torque fields. In response to this forcing, the colloid translates parallel to the interface at velocity $\boldsymbol U$ and rotates on the axis normal to the interface at angular velocity $\boldsymbol \varOmega$. Other motions are prohibited due to contact-line pinning and elevated surface tension. Panel (b) is a similar illustration of an active colloid; we take a motile bacterium as a natural example. Thrust generated by the rotating helical flagellum is balanced by drag due to viscous dissipation and capillary forces.

At the colloidal scale, we may neglect the effects of fluid inertia and assume the flow on either side of the interface is governed by the Stokes equations,

(2.1a,b)\begin{equation} \boldsymbol\nabla\boldsymbol\cdot{\boldsymbol\sigma} ={-} \boldsymbol\nabla p + \mu \nabla^{2} \boldsymbol u = \boldsymbol 0; \quad \boldsymbol\nabla\boldsymbol\cdot{\boldsymbol u} = 0, \end{equation}

where $\boldsymbol \sigma$ is the stress tensor, $\boldsymbol u$ is the fluid velocity, $p$ is the hydrodynamic pressure and $\boldsymbol \nabla$ is the gradient operator. The stress tensor is given by $\boldsymbol \sigma = -p {\boldsymbol{\mathsf{I}}} + \mu [ \boldsymbol \nabla \boldsymbol u + {(\boldsymbol \nabla \boldsymbol u)}^{\mathsf{T}} ]$, where ${\boldsymbol{\mathsf{I}}}$ is the identity tensor. These quantities vary with the position vector $\boldsymbol x = x_1 \hat {\boldsymbol \imath }_1 + x_2 \hat {\boldsymbol \imath }_2 + z \hat {\boldsymbol \imath }_3$. Let $V_1$, $V_2$ and $I$ denote the set of points in fluid 1, fluid 2 and on the interface, respectively. We assume that the viscosity changes abruptly across the interface as $\mu (z) = \mu _1 \mathbb {I}_{\mathbb {R}_+}(z) + \mu _2 \mathbb {I}_{\mathbb {R}_-}(z)$, where the indicator function $\mathbb {I}_P$ is unity if its argument is an element of $P$ but otherwise vanishes (e.g. $\mathbb {I}_{\mathbb {R}_+}$ is equivalent to the Heaviside step function). On the interface, (2.1a,b) satisfies the boundary conditions

(2.2a)\begin{gather}[{\boldsymbol u}]_I = \boldsymbol 0, \end{gather}

(2.2b)\begin{gather}\boldsymbol n \boldsymbol\cdot \boldsymbol u(\boldsymbol x \in I) = \boldsymbol 0 , \end{gather}

(2.2c)\begin{gather}\boldsymbol\nabla_{s}\boldsymbol\cdot{\boldsymbol\varsigma} + \boldsymbol n \boldsymbol\cdot [{\boldsymbol\sigma}]_I \boldsymbol\cdot {\boldsymbol{\mathsf{I}}}_{s} = \boldsymbol 0, \end{gather}

where $\boldsymbol n$ is the unit normal to the interface pointing into fluid 1 and $[\,f]_I (\boldsymbol x \in I) := (\lim _{z \to 0^{+}} - \lim _{z \to 0^{-}}) f(\boldsymbol x)$ denotes the ‘jump’ in some function $f = f(\boldsymbol x)$ across the interface going from fluid 2 to fluid 1. The first two conditions assert that the fluid velocity is continuous across $I$ (2.2a) and that fluid does not pass through the interface (2.2b). The last condition (2.2c) balances tangential stresses. Here, $\boldsymbol \varsigma = \boldsymbol \varsigma (\boldsymbol x \in I)$ is the surface stress tensor, ${\boldsymbol{\mathsf{I}}}_{s} = {\boldsymbol{\mathsf{I}}} - \boldsymbol {nn}$ is the surface projection tensor and $\boldsymbol \nabla _{s} = {\boldsymbol{\mathsf{I}}}_{s} \boldsymbol \cdot \boldsymbol \nabla$ is the surface gradient operator. Note that, since we assume that the interface is planar, $\boldsymbol n = \hat {\boldsymbol \imath }_3$ and $I$ is simply the set of points on $z=0$. Finally, as $|\boldsymbol x| \to \infty$, the fluid velocity and pressure gradient in either volume vanish, i.e. $\boldsymbol u(\boldsymbol x) \to \boldsymbol 0$ and $p(\boldsymbol x) \to p_\infty$.

2.2. Clean interface

We call an interface ‘clean’ if it is free of surfactant molecules. In the absence of temperature gradients, a clean interface is characterized by a uniform surface tension $\gamma _0$, and $\boldsymbol \varsigma (\boldsymbol x) = \gamma _0 {\boldsymbol{\mathsf{I}}}_{s}$. Then, $\boldsymbol \nabla _{s}\boldsymbol \cdot {\boldsymbol \varsigma }$ vanishes and (2.2c) reduces to

(2.3)\begin{equation} \boldsymbol n \boldsymbol\cdot [{\boldsymbol\sigma}]_I \boldsymbol\cdot {\boldsymbol{\mathsf{I}}}_{s} = \boldsymbol 0, \end{equation}

which states that the tangential stress on the fluid is continuous across the interface.

2.3. Incompressible interface

If surfactant is present, gradients in surfactant concentration due to flow exert Marangoni stresses on the surrounding fluids. At interfaces where $\mathinner {Ca} \ll 1$, these gradients need only be infinitesimal to balance viscous stresses due to colloid motion. As a result, the interface is constrained to surface-incompressible motion.

To derive the most conservative estimate for the effects of these Marangoni stresses, consider trace surfactant concentrations, for which the surfactant can be approximated as a two-dimensional ideal gas. We define the surface pressure as ${{\rm \pi} (\boldsymbol x \in I) = \gamma _0 - \gamma (\boldsymbol x \in I)}$. In this case, the dependence of the surface pressure on surfactant concentration ${\varGamma = \varGamma (\boldsymbol x~\in~I)}$ is given by $\partial {\rm \pi}/ \partial \varGamma = k_B T$, where $k_B$ is Boltzmann's constant and $T$ is temperature. Scaling the surface pressure by viscous stresses $\tilde {{\rm \pi} } = {\rm \pi}/ \bar \mu U$, where $\bar \mu = (\mu _1 + \mu _2)/2$ is the average surface viscosity, and letting $\tilde {\varGamma } = \varGamma / \bar \varGamma$, where $\bar \varGamma$ is the average surface concentration over the entire interface, we find

(2.4)\begin{equation} \tilde{\boldsymbol\nabla}_{s} \tilde{\rm \pi} = \frac{k_B T \bar\varGamma}{\bar\mu U} \tilde{\boldsymbol\nabla}_{s} \tilde{\varGamma} = \mathinner{Ma}\, \tilde{\boldsymbol\nabla}_{s} \tilde{\varGamma}, \end{equation}

where $\mathinner {Ma}$ is the dimensionless Marangoni number and $\tilde {\boldsymbol \nabla }_{s} = a\boldsymbol \nabla _{{s}}$. To evaluate $\mathinner {Ma}$, we consider typical parameter values for a colloid moving at $U = 10\ \mathrm {\mu }$m s$^{-1}$ at a hexadecane–water interface ($\gamma _0 \approx 50$ mN m$^{-1}$) in the surface-gaseous state. The surfactant concentration required to produce a 0.1 % decrease in the surface tension is approximately $\bar \varGamma = 2 \times 10^{3}$ molecules $\mathrm {\mu }$m$^{-2}$. Given $\bar \mu \approx 1$ mPa s, we estimate that $\mathinner {Ma} = O(10^{3})$. Thus, very small perturbations in $\varGamma$ generate sufficient Marangoni stress to balance viscous stresses due to motion of the colloid.

The large-$\mathinner {Ma}$ limit has the important consequence that the fluid interface behaves as incompressible layer ($\boldsymbol \nabla _{{s}}\boldsymbol \cdot {\boldsymbol u} = 0$). Assuming bulk-insoluble surfactant, the non-dimensionalized surfactant mass balance on the interface is

(2.5)\begin{equation} \tilde{\varGamma}(\boldsymbol x) \tilde{\boldsymbol\nabla}_{{s}} \boldsymbol\cdot \tilde{\boldsymbol u} + \mathinner{Ma}^{{-}1} (\tilde{\boldsymbol u} \boldsymbol\cdot \tilde{\boldsymbol\nabla}_{{s}}) \tilde{\rm \pi} = {(\mathinner{Ma}\,\mathinner{Pe}_{s})}^{{-}1} \tilde{\nabla}_{{s}}^{2} \tilde{\rm \pi}, \end{equation}

where $\tilde {\boldsymbol u} = \boldsymbol u / U$. Here, $\mathinner {Pe}_{s} = U a /D_{s}$ represents the ‘interfacial’ Péclet number, where ${{\textit {D}}}_{s}$ is the surface diffusivity of the adsorbed surfactant. Equation (2.5) implies that $\tilde {\boldsymbol \nabla }_{{s}} \boldsymbol \cdot \tilde {\boldsymbol u} \ll 1$ if $\mathinner {Ma} \gg 1$ and $\mathinner {Pe}_{s} \gtrsim \mathinner {Ma}^{-1}$. Assuming $a = 10\ \mathrm {\mu }$m and $D_{s} = 10^{2}\ \mathrm {\mu }\textrm {m}^{2}\ \textrm {s}^{-1}$ (a typical value for small molecule surfactants), we have $\mathinner {Pe}_{s} = O(1)$, so surfactant diffusion does not restore compressibility of the interface. At larger surfactant concentrations, the interface, populated by bulk-insoluble surfactants, generally departs from the surface-gaseous state. The interface generally remains incompressible in this case because, excluding phase transitions, $\partial \gamma / \partial \varGamma > k_B T$. Thus, we hereafter assume $\boldsymbol \nabla _{{s}} \boldsymbol \cdot \boldsymbol u = 0$ while discussing interfaces with surfactant. Dilute soluble surfactants also obey this constraint, as mass transport rates between the bulk and the interface are typically negligible. Note that we may express the Marangoni number as $\mathinner {Ma} = E / {\mathinner {Ca}}$, where $E = -(\bar \varGamma / \gamma ) (\partial \gamma / \partial \varGamma )$ is the Gibbs elasticity. Thus, interfacial incompressibility is the typical circumstance for interfacial flow at low capillary number (Bławzdziewicz, Cristini & Loewenberg Reference Bławzdziewicz, Cristini and Loewenberg1999).

Surfactants can also create surface-viscous stresses due to shearing motion of the interface. If we assume Newtonian behaviour, the interfacial stress tensor is given by

(2.6)\begin{equation} \boldsymbol\varsigma(\boldsymbol x) ={-}{\rm \pi}(\boldsymbol x) {\boldsymbol{\mathsf{I}}}_{s} + \mu_{s} [ \boldsymbol\nabla_{{s}}\boldsymbol u + {(\boldsymbol\nabla_{{s}}\boldsymbol u)}^{\mathsf{T}} ], \end{equation}

for $\boldsymbol x \in I$, where $\mu _{s}$ is the surface viscosity. Then, (2.6) and (2.2c) yield the tangential stress balance for an incompressible, surfactant-laden interface

(2.7)\begin{equation} - \boldsymbol\nabla_{{s}}{\rm \pi} + \mu_{s} \nabla_{{s}}^{2} \boldsymbol u + \boldsymbol n \boldsymbol\cdot [{\boldsymbol\sigma}]_I \boldsymbol\cdot {\boldsymbol{\mathsf{I}}}_{s} = \boldsymbol 0. \end{equation}

Equation (2.7) together with the incompressibility condition, $\boldsymbol \nabla _{{s}}\boldsymbol \cdot {\boldsymbol u} = 0$, are the Stokes equations for a two-dimensional Newtonian fluid being externally forced by bulk-viscous stresses.

3.1. Lorentz reciprocal theorem across an interface

The Lorentz reciprocal theorem provides a relation between the velocity and stress fields of two arbitrary Stokes flows. We may extend this theorem to two fluid regions separated by a clean or incompressible Newtonian interface as follows. Consider a region $V_\nu ^{*} \subset V_\nu$ that is fully contained in fluid $\nu$, where $\nu = 1 \text { or } 2$, as illustrated by figure 2. Let $(\boldsymbol u, \boldsymbol \sigma )$ and $(\boldsymbol u', \boldsymbol \sigma ')$ represent the velocity and stress fields of two different solutions to the inhomogeneous Stokes equations,

(3.1a,b)\begin{equation} \boldsymbol\nabla\boldsymbol\cdot{\boldsymbol\sigma} ={-}\boldsymbol f(\boldsymbol x); \quad \boldsymbol\nabla\boldsymbol\cdot{\boldsymbol\sigma'} ={-}\boldsymbol f'(\boldsymbol x), \end{equation}

for $\boldsymbol x \in V^{*}_\nu$, each subject to the conditions given by (2.2) at the interface. Here, the forcing functions $\boldsymbol f$ and $\boldsymbol f'$ aid in our generalization of the reciprocal theorem, as is customary in such derivations (Kim & Karrila Reference Kim and Karrila1991). We will later assert that these quantities vanish. Integration of $\boldsymbol \nabla \boldsymbol \cdot {(\boldsymbol \sigma \boldsymbol \cdot \boldsymbol u' - \boldsymbol \sigma ' \boldsymbol \cdot \boldsymbol u)}$ over $V_\nu ^{*}$ and application of the divergence theorem leads to the identity (see e.g. Kim & Karrila Reference Kim and Karrila1991)

(3.2)\begin{equation} \int_{V_\nu^{*}} [(\boldsymbol{\nabla}\boldsymbol{\cdot}{\boldsymbol{\sigma}}) \boldsymbol{\cdot} \boldsymbol{u}' - (\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{\sigma}') \boldsymbol\cdot \boldsymbol{u}] \,{\textrm{d}}V + \int_{\partial V_\nu^{*}} (\boldsymbol{\sigma} \boldsymbol{\cdot} \boldsymbol{u}' - \boldsymbol{\sigma}' \boldsymbol{\cdot} \boldsymbol{u}) \boldsymbol{\cdot} {\textrm{d}}{\boldsymbol {S}} = 0, \end{equation}

where $\partial V_\nu ^{*}$ denotes the boundary of $V_\nu ^{*}$, and ${\textrm {d}}{\boldsymbol S} = \hat {\boldsymbol {n}} \,{\textrm {d}}S$ points into $V_\nu ^{*}$. Substituting (3.1a,b) into (3.2) gives the Lorentz reciprocal theorem

(3.3)\begin{equation} \int_{V_\nu^{*}} [\,\boldsymbol f(\boldsymbol x) \boldsymbol\cdot \boldsymbol u' - \boldsymbol f'(\boldsymbol x) \boldsymbol\cdot \boldsymbol u] \,{\textrm{d}}V = \int_{\partial V_\nu^{*}} (\boldsymbol\sigma \boldsymbol\cdot \boldsymbol u' - \boldsymbol\sigma' \boldsymbol\cdot \boldsymbol u) \boldsymbol\cdot{\textrm{d}}{\boldsymbol S}. \end{equation}

Figure 2. A colloidal particle, depicted in the centre of the illustration, is surrounded by two arbitrary fluid regions $V^{*}_1 \subset V_1$ and $V^{*}_2 \subset V_2$, which meet at region $I^{*} \subset I$ on the interface. We assign the inward-facing normal vector $\hat {\boldsymbol {n}}$ to the boundaries of both of these regions. The unit normal to the interface $\boldsymbol n$ (sans hat) points in the $+z$ direction. The boundary of $V_1^{*}$ consists of the colloid surface $S_1$, the interfacial region $I^{*}$ and the remaining outer surface $R^{o}_1$, with the boundaries of $V_2^{*}$ being similarly labelled. The boundary of $I^{*}$ (dashed line), denoted $\partial I^{*}$, has the counterclockwise-oriented tangent vector $\hat {\boldsymbol {t}}$, and we define $\hat {\boldsymbol {m}} = \boldsymbol n \times \hat {\boldsymbol {t}}$, which points into $I^{*}$. The three-phase contact line $C$ comprises the inner part of $\partial I^{*}$.

We add the pair of equations given by (3.3) for each of the two fluid phases ($\nu = 1, 2$) to obtain

(3.4)\begin{align} & \int_{V^{*}} [\,\boldsymbol f(\boldsymbol x) \boldsymbol\cdot \boldsymbol u' - \boldsymbol f'(\boldsymbol x) \boldsymbol\cdot \boldsymbol u] \, {\textrm{d}}V \nonumber\\ &\quad = \oint_R (\boldsymbol\sigma \boldsymbol\cdot \boldsymbol u' - \boldsymbol\sigma' \boldsymbol\cdot \boldsymbol u) \boldsymbol\cdot{\textrm{d}}{\boldsymbol S} + \int_{I^{*}} ( [{\boldsymbol\sigma}]_I \boldsymbol\cdot \boldsymbol u'- [{\boldsymbol\sigma'}]_I \boldsymbol\cdot \boldsymbol u ) \boldsymbol\cdot \boldsymbol n \, {\textrm{d}}A, \end{align}

where $V^{*} := V_1^{*} \cup V_2^{*}$ is the union of the fluid volumes in each phase, $I^{*} := \partial V_1 \cap \partial V_2$ is the region (at the fluid interface) where $V_1^{*}$ and $V_2^{*}$ ‘touch’ and $R := \partial V^{*} \setminus I^{*}$ constitutes the remaining boundaries of $V_1^{*}$ and $V_2^{*}$ that are not adjacent to each other. For example, for the fluid region illustrated in figure 2, $R = S_1 \cup S_2 \cup R^{o}_1 \cup R^{o}_2$, which includes both the surfaces of the colloid (the inner surfaces of $V^{*}$) and the outer surfaces of $V^{*}$. Note that $V^{*}_1$ and $V^{*}_2$ are disjoint subsets of $V^{*}$; they do not include points on $I$. We interpret the integral over $V^{*}$ in (3.4) as being a sum of integrations over each of these subsets, and we similarly interpret the integral over $R$. In the integral over $I^{*}$, we have used the fact that the fluid velocities $\boldsymbol u$ and $\boldsymbol u'$ are continuous across the interface (2.2a). This term can be recast using the interfacial stress balance; contracting an arbitrary vector $\boldsymbol t^{*}$ directed tangent to the interface with (2.2c) gives

(3.5)\begin{equation} (\boldsymbol\nabla_{{s}}\boldsymbol\cdot{\boldsymbol\varsigma}) \boldsymbol\cdot \boldsymbol t^{*} + \boldsymbol n \boldsymbol\cdot [{\boldsymbol\sigma}]_I \boldsymbol\cdot \boldsymbol t^{*} + \boldsymbol f_{s} \boldsymbol\cdot \boldsymbol t^{*} = 0, \end{equation}

where we have included an additional external surface force density $\boldsymbol f_{s} = \boldsymbol f_{s}(\boldsymbol x \in I)$ on the interface. Since there is no fluid flux through interface, both $\boldsymbol u$ and $\boldsymbol u'$ are tangent to the interface for $\boldsymbol x \in I$. Thus, (3.4) and (3.5) give, after replacing $\boldsymbol t^{*}$ with $\boldsymbol u$,

(3.6)\begin{align} &\int_{V^{*}} [\, \boldsymbol f \boldsymbol\cdot \boldsymbol u' - \boldsymbol f' \boldsymbol\cdot \boldsymbol u ] \, {\textrm{d}}{V} + \int_{I^{*}} [\, \boldsymbol f_{s} \boldsymbol\cdot \boldsymbol u' - \boldsymbol f_{s}' \boldsymbol\cdot \boldsymbol u ] \, {\textrm{d}}A \nonumber\\ &\quad = \oint_{R} ( \boldsymbol\sigma \boldsymbol\cdot \boldsymbol u' - \boldsymbol\sigma' \boldsymbol\cdot \boldsymbol u ) \boldsymbol\cdot {\textrm{d}}{\boldsymbol S} - \int_{I^{*}} [ (\boldsymbol\nabla_{{s}}\boldsymbol\cdot{\boldsymbol\varsigma}) \boldsymbol\cdot \boldsymbol u' - (\boldsymbol\nabla_{{s}}\boldsymbol\cdot{\boldsymbol\varsigma'}) \boldsymbol\cdot \boldsymbol u ] \, {\textrm{d}}{A}, \end{align}

where $\boldsymbol \varsigma$ and $\boldsymbol \varsigma '$ are the interfacial stress tensors associated with the unprimed and primed flows, respectively.

3.2. Clean interface

For a clean interface, $\boldsymbol \varsigma = -{\boldsymbol{\mathsf{I}}}_{s} \gamma _0$ is constant, so the final integral in (3.6) vanishes:

(3.7)\begin{equation} \int_{V^{*}} [ \,\boldsymbol f \boldsymbol\cdot \boldsymbol u' - \boldsymbol f' \boldsymbol\cdot \boldsymbol u ] \, {\textrm{d}}{V} + \int_{I^{*}} [\, \boldsymbol f_{s} \boldsymbol\cdot \boldsymbol u' - \boldsymbol f_{s}' \boldsymbol\cdot \boldsymbol u ] \, {\textrm{d}}{A} = \oint_{R} ( \boldsymbol\sigma \boldsymbol\cdot \boldsymbol u' - \boldsymbol\sigma' \boldsymbol\cdot \boldsymbol u ) \boldsymbol\cdot {\textrm{d}}{\boldsymbol S}. \end{equation}

If we set $\boldsymbol f_{s} = \boldsymbol f_{s}' = \boldsymbol 0$, then the integral over $I^{*}$ in (3.7) also vanishes, which is the same as (3.3) with $\partial V_\nu ^{*}$ replaced by $R$.

3.3. Incompressible interface

Assuming an incompressible interface with Newtonian behaviour, as described by (2.6), there is a ‘surface’ reciprocal identity for the interface analogous to (3.3) given by

(3.8)\begin{equation} \int_{I^{*}} [ (\boldsymbol{\nabla}_{{s}}\boldsymbol{\cdot}{\boldsymbol{\varsigma}}) \boldsymbol{\cdot} \boldsymbol {u}' - ({\boldsymbol{\nabla}_{s}} \boldsymbol{\cdot}\boldsymbol\varsigma') \boldsymbol\cdot \boldsymbol u ] \, {\textrm{d}}A + \oint_{\partial I^{*}} { (\boldsymbol\varsigma \boldsymbol\cdot \boldsymbol u' - \boldsymbol\varsigma' \boldsymbol\cdot \boldsymbol u) } \boldsymbol\cdot \hat{\boldsymbol{m}} \,{\textrm{d}}C = 0, \end{equation}

where the contour integral is taken over the boundary of $I^{*}$, denoted $\partial I^{*}$. For the system of a particle on an interface illustrated in figure 2, $\partial I^{*}$ includes the contact line on the particle $C$ as its inner boundary. We assign $\partial I^{*}$ the unit tangent vector $\hat {\boldsymbol {t}}$ regarding $I^{*}$ as a counterclockwise-oriented surface. The unit vector $\hat {\boldsymbol {m}} = \boldsymbol n \times \hat {\boldsymbol {t}}$ points into the interfacial region $I^{*}$, meeting $\partial I^{*}$ at a right angle. Equations (3.8) and (3.6) yield

(3.9)\begin{align} & \int_{V^{*}} [\, \boldsymbol f \boldsymbol\cdot \boldsymbol u' - \boldsymbol f' \boldsymbol\cdot \boldsymbol u ] \, {\textrm{d}}{V} + \int_{I^{*}} [\, \boldsymbol f_{s} \boldsymbol\cdot \boldsymbol u' - \boldsymbol f_{s}' \boldsymbol\cdot \boldsymbol u ] \, {\textrm{d}}A \nonumber\\ &\quad = \oint_{R} ( \boldsymbol\sigma \boldsymbol\cdot \boldsymbol u' - \boldsymbol\sigma' \boldsymbol\cdot \boldsymbol u ) \boldsymbol\cdot {\textrm{d}}{\boldsymbol S} + \oint_{\partial I^{*}} ( \boldsymbol\varsigma \boldsymbol\cdot \boldsymbol u' - \boldsymbol\varsigma' \boldsymbol\cdot \boldsymbol u ) \boldsymbol\cdot \hat{\boldsymbol{m}} \,{\textrm{d}}C. \end{align}

Comparing (3.9) to the analogous equation for a clean interface (3.7), we see that the final integral on the right-hand side is new. This contour integral over the boundary of $I^{*}$ accounts for surface pressure gradients, or Marangoni stresses, that enforce the interfacial incompressibility constraint and, if $\mu _{s} > 0$, for surface-viscous dissipation. While we restrict ourselves to planar interfaces, (3.7) and (3.9) hold even if the interface is curved, given that it has the same shape in both the primed and unprimed flow problems.

4.1. Green's function

Due to the linearity of the Stokes equations (2.1a,b) and its boundary conditions for a clean interface (2.3), we may represent the velocity field due to a point force located at $\boldsymbol y = y_1 \hat {\boldsymbol \imath }_1 + y_2 \hat {\boldsymbol \imath }_2 + h \hat {\boldsymbol \imath }_3$ as $\boldsymbol u(\boldsymbol x) = {\boldsymbol{\mathsf{G}}}(\boldsymbol x, \boldsymbol y) \boldsymbol \cdot \boldsymbol F$, where ${\boldsymbol{\mathsf{G}}}$ is the Green's function for two fluids separated by a clean interface. This Green's function satisfies the (inhomogeneous) Stokes equations

(4.1a)\begin{gather} - \boldsymbol\nabla \boldsymbol P({\boldsymbol{\mathsf{G}}}; \boldsymbol x, \boldsymbol y) + \mu(z) \, \nabla^{2} {\boldsymbol{\mathsf{G}}}(\boldsymbol x, \boldsymbol y) = \begin{cases} \boldsymbol 0 & h = 0 \\ -{\boldsymbol{\mathsf{I}}} \delta_{\mathbb{R}^{3}}(\boldsymbol x - \boldsymbol y) & h \neq 0 \end{cases}, \end{gather}

(4.1b)\begin{gather}\boldsymbol\nabla\boldsymbol\cdot{{\boldsymbol{\mathsf{G}}}(\boldsymbol x, \boldsymbol y)} = \boldsymbol 0, \end{gather}

for $\boldsymbol x \in V_1 \cup V_2$. Equation (4.1) is subject to the far-field condition ${\boldsymbol{\mathsf{G}}} \to \boldsymbol 0$ as $|\boldsymbol x| \to \infty$, the interfacial stress balance

(4.2a)\begin{equation} {\boldsymbol{\mathsf{I}}}_{s} \boldsymbol\cdot [{\boldsymbol n \boldsymbol\cdot {\boldsymbol{\mathsf{T}}}({\boldsymbol{\mathsf{G}}}; \boldsymbol x, \boldsymbol y)}]_I = \begin{cases} -{\boldsymbol{\mathsf{I}}}_{s} \delta_{\mathbb{R}^{2}}(\boldsymbol x - \boldsymbol y) & h = 0 \\ \boldsymbol 0 & h \neq 0 \end{cases}, \end{equation}

and the kinematic conditions

(4.2b)\begin{equation} \boldsymbol t^{*} \boldsymbol\cdot [{{\boldsymbol{\mathsf{G}}}(\boldsymbol x, \boldsymbol y)}]_I = \boldsymbol n \boldsymbol\cdot {\boldsymbol{\mathsf{G}}}(\boldsymbol x \in I) = \boldsymbol 0. \end{equation}

Here, $\boldsymbol P({\boldsymbol{\mathsf{G}}};\!)$ is the (vectorial) pressure field associated with ${\boldsymbol{\mathsf{G}}}$, ${\boldsymbol{\mathsf{T}}}({\boldsymbol{\mathsf{G}}};\!)$ is the stress tensor associated with ${\boldsymbol{\mathsf{G}}}$ and $\delta _{\mathbb {R}^{n}}(\boldsymbol x)$ is the Dirac delta in $\mathbb {R}^{n}$. As expressed by (4.2a), for $h = 0$, we consider the point force to be exerted on the interface itself rather than on one of the fluids. Equations (4.1) and (4.2) follow directly from (2.1a,b) and (2.2) after factoring out $\boldsymbol F$ from both sides of each of these equations.

Solving (4.1) and (4.2) yields

(4.3)\begin{equation} {\boldsymbol{\mathsf{G}}}(\boldsymbol x, \boldsymbol y) = \begin{cases} [{\boldsymbol{\mathsf{J}}}(\boldsymbol x - \boldsymbol y) + {\boldsymbol{\mathsf{U}}}(\boldsymbol x, \boldsymbol y^{*})] / \mu(h) & zh \geq 0 \\ {\boldsymbol{\mathsf{V}}}(\boldsymbol x, \boldsymbol y) / \bar\mu & zh \leq 0 \\ {\boldsymbol{\mathsf{J}}}(\boldsymbol x - \boldsymbol y) \boldsymbol\cdot {\boldsymbol{\mathsf{I}}}_{s} / \bar\mu & h = 0 \end{cases}, \end{equation}

where $\boldsymbol y^{*} = (y_1, y_2, -h)$ is the reflection of $\boldsymbol y$ through $z=0$. Equation (4.3) expresses ${\boldsymbol{\mathsf{G}}}$ as a functional of the Oseen tensor ${\boldsymbol{\mathsf{J}}}(\boldsymbol x) = ({\boldsymbol{\mathsf{I}}} / |\boldsymbol x| - \boldsymbol x \boldsymbol x / |\boldsymbol x|^{3}) / 8{\rm \pi}$. The tensors ${\boldsymbol{\mathsf{U}}}$ and ${\boldsymbol{\mathsf{V}}}$ represent the hydrodynamic images necessary to satisfy continuity of tangential stress (4.2a) and continuity of velocity at the interface. These image systems are given by (Aderogba & Blake Reference Aderogba and Blake1978)

(4.4)\begin{gather} {\mathsf {U}}_{ij}(\boldsymbol x, \boldsymbol\xi) = (\delta^{\shortparallel}_{jk} - n_j n_k) {\mathsf {J}}_{ik}(\boldsymbol x - \boldsymbol\xi) - \frac{\mu(\boldsymbol\xi \boldsymbol\cdot \boldsymbol n)}{\bar\mu} {\mathsf {V}}_{ik}(\boldsymbol x, \boldsymbol\xi), \end{gather}

(4.5)\begin{gather}{\mathsf {V}}_{ij}(\boldsymbol x, \boldsymbol\xi) = \left\{ \delta^{\shortparallel}_{jk} + (\delta^{\shortparallel}_{jk} - n_j n_k) \left[ (\boldsymbol\xi \boldsymbol\cdot \boldsymbol n) n_l \frac{\partial}{\partial\xi_k} - \frac 12 (\boldsymbol\xi \boldsymbol\cdot \boldsymbol n)^{2} \delta_{kl} \nabla^{2} \right] \right\} {\mathsf {J}}_{il}(\boldsymbol x - \boldsymbol\xi), \end{gather}

where $\delta _{jk}$ is the Kronecker delta and $\delta ^{\shortparallel }_{jk} = \delta _{jk} - n_j n_k$. The tensor indices $i,j,k,l \in \{1,2,3\}$ follow the Einstein summation convention.

Equation (4.3) partitions ${\boldsymbol{\mathsf{G}}}(\boldsymbol x, \boldsymbol y)$ into three cases: $zh \geq ~0$, where $\boldsymbol x$ and $\boldsymbol y$ are in the same fluid; $zh \leq ~0$, where $\boldsymbol x$ and $\boldsymbol y$ are in different fluids; and $h = 0$ for all $z$, where $\boldsymbol y \in I$. Without loss of generality, if we let the point force at $\boldsymbol y$ be in the upper fluid ($h > 0$), then a Stokeslet, the fundamental solution to the Stokes equations in an unbounded fluid given by ${\boldsymbol{\mathsf{J}}} / 8 {\rm \pi}\mu _1$, is induced at this point. Examining, (4.5), the flow in the lower fluid ($z < 0$) comprises three image flows: a Stokeslet parallel to the interface, a Stokeslet dipole, and a degenerate Stokes quadrupole (a source–sink doublet), all of which have their singular points at $\boldsymbol y$. These images are depicted on the upper right-hand (cyan) side of figure 3. The image system ${\boldsymbol{\mathsf{U}}}$ for the upper fluid (4.4) is similar except that the image singularities are located at the image point $\boldsymbol y^{*}$, depicted on the lower left-hand (yellow) side of figure 3. This image system additionally includes a copy of the original forcing Stokeslet reflected through $z=0$.

Figure 3. A point force of magnitude $F = |\boldsymbol F|$ parallel to the interface induces the image system illustrated above, as expressed by (4.3). The upper left and lower right portions of the figure represent the physical fluid phases while the upper right and lower left are fictitious image domains that contain singularities acting to satisfy the boundary conditions on the interface (4.2). The image singularities in each image domain are depicted separately for clarity, but they all act at the same image point, i.e. $\boldsymbol y^{*}$ for fluid 1 and $\boldsymbol y$ for fluid 2.

Finally, we note two properties of ${\boldsymbol{\mathsf{G}}}$ that will be useful in the analysis that follows. First, it is self-adjoint,

(4.6)\begin{equation} {\boldsymbol{\mathsf{G}}}(\boldsymbol x, \boldsymbol y) = {\boldsymbol{\mathsf{G}}}^{\mathsf{T}}(\,\boldsymbol y, \boldsymbol x), \end{equation}

which may be proven using (3.7) (see appendix A) or directly verified from (4.3). The second property concerns the limiting behaviour of ${\boldsymbol{\mathsf{G}}}$. As $|\boldsymbol x|$ becomes large for fixed $|\boldsymbol y|$, ${{\boldsymbol{\mathsf{G}}}(\boldsymbol x, \boldsymbol y) \boldsymbol \cdot \boldsymbol F}$ effectively appears as a Stokeslet and decays as ${|\boldsymbol x - \boldsymbol y|}^{-1} \sim {|\boldsymbol x|}^{-1}$; the image Stokes dipole and degenerate quadrupole terms, contained in ${\boldsymbol{\mathsf{U}}}$ and ${\boldsymbol{\mathsf{V}}}$, do not affect the far-field behaviour of ${\boldsymbol{\mathsf{G}}}$ because their spatial decay is more rapid than that of the Stokeslet. An exception occurs when $\boldsymbol F$ points directly away from the interface, in which case ${\boldsymbol{\mathsf{G}}} \boldsymbol \cdot \boldsymbol F$ reduces to an effective stresslet of strength $h|\boldsymbol F| \mu (h) / \bar \mu$ for $|\boldsymbol x| \gg |\boldsymbol y|$ and decays as ${|\boldsymbol x|}^{-2}$ (Aderogba & Blake Reference Aderogba and Blake1978). By (4.6), the decay behaviour of ${\boldsymbol{\mathsf{G}}}$ for fixed $\boldsymbol x$ as $\boldsymbol y$ is made large reflects the behaviour for fixed $\boldsymbol y$ as $\boldsymbol x$ is made large; ${\boldsymbol{\mathsf{G}}}(\boldsymbol x, \boldsymbol y) \sim {|\boldsymbol y|}^{-1}$ for $\boldsymbol y \gg \boldsymbol x$.

4.2. Boundary integral equation

Using the Green's function (4.3) as the ‘primed’ flow field in the reciprocal relation (3.7), we may generate a boundary integral equation for an object at the interface. Consider the interfacially trapped colloid, illustrated in figure 2, whose upper surface $S_1$ is in contact with fluid 1 and whose lower surface $S_2$ is in contact with fluid 2. An arbitrary volume of fluid $V^{*} = V^{*}_1 \cup V^{*}_2$ surrounds the colloid, which is bounded by $S_1$ and $S_2$ as well as the outer fluid surfaces $R^{o}_1$ and $R^{o}_2$. We make the following substitutions into (3.7): $\boldsymbol u'(\boldsymbol x) \to {\boldsymbol{\mathsf{G}}}(\boldsymbol x, \boldsymbol y)$, $\boldsymbol \sigma '(\boldsymbol x) \to {\boldsymbol{\mathsf{T}}}({\boldsymbol{\mathsf{G}}}; \boldsymbol x, \boldsymbol y)$, $\boldsymbol f' \to {\boldsymbol{\mathsf{I}}} \delta _{\mathbb {R}^{3}}(\boldsymbol x - \boldsymbol y)$, and $\boldsymbol f_{s}' \to {\boldsymbol{\mathsf{I}}} \delta _{\mathbb {R}^{2}}(\boldsymbol x - \boldsymbol y)$ to find

(4.7)\begin{align} &\int_{V*} [\,{ \boldsymbol f \boldsymbol\cdot {\boldsymbol{\mathsf{G}}}(\boldsymbol x, \boldsymbol y) - {\boldsymbol{\mathsf{I}}} \delta_{\mathbb{R}^{3}}(\boldsymbol x - \boldsymbol y) \boldsymbol\cdot \boldsymbol u(\boldsymbol x) }] \, {\textrm{d}}{V(\boldsymbol x)} + \int_{I^{*}} [\, { \boldsymbol f_{s} \boldsymbol\cdot {\boldsymbol{\mathsf{G}}}(\boldsymbol x, \boldsymbol y) - {\boldsymbol{\mathsf{I}}}_{s} \delta_{\mathbb{R}^{2}}(\boldsymbol x - \boldsymbol y) \boldsymbol\cdot \boldsymbol u }] \, {\textrm{d}}{A(\boldsymbol x)} \nonumber\\ &\quad= \oint_{R} \bigl\{\hat{\boldsymbol{n}}(\boldsymbol{x}) \boldsymbol\cdot {\boldsymbol\sigma} (\boldsymbol{x}) \boldsymbol\cdot \boldsymbol{\mathsf{G}}(\boldsymbol{x}, \boldsymbol{y}) - \boldsymbol{u}(\boldsymbol{x}) \boldsymbol\cdot \bigl[\hat{\boldsymbol{n}}(\boldsymbol{x}) \boldsymbol\cdot \boldsymbol{\mathsf{T}}(\boldsymbol{\mathsf{G}}; \boldsymbol{x}, \boldsymbol{y})\bigr]\bigr\} {\textrm{d}}{S(\boldsymbol{x})}.\end{align}

We assert that the external force densities $\boldsymbol f$ and $\boldsymbol f_{s}$ vanish in (4.7). Using the identity $\int _\varOmega \delta _{\mathbb {R}^{n}}(\boldsymbol x - \boldsymbol y) \, f(\boldsymbol x) \,{\textrm {d}}^{n}{\boldsymbol x} = \mathbb {I}_{\varOmega }(\,\boldsymbol y) \, f(\,\boldsymbol y)$, where $f$ and $\varOmega$ are arbitrary, (4.7) becomes

(4.8)\begin{align} &\mathbb{I}_{V^{*}}(\,\boldsymbol y) \, \boldsymbol u(\,\boldsymbol y) + \mathbb{I}_{I^{*}}(\,\boldsymbol y) \, {\boldsymbol{\mathsf{I}}}_{s} \boldsymbol\cdot \boldsymbol u(\,\boldsymbol y) \nonumber\\ &\quad = - \oint_R \bigl\{\hat{\boldsymbol{n}}(\boldsymbol{x}) \boldsymbol\cdot {\boldsymbol\sigma} (\boldsymbol{x}) \boldsymbol\cdot \boldsymbol{\mathsf{G}}(\boldsymbol{x}, \boldsymbol{y})- \boldsymbol{u}(\boldsymbol{x}) \boldsymbol\cdot \bigl[\hat{\boldsymbol{n}}(\boldsymbol{x}) \boldsymbol\cdot \boldsymbol{\mathsf{T}}(\boldsymbol{\mathsf{G}}; \boldsymbol{x}, \boldsymbol{y}) \bigr] \bigr\} {\textrm{d}}{S(\boldsymbol{x})}. \end{align}

The first term on the left-hand side of (4.8) gives the velocity field (as a function of $\boldsymbol y$) whenever $\boldsymbol y \in V^{*}$ (i.e. $\boldsymbol y$ is in either $V^{*}_1$ or $V^{*}_2$, not including points on the interface) and elsewhere vanishes. The following term is complementary and vanishes unless $\boldsymbol y$ lies right on $I^{*}$; the surface projection ${\boldsymbol{\mathsf{I}}}_{s}$ has no effect here due to the no-penetration condition (2.2b).

In the limit that $V^{*} \to V$ and $I^{*} \to I$, such that the shaded regions in figure 2 grow to fill the entire domain, with $R^{\text o}_1$ and $R^{\text o}_2$ becoming arbitrarily far away from the colloid, we find that (4.8) gives the boundary integral representation for the velocity field,

(4.9)\begin{equation} \boldsymbol u(\boldsymbol x) ={-}\oint_{S_{c}} {\boldsymbol{\mathsf{G}}}(\boldsymbol x, \boldsymbol y) \boldsymbol\cdot [\boldsymbol\sigma \boldsymbol\cdot \hat{\boldsymbol{n}} ](\,\boldsymbol y) \,{\textrm{d}}S(\,\boldsymbol y) + \oint_{S_{c}} {[\boldsymbol u \hat{\boldsymbol{n}}]}(\,\boldsymbol y) \odot {\boldsymbol{\mathsf{T}}} ({\boldsymbol{\mathsf{G}}}; \boldsymbol y, \boldsymbol x) \,{\textrm{d}}S(\,\boldsymbol y), \end{equation}

where $S_{c} = S_1 \cup S_2$ represents the surface of the colloid and the operator ‘$\odot$’ denotes complete contraction of its operands, e.g. $({\boldsymbol{\mathsf{A}}} \odot {\boldsymbol{\mathsf{B}}})_{j_1 \dots j_m} = {\mathsf {A}}_{i_1 \dots i_n} {\mathsf {B}}_{i_n \dots i_1 j_1 \dots j_m}$ if ${\boldsymbol{\mathsf{A}}}$ is the tensor of lower rank and $({\boldsymbol{\mathsf{A}}} \odot {\boldsymbol{\mathsf{B}}})_{j_1 \cdots j_m} = {\mathsf {A}}_{j_1 \cdots j_m i_1 \cdots i_n} {\mathsf {B}}_{i_n \cdots i_1}$ if ${\boldsymbol{\mathsf{B}}}$ is the tensor of lower rank. We have exchanged $\boldsymbol y$ and $\boldsymbol x$ going from (4.8) to (4.9) to make $\boldsymbol x$ be the observation point of $\boldsymbol u(\boldsymbol x)$ and $\boldsymbol y$ be the integration variable. We have also used the self-adjoint property of ${\boldsymbol{\mathsf{G}}}$ (4.6) in the first term on the right-hand side of (4.9). The convergence of (4.8) to (4.9) follows from the decay behaviour of ${\boldsymbol{\mathsf{G}}}(\boldsymbol x, \boldsymbol y)$ and from the quiescent state of the fluid far from the colloid. Equation (4.9) is valid as long as the colloid does not deform in a manner that would distort the flat shape of the pinned contact line.

Equation (4.9) is similar in form and interpretation to the boundary integral equation for Stokes flows that appears in standard textbooks (see e.g. Kim & Karrila Reference Kim and Karrila1991; Pozrikidis Reference Pozrikidis1992). Indeed, (4.9) is derived in an analogous manner using the generalized reciprocal relation (3.7). The key property of (4.9) is that, by construction, integrals over the interface itself do not appear because ${\boldsymbol{\mathsf{G}}}$ and ${\boldsymbol{\mathsf{T}}}$ implicitly account for transmission of hydrodynamic stresses through the interface. This property allows for straightforward generation of the multipole expansion in the following section.

4.3. Multipole expansion

To generate a multipole expansion for $\boldsymbol u(\boldsymbol x)$, we replace ${\boldsymbol{\mathsf{G}}}(\boldsymbol x, \boldsymbol y)$ and ${\boldsymbol{\mathsf{T}}}({\boldsymbol{\mathsf{G}}}; \boldsymbol x, \boldsymbol y)$ in (4.9) with their Taylor series in $\boldsymbol y$ about an point on the interface as near as possible to the centre of the colloid, which we designate as the origin $\boldsymbol 0$. This process is slightly complicated by the piecewise nature of ${\boldsymbol{\mathsf{G}}}$ as $\boldsymbol y$ passes from one side of the interface to the other. In particular, certain components of $\boldsymbol \nabla _{\boldsymbol y} {\boldsymbol{\mathsf{G}}}(\boldsymbol x, \boldsymbol y)$ contain a jump discontinuity over the interface at $z=0$. This difficulty is overcome by separating each integral in (4.9) into one over $S_1$ and another over $S_2$, so that the integrand is continuous over each of these surfaces. Letting $\boldsymbol u^{(1)}$ and $\boldsymbol u^{(2)}$ denote the contributions from integration over $S_1$ and $S_2$, respectively, we may write the expansion as $\boldsymbol u = \boldsymbol u^{(1)} + \boldsymbol u^{(2)}$, where

(4.10)\begin{align} \boldsymbol u^{(1)}(\boldsymbol x) &={-}\sum_{n=0}^{\infty} \frac {1}{n!} \left( \int_{S_1} [\hat{\boldsymbol{n}} \boldsymbol\cdot \boldsymbol\sigma](\,\boldsymbol y)\,\boldsymbol{y}^{\boldsymbol\otimes n} {\textrm{d}}S(\,\boldsymbol y) \right) \odot \left( \lim_{\boldsymbol y \to \boldsymbol 0^{+}} \boldsymbol\nabla_{\boldsymbol y}^{\boldsymbol\otimes n} {\boldsymbol{\mathsf{G}}}^{\mathsf{T}}(\boldsymbol x, \boldsymbol y) \right) \nonumber\\ &\quad + \sum_{n=0}^{\infty} \frac {1}{n!} \left( \int_{S_1} [\boldsymbol u \hat{\boldsymbol{n}}](\,\boldsymbol y)\,\boldsymbol{y}^{\boldsymbol\otimes n} {\textrm{d}}S(\,\boldsymbol y) \right) \odot \left( \lim_{\boldsymbol y \to \boldsymbol 0^{+}} \boldsymbol\nabla_{\boldsymbol y}^{\boldsymbol\otimes n} {\boldsymbol{\mathsf{T}}}({\boldsymbol{\mathsf{G}}}; \boldsymbol y, \boldsymbol x) \right), \end{align}

and

(4.11)\begin{align} \boldsymbol u^{(2)}(\boldsymbol x) &={-}\sum_{n=0}^{\infty} \frac 1{n!} \left( \int_{S_2} [\hat{\boldsymbol{n}} \boldsymbol\cdot \boldsymbol\sigma](\,\boldsymbol y)\,\boldsymbol{y}^{\boldsymbol\otimes n} {\textrm{d}}S(\,\boldsymbol y) \right) \odot \left( \lim_{\boldsymbol y \to \boldsymbol 0^{-}} \boldsymbol\nabla_{\boldsymbol y}^{\boldsymbol\otimes n} {\boldsymbol{\mathsf{G}}}^{\mathsf{T}}(\boldsymbol x, \boldsymbol y) \right) \nonumber\\ &\quad + \sum_{n=0}^{\infty} \frac 1{n!} \left( \int_{S_2} [\boldsymbol u \hat{\boldsymbol{n}}](\,\boldsymbol y)\,\boldsymbol{y}^{\boldsymbol\otimes n} {\textrm{d}}S(\,\boldsymbol y) \right) \odot \left( \lim_{\boldsymbol y \to \boldsymbol 0^{-}} \boldsymbol\nabla_{\boldsymbol y}^{\boldsymbol\otimes n} {\boldsymbol{\mathsf{T}}}({\boldsymbol{\mathsf{G}}}; \boldsymbol y, \boldsymbol x) \right). \end{align}

Here, $\boldsymbol{y}^{\boldsymbol\otimes n} = \boldsymbol y \boldsymbol y \cdots$ ($n$ times) denotes the $n$-fold tensor product and $\boldsymbol \nabla _{\boldsymbol y}^{\boldsymbol \otimes n}$ similarly denotes the $n$-fold gradient operator. Writing ${\boldsymbol{\mathsf{T}}}$ in terms of ${\boldsymbol{\mathsf{G}}}$ as

\[ {\mathsf {T}}_{ijk}({\boldsymbol{\mathsf{G}}}; \boldsymbol y, \boldsymbol x) ={-}\delta_{ij} P_k({\boldsymbol{\mathsf{G}}}; \boldsymbol y, \boldsymbol x) + \mu(h) \left( \frac{\partial {\mathsf {G}}_{kj}(\boldsymbol x, \boldsymbol y)}{\partial y_i} + \frac{\partial {\mathsf {G}}_{ki}(\boldsymbol x, \boldsymbol y)}{\partial y_j} \right), \]

and collecting terms in ${\boldsymbol{\mathsf{G}}}$, ${\boldsymbol \nabla }_{\boldsymbol {y}} {\boldsymbol{\mathsf{G}}}$ and so on for higher-order gradients of ${\boldsymbol{\mathsf{G}}}$, we arrive at the multipole expansion,

(4.12)\begin{equation} \boldsymbol u(\boldsymbol x) = \boldsymbol u^{m0}(\boldsymbol x) + \boldsymbol u^{m1}(\boldsymbol x) + \boldsymbol u^{m2}(\boldsymbol x) + \text{h.o.t}, \end{equation}

where $\boldsymbol u^{m0}$ is the force monopole (zeroth) moment, $\boldsymbol u^{m1}$ is the force-dipole (first) moment, $\boldsymbol u^{m2}$ is the quadrupole (second) moment, and so on for higher-order terms (h.o.t.). In particular, these first three moments are given by

(4.12a)\begin{gather} u^{m0}_i(\boldsymbol x) = F^{(1)}_i {\mathsf {G}}_{ij}(\boldsymbol x, \boldsymbol 0^{+}) + F^{(2)}_i {\mathsf {G}}_{ij}(\boldsymbol x, \boldsymbol 0^{-}), \end{gather}

(4.12b)\begin{gather}u^{m1}_i(\boldsymbol x) = {\mathsf {D}}^{(1)}_{jk} \frac{\partial {\mathsf {G}}_{ij}}{\partial y_k}(\boldsymbol x, \boldsymbol 0^{+}) + {\mathsf {D}}^{(2)}_{jk} \frac{\partial {\mathsf {G}}_{ij}}{\partial y_k}(\boldsymbol x, \boldsymbol 0^{-}), \end{gather}

(4.12c)\begin{gather}u^{m2}_i(\boldsymbol x) = {\mathsf {Q}}^{(1)}_{jkl} \frac{\partial^2 {\mathsf {G}}_{ij}}{\partial y_l \partial y_k}(\boldsymbol x, \boldsymbol 0^{+}) + {\mathsf {Q}}^{(2)}_{jkl} \frac{\partial^2{\mathsf {G}}_{ij}}{\partial y_l \partial y_k}(\boldsymbol x, \boldsymbol 0^{-}), \end{gather}

where $\boldsymbol F^{(\nu )}$, ${\boldsymbol{\mathsf{D}}}^{(\nu )}$ and ${\boldsymbol{\mathsf{Q}}}^{(\nu )}$ are the monopole, dipole and quadrupole coefficients for fluid $\nu \in \{1,2\}$, respectively. The shorthand notation $\boldsymbol 0^{+}$ indicates the limit as $\boldsymbol y$ approaches $\boldsymbol 0$ from above the interface (i.e. from fluid 1). Similarly, $\boldsymbol 0^{-}$ indicates the limit as $\boldsymbol y$ approaches $\boldsymbol 0$ from below. In (4.12b), we have assumed, for simplicity, that the colloid does not grow or shrink in volume so that there is no source or sink flow from the origin. Note that if the colloid is wholly immersed in one fluid, then the multipole coefficients for the other fluid vanish.

At distances far enough from the colloid that points on the colloid surface are virtually indistinguishable from $\boldsymbol 0$, $|\boldsymbol x| \gg a$, the leading terms of (4.12) closely approximate $\boldsymbol u(\boldsymbol x)$. Recall that ${\boldsymbol{\mathsf{G}}}(\boldsymbol x, \boldsymbol y) \sim {|\boldsymbol x|}^{-1}$ for $|\boldsymbol x| \gg |\boldsymbol y|$. It follows that $\boldsymbol u^{m0}(\boldsymbol x) \sim r^{-1}$, where $r = |\boldsymbol x|$. Each successive multipole moment involves a higher-order gradient of ${\boldsymbol{\mathsf{G}}}$. Thus, $\boldsymbol u^{m1}(\boldsymbol x) \sim r^{-2}$, $\boldsymbol u^{m2}(\boldsymbol x) \sim r^{-3}$ and so on for higher-order moments. The lowest-order term with a non-zero coefficient dominates the far-field flow. This behaviour is analogous to that of the multipole expansion for objects in a bulk fluid.

4.3.1. Monopole moment

The monopole moment corresponds to a point force exerted at the interface, which follows intuitively from the fact that at large distances $r \gg a$, the colloid is indistinguishable from a single point at the interface. The functional form of the flow is therefore just that of the Green's function ${\boldsymbol{\mathsf{G}}}$. The prefactors appearing in (4.12a) are given by

(4.13)\begin{equation} \boldsymbol F^{(\nu)} ={-}\int_{S_\nu} \boldsymbol\sigma \boldsymbol\cdot \hat{\boldsymbol{n}} \, {\textrm{d}}S, \end{equation}

which is the force exerted on fluid $\nu \in \{1,2\}$ due to motion of the colloid. There is no need to keep the separate limits on the right-hand side of (4.12a) because ${\boldsymbol{\mathsf{G}}}(\boldsymbol x, \boldsymbol y)$ is continuous as $\boldsymbol y$ is moved across the interface for fixed $\boldsymbol x$. This property is not immediately obvious given the possible viscosity difference between the fluids. Recall, however, the boundary condition (4.2b) that demands continuity of ${\boldsymbol{\mathsf{G}}}$ as $\boldsymbol x$ is brought across the interface for fixed $\boldsymbol y$. Since ${\boldsymbol{\mathsf{G}}}$ is self-adjoint (4.6), continuity in $\boldsymbol x$ implies continuity in $\boldsymbol y$. Indeed, one may verify directly that all three cases in (4.3) are redundant for $h \to 0^{\pm }$.

Equation (4.3) in (4.12a) yields the monopole moment as

(4.14)\begin{equation} u^{m0}_i(\boldsymbol x) = \frac{1}{\bar\mu} F_k \delta^{\shortparallel}_{kj} {\mathsf {J}}_{ij}(\boldsymbol x), \end{equation}

where $\boldsymbol F = \boldsymbol F^{(1)} + \boldsymbol F^{(2)}$ is the total force exerted on both fluids. Equation (4.14) shows that $\boldsymbol u^{m0}$ is indistinguishable from a Stokeslet in an unbounded fluid of viscosity $\bar \mu$ associated with the effective force $\boldsymbol F \boldsymbol \cdot {\boldsymbol{\mathsf{I}}}_{s}$. The component of $\boldsymbol F$ normal to the interface does not contribute to the flow at leading order due to the presence of the interface. The ‘viscosity-averaged’ Stokeslet represented by (4.14) possesses an axis of symmetry lying in the interfacial plane. The tangential shear stress therefore vanishes at $z = 0$, and (2.3) is trivially satisfied. More generally, we will find that any mode with mirror symmetry of the velocity field about the interfacial plane has this property and is therefore a viscosity-averaged flow.

4.3.2. Dipole moment

The dipole moment is the flow generated by a pair of opposite point forces that are displaced by an infinitesimal distance, or, more generally, a linear combination of such force doublets. The functional form of this mode is given by ${\boldsymbol \nabla }_{\boldsymbol {y}} {\boldsymbol{\mathsf{G}}}(\boldsymbol x, \boldsymbol y)$ in the limit that $\boldsymbol y$ approaches $\boldsymbol 0$ from either side of the interface. Its prefactor for phase $\nu$ is given by

(4.15)\begin{equation} {\boldsymbol{\mathsf{D}}}^{(\nu)} = \int_{S_\nu} [ -(\boldsymbol\sigma \boldsymbol\cdot \hat{\boldsymbol{n}}) \boldsymbol y + \mu_\nu (\boldsymbol u \hat{\boldsymbol{n}} + \hat{\boldsymbol{n}} \boldsymbol u) ] \, {\textrm{d}}{S(\,\boldsymbol y)}, \end{equation}

which we decompose as

(4.16)\begin{equation} {\mathsf {D}}^{(\nu)}_{jk} = {\mathsf {S}}^{(\nu)}_{jk} + \tfrac 12 \varepsilon_{jkl} L^{(\nu)}_l + \tfrac 13 {\mathsf {D}}_{ii}^{(\nu)} \delta_{jk}, \end{equation}

where $\boldsymbol \varepsilon$ is the permutation tensor. Here, the irreducible tensor ${\mathsf {S}}^{(\nu )}_{jk} = \frac 12 ( {\mathsf {D}}^{(\nu )}_{jk} + {\mathsf {D}}^{(\nu )}_{kj} ) - \frac 13 {\mathsf {D}}^{(\nu )}_{ii} \delta _{jk}$ is associated with extensional (or contractile) stresses on the fluid, i.e. the stresslet at the interface, and $\boldsymbol L^{(\nu )}$ gives the torque exerted by the colloid on fluid $\nu$. The total torque exerted by the colloid on the outside system is therefore

\[ \boldsymbol L = \boldsymbol L^{(1)} + \boldsymbol L^{(2)} = \boldsymbol\varepsilon \odot ({{\boldsymbol{\mathsf{D}}}^{(1)} + {\boldsymbol{\mathsf{D}}}^{(2)}}) ={-}\left({\int_{S_1} + \int_{S_2}}\right) \boldsymbol y \times (\boldsymbol\sigma \boldsymbol\cdot \hat{\boldsymbol{n}}) \,{\textrm{d}}{S(\,\boldsymbol y)}. \]

The last term of (4.16) is associated with an isotropic stress, which cannot produce flow due to fluid incompressibility (4.1b). Thus, it makes no contribution to $\boldsymbol u^{m1}$.

We may rewrite (4.12b) as

(4.17)\begin{align} u^{{m1}}_i(\boldsymbol x) &= ({{\mathsf {D}}^{(1)}_{{\alpha\beta}} + {\mathsf {D}}^{(2)}_{{\alpha\beta}}}) \frac{\partial {\mathsf {G}}_{i\alpha}}{\partial y_\beta}(\boldsymbol x, \boldsymbol 0) + {\mathsf {D}}^{(1)}_{\alpha 3} \frac{\partial {\mathsf {G}}_{i\alpha}}{\partial h}(\boldsymbol x, \boldsymbol 0^{+}) + {\mathsf {D}}^{(2)}_{\alpha 3} \frac{\partial {\mathsf {G}}_{i\alpha}}{\partial h}(\boldsymbol x, \boldsymbol 0^{-}) \nonumber\\ &\quad + ({{\mathsf {D}}^{(1)}_{3\beta} + {\mathsf {D}}^{(2)}_{3\beta}}) \frac{\partial {\mathsf {G}}_{i3}}{\partial y_\beta}(\boldsymbol x, \boldsymbol 0) + ({{\mathsf {D}}^{(1)}_{33} + {\mathsf {D}}^{(2)}_{33}}) \frac{\partial {\mathsf {G}}_{i3}}{\partial h}(\boldsymbol x, \boldsymbol 0), \end{align}

where we introduce the convention that Greek tensor indices, here $\alpha \in \{1, 2\}$ and $\beta \in \{1, 2\}$, only run over the axes parallel to the interface. We have combined the left and right limits in the first, penultimate, and last terms of (4.17) because gradients of ${\boldsymbol{\mathsf{G}}}$ parallel to the interface are continuous across the interfacial plane by (4.2b). In the case of the last term, continuity follows from (4.1b), (4.2b) and (4.6), which give

(4.18)\begin{equation} \left[\frac{\partial {\mathsf {G}}_{i3}(\boldsymbol x, \boldsymbol y)}{\partial h}\right]_I = \left[\frac{\partial {\mathsf {G}}_{3i}(\,\boldsymbol y, \boldsymbol x)}{\partial h}\right]_I = \left[\frac{\partial {\mathsf {G}}_{\alpha i}(\,\boldsymbol y, \boldsymbol x)}{\partial y_\alpha}\right]_I = 0, \end{equation}

where the usual roles of $\boldsymbol x$ and $\boldsymbol y$ are reversed in the last two equalities. Furthermore, the penultimate term of (4.17) vanishes because ${\mathsf {G}}_{i3}(\boldsymbol x, \boldsymbol 0) = 0$ by (4.2b) and (4.6). We cannot similarly combine limits from the second and third terms of (4.17) because $[{\partial {\mathsf {G}}_{i\alpha } / \partial h}]_I \neq ~0$; the tangential stress balance (4.2a) requires that

(4.19)\begin{equation} \mu_1 \lim_{\boldsymbol x \to \boldsymbol 0^{+}} \frac{\partial {\mathsf {G}}_{\alpha k}(\boldsymbol x, \boldsymbol y)}{\partial z} - \mu_2 \lim_{\boldsymbol x \to \boldsymbol 0^{-}} \frac{\partial {\mathsf {G}}_{\alpha k}(\boldsymbol x, \boldsymbol y)}{\partial z} = 0. \end{equation}

However, (4.6) and (4.19) relate these limits as

(4.20)\begin{equation} \mu_1 \lim_{\boldsymbol y \to \boldsymbol 0^{+}} \frac{\partial {\mathsf {G}}_{k {\alpha}}(\boldsymbol x, \boldsymbol y)}{\partial h} = \mu_2 \lim_{\boldsymbol y \to \boldsymbol 0^{-}} \frac{\partial {\mathsf {G}}_{k {\alpha}}(\boldsymbol x, \boldsymbol y)}{\partial h} ={-}\frac{\mu({-}z)}{\bar\mu} (\delta^{\shortparallel}_{{\alpha} j} n_k + \delta^{\shortparallel}_{{\alpha} k} n_j) \frac{\partial {\mathsf {J}}_{ij}(\boldsymbol x)}{\partial x_k}, \end{equation}

where the last equality follows from differentiation of (4.3).

Combining (4.17) with (4.16) and (4.20) gives

(4.21)\begin{equation} u^{m1}_i (\boldsymbol x) ={-}\frac 1{\bar\mu} \left( {\mathsf {S}}^{\shortparallel}_{{\alpha}{\beta}} + \frac 12 \varepsilon_{{\alpha}{\beta} 3} L_3 - {S}^{{\perp}} \delta^{\shortparallel}_{{\alpha}{\beta}} \right) \frac{\partial {\mathsf {J}}_{i{\alpha}}(\boldsymbol x)}{\partial x_{\beta}} - \frac{A_{\alpha}}{\mu(z)} ( \delta^{\shortparallel}_{{\alpha} j} n_k + \delta^{\shortparallel}_{{\alpha} k} n_j ) \frac{\partial {\mathsf {J}}_{ij}(\boldsymbol x)}{\partial x_k}, \end{equation}

where

(4.22a)\begin{gather} {\mathsf {S}}^{\shortparallel}_{{\alpha}{\beta}} = {\mathsf {S}}^{(1)}_{{\alpha}{\beta}} + {\mathsf {S}}^{(2)}_{{\alpha}{\beta}} - \tfrac 12 ( {\mathsf {S}}^{(1)}_{{\gamma}{\gamma}} + {\mathsf {S}}^{(2)}_{{\gamma}{\gamma}} ) \delta^{\shortparallel}_{{\alpha}{\beta}}, \end{gather}

(4.22b)\begin{gather}S^{{\perp}} = {\mathsf {S}}^{(1)}_{33} + {\mathsf {S}}^{(2)}_{33} - \tfrac 12 ( {\mathsf {S}}^{(1)}_{{\gamma}{\gamma}} + {\mathsf {S}}^{(2)}_{{\gamma}{\gamma}} ), \end{gather}

(4.22c)\begin{gather}\bar\mu A_{\alpha} = \mu_2 {\mathsf {D}}^{(1)}_{{\alpha} 3} + \mu_1 {\mathsf {D}}^{(2)}_{{\alpha} 3} = \mu_2 ({{\mathsf {S}}^{(1)}_{{\alpha} 3} + \tfrac 12 \varepsilon_{{\alpha} 3 {\beta}} L^{(1)}_{\beta}}) + \mu_1 ({{\mathsf {S}}^{(2)}_{{\alpha} 3} + \tfrac 12 \varepsilon_{{\alpha} 3 {\beta}} L^{(2)}_{\beta}}). \end{gather}

Equations (4.21) and (4.22) show that the dipole at a clean interface can be conveniently represented in terms of $\boldsymbol \nabla {\boldsymbol{\mathsf{J}}}$.

Each term in (4.21) makes a distinct contribution to the interfacial dipole moment. We call each contribution by its (tensorial) prefactor and graphically represent them as distributions of Stokes singularities in figure 4. The ${\boldsymbol{\mathsf{S}}}^{\shortparallel }$ mode, given by (4.22a), is a viscosity-averaged stresslet associated with extensional stresses produced by the colloid in the interfacial plane. Similarly, the $S^{\perp }$ mode, given by (4.22b), is the viscosity-averaged stresslet perpendicular to the interface. Furthermore, ${\mathsf {S}}^{(\nu )}_{33} = -{\mathsf {S}}^{(\nu )}_{11} - {\mathsf {S}}^{(\nu )}_{22}$ because ${\boldsymbol{\mathsf{S}}}^{(\nu )}$ is traceless, so $S^{\perp }$ accounts for extensional stress perpendicular to the interface and planar compression of the interface. The $L_3$ mode in (4.21) is a viscosity-averaged rotlet, or point torque, about the $z$ axis of strength $L_3$. These viscosity-averaged flows exhibit mirror symmetry of the velocity field about $z=0$. Therefore, the tangential shear stress due to these modes vanishes on the interface, as is the case for the monopole moment.

Figure 4. Singularity diagrams corresponding to each of the terms in (4.21). Arrows indicate distributions of point forces or torques and the grey shaded region indicates the interface.

The $\boldsymbol A$ mode corresponds to a force dipole where the forces act parallel to the interface and are displaced from one another along the axis normal to the interface. This mode does not produce a viscosity-averaged flow. Instead, the flow speed in one phase differs from that in the opposite phase by a factor of the viscosity ratio; intuitively, the flow is slower in the more viscous phase. On the interface, the fluid velocity vanishes. We see from the last term of (4.21) that the flows in the upper and lower half spaces are equivalent to effective stresslets in an unbounded fluid for $z>0$ and $z<0$, respectively.

Quadrupolar and higher-order terms of (4.12) can be similarly decomposed into two subsets of modes; one whose tangential stress vanishes at the interface and another whose velocity vanishes at the interface. Members of the former subset are mirror-symmetric, viscosity-averaged flows and the latter have velocities that differ by in magnitude by the viscosity ratio on either side of the interface. We do not detail the higher-order modes further. The force monopole (4.14) and force dipole (4.21) describe the leading-order flows due to driven and active colloids, respectively. In many cases, we can infer these modes based on the geometry of a given colloidal particle and its configuration with respect to the interface.

5.1. Green's function

We may define a Green's function ${\boldsymbol{\mathsf{H}}}$ for an incompressible interface that is analogous to that discussed in § 4.1 for a clean interface. The major difference is that the interfacial stress balance (4.2a) is replaced by

(5.1a)\begin{gather} - \boldsymbol\nabla_{{s}} \boldsymbol\varPi + \mu_{s} \nabla_{{s}}^{2} {\boldsymbol{\mathsf{H}}} + {\boldsymbol{\mathsf{I}}}_{s} \boldsymbol\cdot [{\boldsymbol n \boldsymbol\cdot {\boldsymbol{\mathsf{T}}}({\boldsymbol{\mathsf{H}}};\!)}]_I = \begin{cases} -{\boldsymbol{\mathsf{I}}}_{s} \delta_{\mathbb{R}^{2}}(\boldsymbol x - \boldsymbol y) & h = 0 \\ \boldsymbol 0 & h \neq 0 \end{cases}, \end{gather}

(5.1b)\begin{gather}\boldsymbol\nabla_{{s}}\boldsymbol\cdot{{\boldsymbol{\mathsf{H}}}} = 0, \end{gather}

where $\boldsymbol \varPi$ is the (vectorial) surface pressure associated with ${\boldsymbol{\mathsf{H}}}$, which enforces the surface incompressibility constraint (5.1b). Thus, ${\boldsymbol{\mathsf{H}}}$ satisfies (4.1) subject to (4.2b) and (5.1), with ${\boldsymbol{\mathsf{G}}}$ replaced by ${\boldsymbol{\mathsf{H}}}$ in the former two equations. The coupling of bulk-viscous and surface-viscous effects induce a natural length scale in the problem, the Boussinesq length, which is given by ${L_{B}} = \mu _{s} / \bar \mu$. Bulk viscous dissipation dominates at distances $r \gg {L_{B}}$, and surface viscous dissipation dominates when $r \ll {L_{B}}$. It is therefore convenient to define the dimensionless Boussinesq number, $\mathinner {Bq} = {L_{B}}/a$, which quantifies the relative importance of surface-viscous to bulk-viscous effects for a colloid of characteristic size $a$.

The functional form of ${\boldsymbol{\mathsf{H}}}$, derived in full by Bławzdziewicz et al. (Reference Bławzdziewicz, Cristini and Loewenberg1999), is more complicated than that of ${\boldsymbol{\mathsf{G}}}$ owing to non-trivial interfacial stress balance (5.1). However, like ${\boldsymbol{\mathsf{G}}}$, ${\boldsymbol{\mathsf{H}}}$ is self-adjoint (see appendix A) and therefore satisfies

(5.2)\begin{equation} {\boldsymbol{\mathsf{H}}}(\boldsymbol x, \boldsymbol y) = {\boldsymbol{\mathsf{H}}}^{\mathsf{T}}(\,\boldsymbol y, \boldsymbol x). \end{equation}

We may use this property to determine the multipole expansion at points on the interface requiring only knowledge of ${\boldsymbol{\mathsf{H}}}(\boldsymbol x, \boldsymbol y)$ for $\boldsymbol y \in I$, in addition to the previously determined expansion for a clean interface. Letting ${\boldsymbol{\mathsf{H}}}^{0}(\boldsymbol s, z) = {\boldsymbol{\mathsf{H}}}(\boldsymbol x, \boldsymbol 0)$, we find that for a fixed value of ${L_{B}}$ (see appendix B for derivation)

(5.3)\begin{equation} H^{\,0}_{{\alpha}{\beta}}(\boldsymbol s, z) = \frac{1}{4{\rm \pi} \bar\mu} R_0({L_{B}}; s, |z|) \delta_{{\alpha}{\beta}} + \frac{1}{2{\rm \pi} \bar\mu} R_2({L_{B}}; s, |z|) \{ {\hat s_{\alpha} \hat s_{\beta}} \}_0, \end{equation}

and ${\mathsf {H}}^{\,0}_{3i} = {\mathsf {H}}^{\,0}_{i3} = 0$, where $\boldsymbol s = {\boldsymbol{\mathsf{I}}}_{s} \boldsymbol \cdot \boldsymbol x = x_1 \hat {\boldsymbol \imath }_1 + x_2 \hat {\boldsymbol \imath }_2$, $s = |\boldsymbol s|$, $\hat {\boldsymbol {s}} = |\boldsymbol s| / s$ and $\{ {\cdot } \}_0$ denotes the irreducible (traceless, symmetric) part of the enclosed tensor. Here, e.g. $\{ {\hat s_{\alpha} \hat s_{\beta} } \}_0 = \hat s_{\alpha} \hat s_{\beta} - \frac 12 \delta _{{\alpha} {\beta} }$, where we regard $\hat {\boldsymbol {s}}$ as a two-dimensional vector and the $\{\cdot \}_0$ operation as being on a two-dimensional tensor. The functions $R_0$ and $R_2$, given by (B16), depend only on the magnitudes of $\boldsymbol s$ and $z$. Equation (5.2) equips ${\boldsymbol{\mathsf{H}}}^{0}$ with dual interpretations; ${\mathsf {H}}^{\,0}_{ij}(\boldsymbol s, z)\, F_j$ is the velocity field due to the point force $\boldsymbol F$ on the interface at $\boldsymbol y = \boldsymbol 0$, and ${\mathsf {H}}^{\,0}_{ij}(\boldsymbol s, h)\, F_j$ is the velocity field on the interface induced by the same point force exerted at $\boldsymbol y = h \hat {\boldsymbol \imath }_3$ (within either of the fluids).

The velocity field represented by ${\boldsymbol{\mathsf{H}}}^{0}(\boldsymbol s, z)$ is everywhere parallel to the interface. As noted by Stone & Masoud (Reference Stone and Masoud2015), this feature is generally expected of Stokes flow driven by arbitrary motion of an incompressible plane. The $z$ velocities of the fluids vanish at the interface as do their $z$ derivatives due to the incompressibility of the interface and the surrounding fluids. These quantities also vanish as $|\boldsymbol x| \to \infty$. As a Stokes flow, $\boldsymbol u$ is biharmonic, and hence $\nabla ^{4} u_3 = 0$. It follows from the homogeneous behaviour of $u_3$ on the interface and at infinity that $u_3 = 0$ everywhere. The vanishing behaviour of ${\mathsf {H}}^{\,0}_{3j}$ reflects this property.

When the interface is inviscid, ${L_{B}} = 0$, $R_n$ admits the closed form given by (B17), and (5.3) reduces to

(5.4)\begin{equation} {{\mathsf {H}}^{\,0}_{{\alpha}{\beta}}}|_{{L_{B}}=0} = \frac{\delta_{{\alpha}{\beta}}}{8{\rm \pi}\bar\mu r} + \frac{(r - |z|)^{2}}{4{\rm \pi} \bar\mu r s^{2}} \{ {\hat s_{\alpha} \hat s_{\beta}} \}_0. \end{equation}

Thus, Marangoni stresses alone do not change the $r^{-1}$ rate of decay of the fluid velocity from the singular point at the origin found for a clean interface. The flow on the interface is purely radial and is given by

\[ {{\mathsf {H}}^{\,0}_{{\alpha}{\beta}}(\boldsymbol x \in I)}|_{\mathinner{Bq}=0} = \frac{\hat s_{\alpha} \hat s_{\beta}}{8{\rm \pi}\bar\mu r}. \]

For finite ${L_{B}}$, ${\boldsymbol{\mathsf{H}}}$ approaches the form given by (5.4) for $r \gg {L_{B}}$. In the opposite limit where ${L_{B}} \to \infty$, we recover the equations governing a two-dimensional Stokes flow from (5.1), as viscous stresses from the two fluid phases become negligible. Therefore, ${\boldsymbol{\mathsf{H}}}^{0}$ coincides with a Stokeslet in a two-dimensional fluid (up to an arbitrary constant) (Saffman & Delbrück Reference Saffman and Delbrück1975),

(5.5)\begin{equation} {{\mathsf {H}}^{\,0}_{{\alpha}{\beta}}}|_{{L_{B}}\to\infty} \sim \frac{\hat s_{\alpha} \hat s_{\beta} - \delta_{{\alpha}{\beta}} \ln s}{4{\rm \pi}\mu_{s}}, \end{equation}

which is constant in $z$ (as a three-dimensional flow field) and diverges logarithmically as $s$ is made large. For finite ${L_{B}}$, ${\boldsymbol{\mathsf{H}}}$ approaches the form given by (5.5) for $r \ll {L_{B}}$. Clearly, (5.5) does not satisfy the homogeneous boundary conditions imposed on ${\boldsymbol{\mathsf{H}}}$ as $r \to \infty$. Of course, this is Stokes’ paradox; for large but finite ${L_{B}}$, (5.5) is not valid for $r \gtrsim {L_{B}}$, where bulk-viscous effects inevitably become important. Returning to the general expression (5.3), we find that the $R_n$ functions transition the flow field between the surface-viscosity-dominated, logarithmically divergent behaviour at distances $r \ll {L_{B}}$ from the colloid to the convergent, $1/r$ decay at distances $r \gg {L_{B}}$, where surface viscosity has a negligible effect.

The surface pressure $\boldsymbol \varPi = \boldsymbol \varPi (\boldsymbol x, \boldsymbol y)$ associated with ${\boldsymbol{\mathsf{H}}}(\boldsymbol x, \boldsymbol y)$ is independent of ${L_{B}}$, as has been previously found (Bławzdziewicz et al. Reference Bławzdziewicz, Cristini and Loewenberg1999; Fischer et al. Reference Fischer, Dhar and Heinig2006), and is given in our notation by (see appendix B for derivation)

(5.6)\begin{equation} \varPi_i(\boldsymbol s, h \hat{\boldsymbol\imath}_3) = \frac{s_i}{4{\rm \pi} s^{2}} \left({1 - \frac{|h|}{\sqrt{s^{2} + h^{2}}}}\right) - \frac{|h| (s_i - h n_i)}{4{\rm \pi} {(s^{2} + h^{2})}^{3/2}}. \end{equation}

Equation (5.6) shows that $|\boldsymbol \varPi | \sim 1/s$ for $h$ fixed and $s \gg h$, and $|\boldsymbol \varPi | \sim 1/h$ for $s$ fixed and $h \gg s$. For $h = 0$, (5.6) reduces to the harmonic pressure field $\boldsymbol \varPi (\boldsymbol s, \boldsymbol 0) = \boldsymbol s / 4{\rm \pi} s^{2}$.

5.2. Multipole expansion

5.2.1. Expansion of the boundary integral equation

Using the reciprocal relation (3.9) for two fluids separated by an incompressible interface and following a procedure similar to that described in § 4.2, we obtain the boundary integral representation for $\boldsymbol u(\boldsymbol x)$ as

(5.7)\begin{align} u_k(\boldsymbol x) &={-} \oint_{S_{c}} { {\mathsf {H}}_{kj}(\boldsymbol x, \boldsymbol y) \, [\hat n_i \sigma_{ij}](\,\boldsymbol y) }\,{\textrm{d}}S(\,\boldsymbol y) + \oint_{S_{c}} { [\hat n_i u_j](\,\boldsymbol y) \, {\mathsf {T}}_{ijk}({\boldsymbol{\mathsf{H}}}; \boldsymbol y, \boldsymbol x) }\,{\textrm{d}}S(\,\boldsymbol y), \nonumber\\ &\quad - \oint_C { {\mathsf {H}}_{k\beta}(\boldsymbol x, \boldsymbol y)\, [\hat m_\alpha \varsigma_{\alpha\beta}](\,\boldsymbol y) } \,{\textrm{d}}C(\,\boldsymbol y) + \oint_C { [\hat m_\alpha u_\beta](\,\boldsymbol y) \, \varSigma_{\alpha\beta k}({\boldsymbol{\mathsf{H}}}; \boldsymbol y, \boldsymbol x) } \,{\textrm{d}}C(\,\boldsymbol y), \end{align}

where $\hat {\boldsymbol {m}} = \hat {\boldsymbol \imath }_3 \times \hat {\boldsymbol {t}}$, $C$ is the curve in the $z=0$ plane that runs along the three-phase contact line (see figure 2), and $\boldsymbol \varSigma ({\boldsymbol{\mathsf{H}}};\!)$ is the surface stress tensor associated with ${\boldsymbol{\mathsf{H}}}$, which is given by

\[ \varSigma_{{\alpha}{\beta} k}({\boldsymbol{\mathsf{H}}}; \boldsymbol y, \boldsymbol x) ={-}\delta^{\shortparallel}_{{\alpha}{\beta}} \varPi_k(\,\boldsymbol y, \boldsymbol x) + \mu_s \left( \frac{\partial {\mathsf {H}}_{k{\beta}}(\boldsymbol x, \boldsymbol y)}{\partial y_{\alpha}} + \frac{\partial {\mathsf {H}}_{k{\alpha}}(\boldsymbol x, \boldsymbol y)}{\partial y_{\beta}} \right), \]

for $\boldsymbol y \in I$. Equation (5.7) is valid as long as ${L_{B}}$ is finite. We require the $1/r$ spatial decay of ${\boldsymbol{\mathsf{H}}}$ at distances $r \gg L_B$ from the colloid for the integrals in this equation to converge unconditionally. Comparing (5.7) to (4.9), the last two terms of (5.7) are new and account for Marangoni forces and surface-viscous stresses at the contact line, respectively.

As before, we may generate a multipole expansion for $\boldsymbol u(\boldsymbol x)$ by replacing ${\boldsymbol{\mathsf{H}}}$, ${\boldsymbol{\mathsf{T}}}({\boldsymbol{\mathsf{H}}};\!)$, and $\boldsymbol \varSigma ({\boldsymbol{\mathsf{H}}};\!)$ in (5.7) with their respective Taylor series about $\boldsymbol y = \boldsymbol 0$, where $\boldsymbol 0$ is placed at an appropriate point on the interface. We may write the expansion as $\boldsymbol u = \boldsymbol u^{(1)} + \boldsymbol u^{(2)} + \boldsymbol u^{(\text i)}$, where

(5.8)\begin{gather} \boldsymbol u^{(1)} = \text{same as right-hand side of (4.10) with ${\boldsymbol{\mathsf{G}}}$ replaced by ${\boldsymbol{\mathsf{H}}}$,} \end{gather}

(5.9)\begin{gather}\boldsymbol u^{(2)} = \text{same as right-hand side of (4.11) with ${\boldsymbol{\mathsf{G}}}$ replaced by ${\boldsymbol{\mathsf{H}}}$,} \end{gather}

and

(5.10)\begin{align} u^{(\text i)}_k (\boldsymbol x) &={-}\sum_{n=0}^{\infty} { \frac 1{n!} \left( \int_C [\hat m_{\alpha} \varsigma_{{\alpha}{\beta}}](\,\boldsymbol y)\,y_{{\gamma}_1} \cdots y_{{\gamma}_n} {\textrm{d}}C(\,\boldsymbol y) \right) \left. \frac{\partial^{n} {\mathsf {H}}_{k{\beta}}(\boldsymbol x, \boldsymbol y \in I)}{\partial y_{{\gamma}_1} \cdots \partial y_{{\gamma}_n}} \right|_{\boldsymbol y = \boldsymbol 0} } \nonumber\\ &\quad +\sum_{n=0}^{\infty} { \frac 1{n!} \left( \int_C [u_{\beta} \hat m_{\alpha}](\,\boldsymbol y)\,y_{{\gamma}_1} \cdots y_{{\gamma}_n} {\textrm{d}}C(\,\boldsymbol y) \right) \left. \frac{\partial^{n} \varSigma_{{\alpha}{\beta} k}({\boldsymbol{\mathsf{H}}}; \boldsymbol y \in I, \boldsymbol x)}{\partial y_{{\gamma}_1} \cdots \partial y_{{\gamma}_n}} \right|_{\boldsymbol y = \boldsymbol 0} }. \end{align}

Collecting terms from (5.8) to (5.10), we may write $\boldsymbol u$ as a multipole expansion analogous to that given by (4.11),

(5.11)\begin{equation} \boldsymbol u(\boldsymbol x) = \boldsymbol u^{m0}(\boldsymbol x) + \boldsymbol u^{m1}(\boldsymbol x) + \boldsymbol u^{m2}(\boldsymbol x) + \text{h.o.t}, \end{equation}

where

(5.11a)\begin{gather} u^{m0}_i(\boldsymbol x) = F^{(1)}_j {\mathsf {H}}_{ij}(\boldsymbol{x}, \boldsymbol{0}^+) + F^{(2)}_j {\mathsf {H}}_{ij}(\boldsymbol{x}, \boldsymbol{0}^-) + F^{(\text i)}_{\beta} {\mathsf {H}}_{i{\beta}}(\boldsymbol x, \boldsymbol 0), \end{gather}

(5.11b)\begin{gather}u^{m1}_i(\boldsymbol x) = {\mathsf {D}}^{(1)}_{jk} \frac{\partial {\mathsf {H}}_{ij}}{\partial y_k}(\boldsymbol{x}, \boldsymbol{0}^+) + {\mathsf {D}}^{(2)}_{jk} \frac{\partial {\mathsf {H}}_{ij}}{\partial y_k}(\boldsymbol{x}, \boldsymbol{0}^-) + {\mathsf {D}}^{(\text i)}_{{\beta}{\gamma}} \frac{\partial {\mathsf {H}}_{i{\beta}}}{\partial y_{\gamma}}(\boldsymbol x, \boldsymbol 0). \end{gather}

Equations (5.11a) and (5.11b) are analogous to (4.12a) and (4.12b), respectively, and each of these equations contain an additional term that accounts for stresses exerted by the colloid on the interface. Equation (5.11b) assumes that the hole in the interface created by the colloid is of constant surface area and that the volume of the colloid is also constant.

5.2.2. Monopole moment

At an incompressible interface, the monopole moment retains its interpretation as the leading-order flow due to a colloid that is driven by an external force. Indeed, ${\boldsymbol{\mathsf{H}}}(\boldsymbol x, \boldsymbol y)$ is continuous as $\boldsymbol y$ is moved across the interface, so ${\boldsymbol u}^{m0}(\boldsymbol x)$ reduces to

(5.12)\begin{equation} u^{m0}_i(\boldsymbol x) = F_j {\mathsf {H}}^{\,0}_{ij}(\boldsymbol x) = ( F^{(1)}_j + F^{(2)}_j + F^{(\text i)}_\beta \delta^{\shortparallel}_{\beta j} ) {\mathsf {H}}^{\,0}_{ij}(\boldsymbol x), \end{equation}

where $\boldsymbol F$ is the total force exerted on the surrounding system: the two fluids and the interface. Compared with the analogous expression for a clean interface (4.14), the surface-incompressible monopole has the additional contribution to its prefactor given by $F^{(\text i)}_{{\beta} } = -\oint _C \hat m_\alpha \varsigma _{\alpha \beta } \,{\textrm {d}}C,$ which is the force exerted on the interface by the colloid at the contact line due to Marangoni and surface-viscous stresses. Like the clean-interface monopole, the surface-incompressible monopole given by (5.12) is independent of the viscosity contrast between the two fluids and is mirror symmetric with respect to the $z=0$ plane. However, the surface-incompressible monopole lacks the axisymmetry of its clean-interface analogue. From (5.12), we find that $\boldsymbol u^{m0}$ only contributes to velocity components parallel to the interface because ${\mathsf {H}}^{\,0}_{i3} = 0$. Hence, this mode is associated with ‘lamellar’ fluid motion, a feature not shared with its clean-interface analogue.

Figure 7 shows the interfacial velocity and surface pressure profiles of the force monopole (5.12). For ${L_{B}} = 0$ (figure 7a), the flow is quite different than that for a clean interface (figure 7d), although in both cases the velocity magnitude decays as $1/r$. Here, the flow is modified from that at a clean interface by Marangoni stresses alone. Figure 7(a) is representative of the leading-order flow for a driven colloid whose characteristic size far exceeds ${L_{B}}$ ($\mathinner {Bq} \ll 1$), where surface-viscous forces are weak everywhere on the interface. The moderate-$\mathinner {Bq}$ case is shown in figure 7(b). The more complicated flow pattern owes to the interplay between bulk viscous, surface-viscous and Marangoni stresses, which are all comparable in magnitude for $r \sim {L_{B}}$. Figure 7(c) shows the large-$\mathinner {Bq}$ regime, where dissipation is dominated by the surface viscosity except at very large distances of $r \geq O(a\mathinner {Bq})$. For smaller $O(\textit {a})$ distances, the flow profile has the same angular dependence about the $z$ axis as is found for a clean interface, but the magnitude of the velocity behaves as $-\ln (r)$ rather than as $1/r$. The surface pressure field associated with $\boldsymbol u^{m0}$, $\boldsymbol F \boldsymbol \cdot \boldsymbol \varPi |_{h=0}$ (figure 7e), does not vary with ${L_{B}}$.

Figure 7. The velocity field on the interfacial plane due to a surface-incompressible force monopole is shown for different values of ${L_{B}}$ (a–c). The result for a clean interface (d) is also shown for comparison. In (b), the distance $r = {L_{B}}$ is indicated by the dashed circle. The arrows show the location and direction of the point of forcing. The surface pressure profile (e) associated with (a–c) is independent of ${L_{B}}$. The logarithmically spaced contours correspond to velocity magnitude (a–d) and surface pressure (e), where lighter coloured contours indicate larger values and darker colours approach zero. Panels (a), (b) and (c), represent the leading-order velocity fields due to a colloid driven by an external force for small-, moderate- and large-$\mathinner {Bq}$, respectively. (a) $L_B=0$, (b) finite $L_B$, (c) $L_B \rightarrow \infty$, (d) clean and (e) surface pressure.

5.2.3. Dipole moment

Similar to the monopole moment, the dipole moment has an additional contribution due to interfacial stresses given by the final term in (5.11b), whose prefactor is

(5.13)\begin{equation} {\mathsf {D}}^{(\text i)}_{\beta\gamma} = \oint_C [ -\hat m_\alpha \varsigma_{\alpha\beta} y_\gamma + \mu_{s} (\hat m_\beta u_\gamma + u_\gamma \hat m_\beta) ] \, {\textrm{d}}C(\,\boldsymbol y), \end{equation}

where $\boldsymbol m$, $\boldsymbol \varsigma$ and $\boldsymbol u$ are regarded as functions of $\boldsymbol y$. We may rewrite (5.11b) as

(5.14)\begin{equation} u^{{m1}}_i(\boldsymbol x) = ({{\mathsf {D}}^{(1)}_{{\alpha\beta}} + {\mathsf {D}}^{(2)}_{{\alpha\beta}} + {\mathsf {D}}^{(\text i)}_{{\alpha\beta}}}) \frac{\partial {\mathsf {H}}_{i\alpha}}{\partial y_\beta}(\boldsymbol x, \boldsymbol 0) + {\mathsf {D}}^{(1)}_{\alpha 3} \frac{\partial {\mathsf {H}}_{i\alpha}}{\partial h}(\boldsymbol x, \boldsymbol 0^{+}) + {\mathsf {D}}^{(2)}_{\alpha 3} \frac{\partial {\mathsf {H}}_{i\alpha}}{\partial h}(\boldsymbol x, \boldsymbol 0^{-}). \end{equation}

Equation (5.14) omits terms involving derivatives of ${\mathsf {H}}_{i3}$, which vanish because the interface is impenetrable and because incompressibility of the interface and the surrounding fluid imply that

(5.15)\begin{equation} 0 = \left.{\frac{\partial {\mathsf {H}}_{\alpha j}(\boldsymbol x, \boldsymbol y)}{\partial x_\alpha}}\right|_{z = 0} = \left.{\frac{\partial {\mathsf {H}}_{3j}(\boldsymbol x, \boldsymbol y)}{\partial z}}\right|_{z = 0} = \left.{\frac{\partial {\mathsf {H}}_{j3}(\boldsymbol x, \boldsymbol y)}{\partial h}}\right|_{h = 0}, \end{equation}

where the final equality follows from (5.2).

We may decompose ${\boldsymbol{\mathsf{D}}}^{(1)}$ and ${\boldsymbol{\mathsf{D}}}^{(2)}$ into irreducible tensors as before (see (4.16)). A similar decomposition of the two-dimensional tensor ${\boldsymbol{\mathsf{D}}}^{(\text i)}$ is given by

(5.16)\begin{equation} {\mathsf {D}}^{(\text i)}_{{\alpha}{\beta}} = {\mathsf {S}}^{(\text i)}_{{\alpha}{\beta}} + \tfrac 12 \varepsilon_{{\alpha}{\beta} 3} L^{(\text i)} + \tfrac 12 \delta_{{\alpha}{\beta}} {\mathsf {D}}^{(\text i)}_{{\gamma}{\gamma}}, \end{equation}

where the irreducible part of ${\boldsymbol{\mathsf{D}}}^{(\text i)}$ is given by

\[ {\mathsf {S}}^{(\text i)}_{{\alpha}{\beta}} = \tfrac 12 ({{\mathsf {D}}^{(\text i)}_{{\alpha}{\beta}} + {\mathsf {D}}^{(\text i)}_{{\beta}{\alpha}}}) - \tfrac 12 \delta_{{\alpha}{\beta}} {\mathsf {D}}^{(\text i)}_{{\gamma}{\gamma}}, \]

which represents the stresslet on the interface due to forces on the contact line. Similarly, the pseudoscalar $L^{(\text i)} = -\hat {\boldsymbol \imath }_3 \boldsymbol \cdot \oint _C { \boldsymbol y \times [\hat {\boldsymbol {m}} \boldsymbol \cdot \boldsymbol \varsigma ](\,\boldsymbol y) } \,{\textrm {d}}C(\,\boldsymbol y)$ is the torque (about the $z$ axis) exerted on the interface by the colloid. The total torque exerted on the surrounding system (both fluids and the interface) is therefore $\boldsymbol L = \boldsymbol L^{(1)} + \boldsymbol L^{(2)} + L^{(\text i)} \hat {\boldsymbol \imath }_3$. From (2.6), it is readily shown that the surface pressure makes no contribution to $L^{(\text i)}$, and therefore $L^{(\text i)} = 0$ if $\mu _{s} = 0$.

Finally, (5.1b), (5.2) and (5.15) imply that $[{\partial {\mathsf {H}}_{j \alpha }(\boldsymbol x, \boldsymbol y)} / {\partial y_\alpha }]_{h = 0} = 0$. Comparing this result with (5.14) reveals that the ‘surface’ traces of ${\boldsymbol{\mathsf{D}}}^{(1)}$, ${\boldsymbol{\mathsf{D}}}^{(2)}$ and ${\boldsymbol{\mathsf{D}}}^{(\text i)}$, i.e. ${\mathsf {D}}^{(\nu )}_{{\gamma} {\gamma} }$, are of no dynamical significance because the interface is incompressible. It follows from the bulk incompressibility of the surrounding fluids that ${\mathsf {S}}^{(\text 1)}_{33}$ and ${\mathsf {S}}^{(\text 2)}_{33}$ also have no effect on the flow. Recall that, for a clean interface, the modes associated with these components of the stresslet (the $S^{\perp }$ mode) produced a radially symmetric flow associated with local expansion or compression of the interface (see figure 6b). It is no surprise that these source or sink flows vanish for incompressible interfaces. One may easily verify that there exists no radially symmetric vector field on the interface that both satisfies $\boldsymbol \nabla _{{s}}\boldsymbol \cdot {\boldsymbol u} = 0$ and vanishes at infinity.

To further reduce (5.14), we break $\boldsymbol u^{m1}$ into three constituent modes as

(5.17)\begin{align} u^{{m1}}_i(\boldsymbol x) &= ({{\mathsf {S}}^{\shortparallel}_{{\alpha}{\beta}} + \tfrac 12 L_3 \varepsilon_{3{\alpha}{\beta}}}) \varDelta^{\shortparallel}_{i{\alpha}{\beta}}(\boldsymbol x) \nonumber\\ &\quad + \tfrac 12 ({{\mathsf {D}}_{{\alpha} 3}^{(1)} + {\mathsf {D}}^{(2)}_{{\alpha} 3}}) \varDelta^{{\ddagger}}_{i{\alpha}}(\boldsymbol x) + \tfrac 12 ({{\mathsf {D}}_{{\alpha} 3}^{(1)} - {\mathsf {D}}^{(2)}_{{\alpha} 3}}) \varDelta^{{\pm}}_{i{\alpha}}(\boldsymbol x). \end{align}

The first mode $\boldsymbol \varDelta ^{\shortparallel }(\boldsymbol x)$ is given by

(5.18)\begin{equation} \varDelta^{\shortparallel}_{i{\alpha}{\beta}}(\boldsymbol x) = \left.{\frac{\partial {\mathsf {H}}_{i{\alpha}}(\boldsymbol x, \boldsymbol y)}{\partial y_{\beta}}}\right|_{\boldsymbol y = \boldsymbol 0} ={-}\frac{\partial {\mathsf {H}}^{\,0}_{i{\alpha}}(\boldsymbol s, z)}{\partial x_{\beta}}, \end{equation}

where the second equality follows from the translation invariance of ${\boldsymbol{\mathsf{H}}}(\boldsymbol x, \boldsymbol y)$ with respect to components of the singular point $\boldsymbol y$ parallel to the interface. Carrying out the differentiation of ${\boldsymbol{\mathsf{H}}}^{0}$ in (5.18), we find that

(5.19)\begin{equation} \varDelta^{\shortparallel}_{{\alpha}{\beta}{\gamma}}(\boldsymbol x) = \frac{R_1'(L_B; s, |z|)}{8{\rm \pi} \bar\mu} ( \hat s_{\alpha} \delta_{{\beta}{\gamma}} + \hat s_{\beta} \delta_{{\gamma}{\alpha}} - 3\hat s_{\gamma} \delta_{{\alpha}{\beta}} ) - \frac{R_3'(L_B; s, |z|)}{2{\rm \pi} \bar\mu} \{ {\hat s_{\alpha} \hat s_{\beta} \hat s_{\gamma}} \}_0, \end{equation}

and $\varDelta ^{\shortparallel }_{3{\beta} {\gamma} } = 0$, where the functions $R'_n$ are given by (B19). The first tensorial prefactor of $\boldsymbol \varDelta ^{\shortparallel }$ in (5.17) is given by

(5.20)\begin{equation} {\mathsf {S}}^{\shortparallel}_{{\alpha}{\beta}} = ( \delta_{{\alpha}{\gamma}} \delta_{{\beta}{\delta}} - \tfrac 12 \delta_{{\alpha}{\beta}} \delta_{{\delta}{\gamma}} ) ( {\mathsf {S}}^{(1)}_{{\gamma}{\delta}} + {\mathsf {S}}^{(2)}_{{\gamma}{\delta}} + {\mathsf {S}}^{(\text i)}_{{\gamma}{\delta}} ), \end{equation}

which is the effective stresslet strength in the plane of the interface.

The second mode of (5.17), $\boldsymbol \varDelta ^{\ddagger }(\boldsymbol x)$, describes an equal pair of force dipoles, one just above the interface and the other just below, of strength $\boldsymbol F \boldsymbol n$ for $\boldsymbol F$ parallel to the interface. This mode corresponds to the ‘symmetric’ part of the second and third terms of (5.14) and is given by

(5.21)\begin{equation} \varDelta^{{\ddagger}}_{i{\alpha}}(\boldsymbol x) = \left({{\lim_{\boldsymbol y \to \boldsymbol 0^{+}}} + {\lim_{\boldsymbol y \to \boldsymbol 0^{-}}}}\right) \frac{\partial {\mathsf {H}}_{i\alpha}(\boldsymbol x, \boldsymbol y)}{\partial h}. \end{equation}

A convenient property of this mode is that it vanishes on $z = 0$; differentiation of (5.3) yields

\[ \lim_{h \to 0^{{\pm}}} \frac{\partial {\mathsf {H}}^{\,0}_{{\alpha}{\beta}}(\boldsymbol s, h)}{\partial h} ={\pm} \left[\frac{R'_0({L_{B}}; \boldsymbol s, 0)}{4{\rm \pi}\bar\mu} \delta_{{\alpha}{\beta}} + \frac{R'_2({L_{B}}; \boldsymbol s, 0)}{2{\rm \pi}\bar\mu} \left( \hat s_{\alpha} \hat s_{\beta} - \frac 12 \delta_{{\alpha}{\beta}} \right) \right]. \]

The analogous mode for a clean interface $\boldsymbol \varDelta ^{\ddagger }_{clean}$, given by replacing ${\boldsymbol{\mathsf{H}}}$ with ${\boldsymbol{\mathsf{G}}}$ in (5.21), shares the same property. In fact, we find that ${\lim _{h \to 0} [{\partial {\mathsf {G}}_{{\alpha} {\beta} }(\boldsymbol x, \boldsymbol y)}/{\partial h}]}_{z = 0} = 0.$ Therefore, both $\boldsymbol \varDelta ^{\ddagger }$ and $\boldsymbol \varDelta ^{\ddagger }_{clean}$ share the same boundary velocities (which vanish at infinity and on the interface) and external forcing (a pair of equal dipoles on either side of the interface). Due to uniqueness of solutions to the Stokes equations in each phase, we conclude that $\boldsymbol \varDelta ^{\ddagger } = \boldsymbol \varDelta ^{\ddagger }_{clean}$, and hence ${\boldsymbol{\mathsf{H}}}$ may be replaced with ${\boldsymbol{\mathsf{G}}}$ in (5.21) without loss of generality. We then find from (4.20) that

(5.22)\begin{equation} \varDelta^{{\ddagger}}_{i{\alpha}}(\boldsymbol x) ={-}\frac{2}{\mu(z)} ( \delta^{\shortparallel}_{{\alpha} j} n_k + \delta^{\shortparallel}_{{\alpha} k} n_j ) \frac{\partial {\mathsf {J}}_{ij}(\boldsymbol x)}{\partial x_k} = \frac{1}{\mu(z)} \left( \frac{3 x_i s_{\alpha} z}{2{\rm \pi} r^{5}} \right). \end{equation}

Comparison of (5.22) to the last term of (4.21) shows that $\boldsymbol \varDelta ^{\ddagger }$ assumes the same functional form as the ‘$\boldsymbol A$ mode’ discussed for clean interfaces in § 4.

The third and final dipolar mode is the antisymmetric compliment of $\boldsymbol \varDelta ^{\ddagger }$,

(5.23)\begin{equation} \varDelta^{{\pm}}_{i{\alpha}}(\boldsymbol x) = \left({{\lim_{\boldsymbol y \to \boldsymbol 0^{+}}} - {\lim_{\boldsymbol y \to \boldsymbol 0^{-}}}}\right) \frac{\partial {\mathsf {H}}_{i\alpha}(\boldsymbol x, \boldsymbol y)}{\partial h}, \end{equation}

which describes a similar pair of force dipoles of opposite sign as they approach the interface from above and below. To deduce the form of $\varDelta ^{\pm }_{i{\alpha} }$, we use (5.2) to reinterpret the tangential stress balance for ${\boldsymbol{\mathsf{H}}}$ (5.1) as the velocity profile ${\boldsymbol{\mathsf{W}}}$ due to a pair of force dipoles similar to that represented by $\boldsymbol \varDelta ^{\pm }$, except with each dipole weighted by the viscosity of the phase in which it resides. This velocity profile is given by

(5.24)\begin{align} \bar\mu {\mathsf {W}}_{i{\alpha}}(\boldsymbol x) &= \left({ \mu_1 {\lim_{\boldsymbol y \to \boldsymbol 0^{+}}} - {\mu_2 \lim_{\boldsymbol y \to \boldsymbol 0^{-}}} }\right) \frac{\partial {\mathsf {H}}_{i{\alpha}}(\boldsymbol x, \boldsymbol y)}{\partial h} \nonumber\\ &= \left[\frac{\partial \varPi_k(\,\boldsymbol y_s, \boldsymbol x)}{\partial y_\beta} - \mu_s \frac{\partial^{2} {\mathsf {H}}_{i{\alpha}}(\boldsymbol x, \boldsymbol y_s)}{\partial y^{2}_{\alpha}}\right]_{\boldsymbol y_s = \boldsymbol 0} \nonumber\\ &= \bar\mu {\mathsf {q}}_{i{\alpha}}(\boldsymbol x) - \mu_s \nabla^{2} {\mathsf {H}}^{\,0}_{i{\alpha}}(\boldsymbol x), \end{align}

for $\boldsymbol x \neq \boldsymbol 0$, where $\boldsymbol y_s = y_1 \hat {\boldsymbol \imath }_1 + y_2 \hat {\boldsymbol \imath }_2$ is the surface projection of $\boldsymbol y$ and

(5.25)\begin{align} {\mathsf {q}}_{i{\alpha}}(\boldsymbol x) &= \frac{1}{\bar\mu} \left.{ \frac{\partial \varPi_i(\,\boldsymbol y_s, \boldsymbol x)}{\partial y_{\alpha}} }\right|_{\boldsymbol y_s = \boldsymbol 0} \nonumber\\ &=\frac{s_i s_{\alpha}}{4{\rm \pi}\bar\mu} \left({ \frac{3|z|}{r^{5}} + \frac{2|z|}{s^{4} r} + \frac{|z|}{s^{2} r^{3}} - \frac{2}{s^{4}} }\right) + \frac{\delta^{\shortparallel}_{i{\alpha}}}{4{\rm \pi}\bar\mu} \left( { -\frac{|z|}{r^{3}} - \frac{|z|}{s^{2} r} + \frac{1}{s^{2}} }\right) - \frac{3 z |z| s_{\alpha} \delta_{i3}}{4{\rm \pi}\bar\mu r^{5}}. \end{align}

The mode analogous to ${\boldsymbol{\mathsf{W}}}$ for a clean interface is null due to continuity of tangential stress (4.2a). Equation (5.24) shows that the gradient of $\boldsymbol \varPi$ can be reinterpreted as the velocity field ${\boldsymbol{\mathsf{q}}}(\boldsymbol x)$ (5.25). This velocity field decays as $1/r^{2}$ for all $r$ and is mirror symmetric about $z=0$. On the $z=0$ plane, ${\boldsymbol{\mathsf{q}}}$ takes the same functional form as a two-dimensional source–sink doublet, ${\mathsf {q}}_{\alpha\beta}(\boldsymbol s)(\boldsymbol s) = -\{s_{\alpha} s_{\beta} \}_0 / 2{\rm \pi} \bar \mu s^{2}$.

We may rewrite the first equality of (5.24) as

(5.26)\begin{align} \bar\mu {\mathsf {W}}_{k{\beta}}(\boldsymbol x) &= \left[\frac{\mu_1 + \mu_2}{2} \left( { {\lim_{\boldsymbol y \to \boldsymbol 0^{+}}} - {\lim_{\boldsymbol y \to \boldsymbol 0^{-}}} }\right) + \frac{\mu_1 - \mu_2}{2} \left( {\lim_{\boldsymbol y \to \boldsymbol 0^{+}}} + {\lim_{\boldsymbol y \to \boldsymbol 0^{-}}} \right) \right] \frac{\partial {\mathsf {H}}_{k{\beta}}(\boldsymbol x, \boldsymbol y)}{\partial h} \nonumber\\ &= \bar\mu \varDelta^{{\pm}}_{k{\beta}}(\boldsymbol x) + \frac{\mu_1 - \mu_2}{2} \varDelta^{{\ddagger}}_{k{\beta}}(\boldsymbol x). \end{align}

Solving (5.26) for $\boldsymbol \varDelta ^{\pm }$ and using the final equality of (5.24), we find that

(5.27)\begin{equation} \varDelta^{{\pm}}_{i{\alpha}}(\boldsymbol x) = {\mathsf {q}}_{i{\alpha}}(\boldsymbol x) - {L_{B}} \nabla^{2} {\mathsf {H}}^{\,0}_{i{\alpha}}(\boldsymbol x) - \frac{\mu_1 - \mu_2}{2\bar\mu}\varDelta^{{\ddagger}}_{i{\alpha}}(\boldsymbol x). \end{equation}

The appearance of $\nabla ^{2} {\boldsymbol{\mathsf{H}}}^{0}$ in (5.27) is peculiar because $\nabla ^{2} {\boldsymbol{\mathsf{H}}}^{0}(\boldsymbol x) = \nabla ^{2} {\boldsymbol{\mathsf{H}}}(\boldsymbol x, \boldsymbol y)|_{\boldsymbol y = \boldsymbol 0}$ has the functional form of a degenerate quadrupole, whose appearance we anticipate in the quadrupolar term $\boldsymbol u^{m2}$ of (5.11) but not the dipolar term $\boldsymbol u^{m1}$. Indeed, $\nabla ^{2} {\boldsymbol{\mathsf{H}}}^{0}(\boldsymbol x)$ decays as $1/r^{3}$ for $r \gg {L_{B}}$, whereas all other contributions to $\boldsymbol u^{m1}$ in (5.17) decay as $1/r^{2}$. However, $\boldsymbol \varDelta ^{\pm }$ describes a mode that is conceptually similar to a ‘usual’ quadrupole: two dipoles of opposite sign separated by a vanishingly small distance. The key difference here is that these dipoles straddle the interface, creating a Marangoni-driven flow that decays from the singular point as $1/r^{2}$ in addition to a quadrupolar flow. This result suggests an adjustment to our original definition of the dipole given by (5.14) where we exclude the $\nabla ^{2} {\boldsymbol{\mathsf{H}}}^{0}$ term in $\boldsymbol \varDelta ^{\pm }$, which is instead incorporated into a modified quadrupolar term ($\boldsymbol u^{m2}$) in (5.11). Denoting the modified dipole as ${\boldsymbol u^{m1}}'$, we combine (5.17) and (5.27) and subtract the degenerate quadrupole from the result to give

(5.28)\begin{equation} {u^{{m1}}_i}' (\boldsymbol x) = ({{\mathsf {S}}^{\shortparallel}_{{\alpha}{\beta}} + \tfrac 12 L_3 \varepsilon_{3{\alpha}{\beta}}}) \varDelta^{\shortparallel}_{i{\alpha}{\beta}}({L_{B}}; \boldsymbol x) + \tfrac 12 A_{\alpha} \varDelta^{{\ddagger}}_{i{\alpha}}(\boldsymbol x) + \tfrac 12 B_{\alpha} {\mathsf {q}}_{i{\alpha}}(\boldsymbol x), \end{equation}

where $\boldsymbol A$ is defined by (4.22c), as it was for a clean interface, and $B_{\alpha} = ({\mathsf {D}}^{(1)}_{{\alpha} 3} - {\mathsf {D}}^{(2)}_{{\alpha} 3})$.

Each term of (5.28) represents a distinct contribution to ${\boldsymbol u^{m1}}'$. Only the first term of ${\boldsymbol u^{m1}}'$ in (5.28) is sensitive to surface viscosity, hence our inclusion of ${L_{B}}$ as a argument to $\boldsymbol \varDelta ^{\shortparallel }$ here. Like the monopolar velocity field, $\boldsymbol \varDelta ^{\shortparallel }$ produces a ‘lamellar’ fluid motion, with no flow normal to the interface. Its coefficients ${\boldsymbol{\mathsf{S}}}^{\shortparallel }$ and $L_3$ are the tensorial strengths of the stresslet parallel to the interface and the rotlet normal to the interface, respectively. The associated modes differ significantly from their counterparts on a clean interface. The ${\boldsymbol{\mathsf{S}}}^{\shortparallel }$ (stresslet) mode is affected by Marangoni and surface-viscous stresses; it is distinct from the analogous mode on a clean interface even if ${L_{B}} = 0$. Figure 8 visualizes the ${\boldsymbol{\mathsf{S}}}^{\shortparallel }$ contribution to ${\boldsymbol u^{m1}}'$ for different values of ${L_{B}}$. On the other hand, the $L_3$ (rotlet) mode does not generate a surface pressure gradient because $\boldsymbol \varPi (\boldsymbol s, h=0)$ is harmonic. It is therefore affected by surface viscosity but does not contribute to Marangoni stresses. For an interface with vanishing surface viscosity, the $L_3$ mode for an incompressible interface does not differ from that of a clean interface; both assume the functional form of a rotlet in a bulk fluid of viscosity $\bar \mu$.

Figure 8. The interfacial velocity and surface pressure fields associated with the ${\boldsymbol{\mathsf{S}}}^{\shortparallel }$ mode are shown for the case of an extensile force dipole. The panels are analogous to those shown in figure 7. The arrows indicate the orientation of the force dipole. Panels (a), (b) and (c) represent one of three leading-order flow modes induced by self-propelled (force- and torque-free) colloids for small-, moderate- and large-$\mathinner {Bq}$, respectively. (a) $L_B=0$, (b) finite $L_B$, (c) $L_B \rightarrow \infty$, (d) clean and (e) surface pressure.

Examining the behaviour of $\boldsymbol \varDelta ^{\shortparallel }$ for ${L_{B}} = 0$, (5.19) and (B20) give

(5.29)\begin{equation} {\varDelta^{\shortparallel}_{{\alpha}{\beta}{\gamma}}(\boldsymbol x)}|_{{L_{B}}=0} = \frac{(3r + |z|) \{ {s_{\alpha} s_{\beta} s_{\gamma}} \}_0}{4{\rm \pi} \bar\mu r^{3} (r + |z|)^{3}} - \frac{s_{\alpha} \delta_{{\beta}{\gamma}} + s_{\beta} \delta_{{\gamma}{\alpha}} - 3 s_{\gamma} \delta_{{\alpha}{\beta}}} {16 {\rm \pi}\bar\mu r^{3}}, \end{equation}

which decays spatially as $1/r^{2}$, similar to the analogous ${\boldsymbol{\mathsf{S}}}^{\shortparallel }$ and $L_3$ modes for a clean interface. As ${L_{B}} \to \infty$, $\boldsymbol \varDelta ^{\shortparallel }$ approaches the functional form of a two-dimensional Stokeslet dipole, given by the gradient of equation (5.5),

(5.30)\begin{equation} \varDelta^{\shortparallel}_{{\alpha}{\beta}{\gamma}}|_{{L_{B}}\to\infty} \sim \frac{ 2 \hat s_{\alpha} \hat s_{\beta} \hat s_{\gamma} - \hat s_{\alpha} \delta_{{\beta}{\gamma}} - \hat s_{\beta} \delta_{{\gamma}{\alpha}} + \hat s_{\gamma} \delta_{{\alpha}{\beta}} }{4{\rm \pi}\mu_{s}s}. \end{equation}

Thus, this dipolar mode decays as $1/s$ as $s \to \infty$, as opposed to the logarithmically divergent behaviour of the monopole in the same limit. However, as a three-dimensional flow, (5.30) is constant along the $z$ axis, in conflict with the condition that $\boldsymbol u$ vanishes as $z \to \infty$. This behaviour is apparently another manifestation of Stokes’ paradox. Physically, if we consider a colloid of size $a$ and $\mathinner {Bq}$ large but finite, bulk-viscous effects eventually dominate over surface-viscous effects at distances $r/a \gg \mathinner {Bq}$ from the colloid, and a $1/r^{2}$ velocity decay is recovered in the far field.

The second term in (5.28), the $\boldsymbol A$ mode, is unaltered from the expression for $\boldsymbol u^{m1}$ for a clean interface (4.21). Of the three terms, only this one is sensitive to the viscosity contrast. We refer the reader back to figure 6(c) for a depiction of the associated velocity field. Marangoni and surface-viscous stresses have no effect on this mode, which fails to induce motion of the interface. Its interpretation regarding colloid asymmetry is therefore the same as discussed in § 4.

The final term of (5.17), the $\boldsymbol B$ mode, has no analogue on a clean interface. Physically, the $\boldsymbol B$ mode accounts for the coupling of off-interface forcing of the fluid by the colloid to Marangoni forces that react to enforce interfacial incompressibility. Interestingly, the coefficient $\boldsymbol B$ does not explicitly involve forces on the contact line, in contrast to ${\boldsymbol{\mathsf{S}}}^{\shortparallel }$ and $L_3$, but the resulting flow is nonetheless driven by the surface pressure according to (5.25). The interfacial flow field due to this mode is harmonic, $\nabla _{s}^{2} {\boldsymbol{\mathsf{q}}}(\boldsymbol s) = \boldsymbol 0$ (figure 9a). By (2.7), it therefore does not induce surface-viscous stresses, explaining why ${\boldsymbol{\mathsf{q}}}$ is independent of ${L_{B}}$. Like the ${\boldsymbol{\mathsf{S}}}^{\shortparallel }$ and $L_3$ modes, the ${\boldsymbol{\mathsf{B}}}$ mode produces a mirror-symmetric velocity field about the interface. However, while the ${\boldsymbol{\mathsf{S}}}^{\shortparallel }$ and $L_3$ modes produce a lamellar velocity field, the velocity field due to the $\boldsymbol B$ mode is fully three-dimensional (figure 9b). The surface pressure associated with ${\mathsf {q}}_{i{\alpha} }(\boldsymbol x)$ is given by ${\partial \varPi _{\alpha} (\boldsymbol s, h \hat {\boldsymbol \imath }_3)} / {\partial h} = - s_{\alpha} / 2{\rm \pi} s^{3}$ (figure 9c).

Figure 9. The velocity field $\boldsymbol B \boldsymbol \cdot {\boldsymbol{\mathsf{q}}}(\boldsymbol x)$ is visualized for the vector $\boldsymbol B$ pointing rightward. This mode is associated with a (viscosity-weighted) pair of force couplets indicated by the arrows. The flow on the interface is shown in (a), and a perpendicular cross-section of the off-interface flow is shown in (b). The associated surface pressure is shown in (c).

Both the $\boldsymbol B$ and $\boldsymbol A$ modes can be attributed to the finite protrusion of an adhered colloid from the interface. As the protrusion distance becomes small (or as the geometry of the colloid becomes flat) $\boldsymbol A$ and $\boldsymbol B$ vanish. It is interesting that finite-size effects enter at dipolar order in this manner. Consider again the velocity profile due to a sphere straddling a clean interface with a 90$^{\circ }$ contact angle. Here, the size of the sphere matters only at quadrupolar order, through the $a^{2} \nabla ^{2} {\boldsymbol{\mathsf{J}}}$ term in (4.25). In contrast to the $1/r^{3}$ decay of this term, the dipolar $\boldsymbol A$ and $\boldsymbol B$ modes decay as $1/r^{2}$. On an incompressible (or clean) interface, $\boldsymbol A$ vanishes by symmetry for this particular case, but we expect $\boldsymbol B \sim a \boldsymbol F$, where $\boldsymbol F$ is the force driving the colloid. It is intriguing how this finite-protrusion effect arises from interfacial Marangoni forces.