2.1 Autophoretic propulsion

A phoretic swimmer achieves propulsion by catalysing a chemical reaction in the surrounding solute fuel, denoted by its ‘activity’ ${\mathcal{A}}(\boldsymbol{x})$, for $\boldsymbol{x}$ a point on the swimmer surface. The activity represents concentration flux and may vary across the swimmer, with ${\mathcal{A}}>0$, ${\mathcal{A}}<0$ corresponding to release and or consumption of solute respectively. Local concentration gradients arise from spatial variation in the activity and confinement effects. These gradients result in local pressure imbalances in a thin boundary layer at the swimmer surface, driving a surface slip flow (Anderson Reference Anderson1989; Michelin & Lauga Reference Michelin and Lauga2014). This slip flow is locally proportional to the concentration gradient, with the swimmer’s ‘mobility’ ${\mathcal{M}}(\boldsymbol{x})$ as the (spatially varying) constant of proportionality. The mobility may be positive or negative, depending on whether the slip flow moves up or down the concentration gradient.

We consider neutral solute self-diffusiophoresis, where electrokinetic effects are absent (Brown & Poon Reference Brown and Poon2014; Ebbens et al. Reference Ebbens, Gregory, Dunderdale, Howse, Ibrahim, Liverpool and Golestanian2014), and work in the limit of zero Péclet number (Golestanian, Liverpool & Ajdari Reference Golestanian, Liverpool and Ajdari2007) where diffusion dominates and advection of the solute due to flow can be neglected. This limit is appropriate provided that the particle is smaller than $r_{c}=D/U$, for $D$ solute diffusivity and $U$ the typical phoretic velocity (Michelin & Lauga Reference Michelin and Lauga2014). For platinum-coated Janus particles in hydrogen peroxide solution, this critical radius corresponds to $r_{c}\approx 10{-}100~\unicode[STIX]{x03BC}\text{m}$ (Howse et al. Reference Howse, Jones, Ryan, Gough, Vafabakhsh and Golestanian2007). For a detailed discussion of propulsion at finite Péclet number, see Michelin & Lauga (Reference Michelin and Lauga2014).

At zero Péclet number, the solute dynamics decouples from the flow dynamics at any instant, so that the concentration field $c(\boldsymbol{x},t)$ is found by solving Laplace’s equation

(2.1)$$\begin{eqnarray}D\unicode[STIX]{x1D6FB}^{2}c=0,\end{eqnarray}$$

in the region outside the swimmer, with surface $S$, subject to the boundary condition

(2.2)$$\begin{eqnarray}-D\boldsymbol{n}_{f}\boldsymbol{\cdot }\unicode[STIX]{x1D735}c|_{S}={\mathcal{A}}(\boldsymbol{x})|_{S},\end{eqnarray}$$

with $\boldsymbol{n}_{f}$ the outward normal to the swimmer’s surface, pointing into the fluid, and $D$ the solute diffusivity. The solution of the diffusion problem is then used to calculate the surface slip velocity,

(2.3)$$\begin{eqnarray}\boldsymbol{v}_{slip}|_{S}={\mathcal{M}}(\boldsymbol{x})(\mathbf{1}-\boldsymbol{n}_{f}\boldsymbol{n}_{f})\boldsymbol{\cdot }\unicode[STIX]{x1D735}c,\end{eqnarray}$$

which gives the body-frame boundary conditions for solving the Stokes flow problem

(2.4)$$\begin{eqnarray}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}-\unicode[STIX]{x1D735}p=\mathbf{0},\quad \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0.\end{eqnarray}$$

This slip flow will in general propel the swimmer with translational velocity $\boldsymbol{U}_{sw}$, and angular velocity $\unicode[STIX]{x1D734}_{sw}$, which are found by enforcing the constraints that no net force or torque acts on the swimmer.

Thus, given an activity and mobility, the task of this paper is to find a convenient slender body approximation for the concentration field, and slip flow that results from its surface gradient. This flow will drive the kinematics of the filament, which will change depending on its centreline shape. The set-up of the problem and governing equations are shown in figure 1.

Figure 1.

A schematic of a slender phoretic filament in three dimensions, in the shape of a helix. Surface lines are used to emphasise curvature and torsion. Laplace’s equation is solved in the bulk, with flux boundary condition on the surface of the filament $S$ determined by its activity ${\mathcal{A}}(\boldsymbol{x})|_{S}$. In later examples, we consider axisymmetric activity in the form of catalytic end caps (demarcated here by the red) with an inert midpiece, so that ${\mathcal{A}}(s)=-1$, $|s|>s_{c}$, ${\mathcal{A}}(s)=0$ otherwise.

2.2 Filament geometry

We begin by describing the filament geometry, following the approach of Koens & Lauga (Reference Koens and Lauga2018). The filament centreline $\boldsymbol{r}(s)$ is parametrised by its arclength $s\in [-l,l]$, where $2l$ is the total contour length. The centreline tangent $\hat{\boldsymbol{t}}(s)$, normal $\hat{\boldsymbol{n}}(s)$, and binormal $\hat{\boldsymbol{b}}(s)$ satisfy the Serret–Frenet equations,

(2.5a-c)$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\hat{\boldsymbol{t}}}{\unicode[STIX]{x2202}s}=\unicode[STIX]{x1D705}\hat{\boldsymbol{n}},\quad \frac{\unicode[STIX]{x2202}\hat{\boldsymbol{n}}}{\unicode[STIX]{x2202}s}=-\unicode[STIX]{x1D705}\hat{\boldsymbol{t}}+\unicode[STIX]{x1D70F}\hat{\boldsymbol{b}},\quad \frac{\unicode[STIX]{x2202}\hat{\boldsymbol{b}}}{\unicode[STIX]{x2202}s}=-\unicode[STIX]{x1D70F}\hat{\boldsymbol{n}},\end{eqnarray}$$

where $\unicode[STIX]{x1D705}(s)$ and $\unicode[STIX]{x1D70F}(s)$ are the curvature and torsion of the filament centreline, respectively.

2.2.1 Parametrising the surface

The filament cross-sectional radius, which may vary along the filament, takes the value $r_{f}\unicode[STIX]{x1D70C}(s)$ at $s$, where $r_{f}$ is the maximal radius and $\unicode[STIX]{x1D70C}(s)\in [0,1]$. The surface of the filament is then parametrised by $s,r_{f}\unicode[STIX]{x1D70C}(s)$, and the azimuthal angle of the cross-section, $\unicode[STIX]{x1D703}\in [-\unicode[STIX]{x03C0},\unicode[STIX]{x03C0}]$, by

(2.6)$$\begin{eqnarray}\boldsymbol{S}(s,\unicode[STIX]{x1D703})=\boldsymbol{r}(s)+r_{f}\unicode[STIX]{x1D70C}(s)\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(s,\unicode[STIX]{x1D703}).\end{eqnarray}$$

The local radial unit vector perpendicular to the centreline tangent, $\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(s,\unicode[STIX]{x1D703})$, is given by

(2.7)$$\begin{eqnarray}\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(s,\unicode[STIX]{x1D703})=\cos \unicode[STIX]{x1D703}_{m}(s,\unicode[STIX]{x1D703})\hat{\boldsymbol{n}}(s)+\sin \unicode[STIX]{x1D703}_{m}(s,\unicode[STIX]{x1D703})\hat{\boldsymbol{b}}(s),\end{eqnarray}$$

where $\unicode[STIX]{x1D703}_{m}(s,\unicode[STIX]{x1D703})=\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{i}(s)$, with $\unicode[STIX]{x1D703}_{i}(s)$, accounting for the torsion of the curve, chosen to satisfy

(2.8)$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}_{i}}{\unicode[STIX]{x2202}s}=\unicode[STIX]{x1D70F}(s),\end{eqnarray}$$

as in Koens & Lauga (Reference Koens and Lauga2018). With this choice, the derivative of $\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}$ with respect to $s$ simplifies to

(2.9)$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x2202}s}=-\unicode[STIX]{x1D705}(s)\cos \unicode[STIX]{x1D703}_{m}(s,\unicode[STIX]{x1D703})\hat{\boldsymbol{t}}(s).\end{eqnarray}$$

2.2.2 Surface elements

To obtain the surface element, used later in our integrals, we calculate the derivatives

(2.10a)$$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{S}}{\unicode[STIX]{x2202}s} & = & \displaystyle \hat{\boldsymbol{t}}(s)[1-r_{f}\unicode[STIX]{x1D70C}(s)\unicode[STIX]{x1D705}(s)\cos \unicode[STIX]{x1D703}_{m}(s,\unicode[STIX]{x1D703})]\nonumber\\ \displaystyle & & \displaystyle +\,r_{f}\frac{\text{d}\unicode[STIX]{x1D70C}(s)}{\text{d}s}[\cos \unicode[STIX]{x1D703}_{m}(s,\unicode[STIX]{x1D703})\hat{\boldsymbol{n}}(s)+\sin \unicode[STIX]{x1D703}_{m}(s,\unicode[STIX]{x1D703})\hat{\boldsymbol{b}}(s)],\end{eqnarray}$$

(2.10b)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{S}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}=r_{f}\unicode[STIX]{x1D70C}(s)[-\text{sin}\,\unicode[STIX]{x1D703}_{m}(s,\unicode[STIX]{x1D703})\hat{\boldsymbol{n}}(s)+\cos \unicode[STIX]{x1D703}_{m}(s,\unicode[STIX]{x1D703})\hat{\boldsymbol{b}}(s)], & \displaystyle\end{eqnarray}$$

which gives the surface element as

(2.11)$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\boldsymbol{S}}{\unicode[STIX]{x2202}s}\times \frac{\unicode[STIX]{x2202}\boldsymbol{S}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}=r_{f}\unicode[STIX]{x1D70C}(s)\left(r_{f}\frac{\text{d}\unicode[STIX]{x1D70C}(s)}{\text{d}s}\hat{\boldsymbol{t}}(s)-[1-r_{f}\unicode[STIX]{x1D70C}(s)\unicode[STIX]{x1D705}(s)\cos \unicode[STIX]{x1D703}_{m}(s,\unicode[STIX]{x1D703})]\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(s,\unicode[STIX]{x1D703})\right),\end{eqnarray}$$

with magnitude

(2.12)$$\begin{eqnarray}\left|\frac{\unicode[STIX]{x2202}\boldsymbol{S}}{\unicode[STIX]{x2202}s}\times \frac{\unicode[STIX]{x2202}\boldsymbol{S}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}\right|=r_{f}\unicode[STIX]{x1D70C}(s)\sqrt{\left[r_{f}\frac{\text{d}\unicode[STIX]{x1D70C}(s)}{\text{d}s}\right]^{2}+[1-r_{f}\unicode[STIX]{x1D70C}(s)\unicode[STIX]{x1D705}(s)\cos \unicode[STIX]{x1D703}_{m}(s,\unicode[STIX]{x1D703})]^{2}}.\end{eqnarray}$$

2.3 Boundary integral equation for the diffusion equation

We begin with the well-known Green’s function for Laplace’s equation in an unconfined, three-dimensional region,

(2.13)$$\begin{eqnarray}{\mathcal{G}}(\boldsymbol{x},\tilde{\boldsymbol{x}})=\frac{1}{4\unicode[STIX]{x03C0}|\boldsymbol{x}-\tilde{\boldsymbol{x}}|},\end{eqnarray}$$

which solves Laplace’s equation, forced by a point sink at $\tilde{\boldsymbol{x}}$,

(2.14)$$\begin{eqnarray}\unicode[STIX]{x1D6FB}^{2}{\mathcal{G}}(\boldsymbol{x},\tilde{\boldsymbol{x}})=-\unicode[STIX]{x1D6FF}(\boldsymbol{x}-\tilde{\boldsymbol{x}}).\end{eqnarray}$$

Since in our unbounded domain we have translational invariance, we use the notation ${\mathcal{G}}(\boldsymbol{x},\tilde{\boldsymbol{x}})={\mathcal{G}}(\boldsymbol{x}-\tilde{\boldsymbol{x}})=1/4\unicode[STIX]{x03C0}|\boldsymbol{R}|$, where $\boldsymbol{R}=\boldsymbol{x}-\tilde{\boldsymbol{x}}$.

Using Green’s second identity for the functions $c(\tilde{\boldsymbol{x}})$ and ${\mathcal{G}}(\tilde{\boldsymbol{x}}-\boldsymbol{x})$, in the body of the fluid outside the filament $V$, bounded by filament surface $S$, with normal $\boldsymbol{n}_{f}$ pointing out of the filament, and the notation $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}n_{f}=\boldsymbol{n}_{f}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{\tilde{\boldsymbol{x}}}$, we have

(2.15)$$\begin{eqnarray}\displaystyle & & \displaystyle \int _{V}(c(\tilde{\boldsymbol{x}})\unicode[STIX]{x1D6FB}_{\tilde{\boldsymbol{x}}}^{2}{\mathcal{G}}(\tilde{\boldsymbol{x}}-\boldsymbol{x})-{\mathcal{G}}(\tilde{\boldsymbol{x}}-\boldsymbol{x})\unicode[STIX]{x1D6FB}_{\tilde{\boldsymbol{x}}}^{2}c(\tilde{\boldsymbol{x}}))\,\text{d}V(\tilde{\boldsymbol{x}})\nonumber\\ \displaystyle & & \displaystyle \quad =-\int _{S}\left(c(\tilde{\boldsymbol{x}})\boldsymbol{n}_{f}\boldsymbol{\cdot }\frac{\unicode[STIX]{x2202}{\mathcal{G}}(\tilde{\boldsymbol{x}}-\boldsymbol{x})}{\unicode[STIX]{x2202}\tilde{\boldsymbol{x}}}-{\mathcal{G}}(\tilde{\boldsymbol{x}}-\boldsymbol{x})\boldsymbol{n}_{f}\boldsymbol{\cdot }\frac{\unicode[STIX]{x2202}c(\tilde{\boldsymbol{x}})}{\unicode[STIX]{x2202}\tilde{\boldsymbol{x}}}\right)\text{d}S(\tilde{\boldsymbol{x}}),\end{eqnarray}$$

which simplifies to the classic boundary integral formulation

(2.16)$$\begin{eqnarray}\unicode[STIX]{x1D706}c(\boldsymbol{x})=\frac{1}{4\unicode[STIX]{x03C0}}\int _{S}\left[\frac{c(\tilde{\boldsymbol{x}})\boldsymbol{n}_{f}\boldsymbol{\cdot }(\boldsymbol{x}-\tilde{\boldsymbol{x}})}{|\boldsymbol{x}-\tilde{\boldsymbol{x}}|^{3}}+\frac{{\mathcal{A}}(\tilde{\boldsymbol{x}})}{D|\boldsymbol{x}-\tilde{\boldsymbol{x}}|}\right]\text{d}S(\tilde{\boldsymbol{x}}),\end{eqnarray}$$

where $\unicode[STIX]{x1D706}=1/2$ for $\boldsymbol{x}\in S$, $\unicode[STIX]{x1D706}=1$ for $\boldsymbol{x}\in V$, and $\unicode[STIX]{x1D706}=0$ for $\boldsymbol{x}\notin V$ (Pozrikidis Reference Pozrikidis1992).

We now substitute the filament geometry into (2.16). For $\boldsymbol{x}=\boldsymbol{S}(s,\unicode[STIX]{x1D703})$ and $\tilde{\boldsymbol{x}}=\boldsymbol{S}(\tilde{s},\tilde{\unicode[STIX]{x1D703}})$ points on the filament surface, we set $\unicode[STIX]{x1D706}=1/2$, and using the notation,

(2.17)$$\begin{eqnarray}\boldsymbol{R}\equiv \boldsymbol{S}(s,\unicode[STIX]{x1D703})-\boldsymbol{S}(\tilde{s},\tilde{\unicode[STIX]{x1D703}})=\boldsymbol{R}_{0}(s,\tilde{s})+r_{f}[\unicode[STIX]{x1D70C}(s)\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(s,\unicode[STIX]{x1D703})-\unicode[STIX]{x1D70C}(\tilde{s})\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(\tilde{s},\tilde{\unicode[STIX]{x1D703}})],\end{eqnarray}$$

with $\boldsymbol{R}_{0}\equiv \boldsymbol{r}(s)-\boldsymbol{r}(\tilde{s})$, and $\boldsymbol{n}_{f}\,\text{d}S_{\tilde{\boldsymbol{x}}}=-(\unicode[STIX]{x2202}\boldsymbol{S}/\unicode[STIX]{x2202}\tilde{s})\times (\unicode[STIX]{x2202}\boldsymbol{S}/\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D703}})\,\text{d}\tilde{\unicode[STIX]{x1D703}}\,\text{d}\tilde{s}$ (where the sign is due to definition of $\boldsymbol{n}_{f}$ pointing out of the filament), we can write (2.16) as

(2.18)$$\begin{eqnarray}2\unicode[STIX]{x03C0}c(s,\unicode[STIX]{x1D703})=\int _{-l}^{l}\int _{-\unicode[STIX]{x03C0}}^{\unicode[STIX]{x03C0}}\left[\frac{{\mathcal{A}}(\tilde{s},\tilde{\unicode[STIX]{x1D703}})}{D|\boldsymbol{R}|}\left|\frac{\unicode[STIX]{x2202}\boldsymbol{S}}{\unicode[STIX]{x2202}\tilde{s}}\times \frac{\unicode[STIX]{x2202}\boldsymbol{S}}{\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D703}}}\right|-\frac{c(\tilde{s},\tilde{\unicode[STIX]{x1D703}})\boldsymbol{R}}{|\boldsymbol{R}|^{3}}\boldsymbol{\cdot }\left(\frac{\unicode[STIX]{x2202}\boldsymbol{S}}{\unicode[STIX]{x2202}\tilde{s}}\times \frac{\unicode[STIX]{x2202}\boldsymbol{S}}{\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D703}}}\right)\right]\text{d}\tilde{\unicode[STIX]{x1D703}}\,\text{d}\tilde{s},\end{eqnarray}$$

with the surface element and its magnitude given by (2.11) and (2.12) respectively.

While the boundary integral (2.18) now includes the filament geometry, it does not yet use the approximation that the filament is slender. This approximation will allow us, after performing matched asymptotics (details in the appendices), to write the double integral equation (2.18) into a single integral formula for evaluating the concentration on the filament surface.

We non-dimensionalise lengths by $r_{f}$, activity by a typical activity $[{\mathcal{A}}]$ and concentration by a typical concentration $[c]$ taken as $[c]\equiv [{\mathcal{A}}]r_{f}/D$. The last choice comes from considering the boundary condition $-D\boldsymbol{n}_{f}\boldsymbol{\cdot }\unicode[STIX]{x1D735}c|_{S}={\mathcal{A}}|_{S}$, and noting that, since $\boldsymbol{n}_{f}$ is mostly aligned with $\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}$, $\boldsymbol{n}_{f}\boldsymbol{\cdot }\unicode[STIX]{x1D735}c$ scales as $[c]/r_{f}$. For a typical mobility scale $[{\mathcal{M}}]$, using that $[c]\sim [{\mathcal{A}}]r_{f}/D$, and scaling $\boldsymbol{v}_{slip}\sim [{\mathcal{M}}][{\mathcal{A}}]r_{f}/(Dl)$, (i.e. $[\unicode[STIX]{x2202}c/\unicode[STIX]{x2202}s]\sim [c]/l$). All quantities are henceforth non-dimensional, unless otherwise stated.

2.4 Azimuthal versus longitudinal slip flows in the slender limit

Before deriving the slender body approximation of (2.18), we sketch out the relevant terms in the theory, and the assumptions we will make. Defining $\unicode[STIX]{x1D716}=r_{f}/l$, the slenderness parameter, we are interested in the leading-order swimming velocity, which is determined by the leading-order slip velocity.

For the majority of the filament, this slip velocity is equal to ${\mathcal{M}}\unicode[STIX]{x2202}c/\unicode[STIX]{x2202}s$ longitudinally, and ${\mathcal{M}}(\unicode[STIX]{x1D716}\unicode[STIX]{x1D70C})^{-1}\unicode[STIX]{x2202}c/\unicode[STIX]{x2202}\unicode[STIX]{x1D703}$ azimuthally. For non-axisymmetric chemical patterning ${\mathcal{A}}(s,\unicode[STIX]{x1D703})$, we might expect variations of $c(s,\unicode[STIX]{x1D703})$ in $s$ and $\unicode[STIX]{x1D703}$ to be of the same order, so that azimuthal slip flows will be $O(\unicode[STIX]{x1D716}^{-1})$ and dominate the dynamics.

For axisymmetric chemical patterning ${\mathcal{A}}(s)$, a straight rod will have no azimuthal concentration variation, by symmetry. However, a curved rod will have small azimuthal variations in concentration arising from geometric confinement. We will show that these variations are in general $O(\unicode[STIX]{x1D716})$, and as a consequence contribute to the slip flow at the same order as $\unicode[STIX]{x2202}c/\unicode[STIX]{x2202}s$. As such, a consistent leading-order SPT expansion of the velocity requires $O(\unicode[STIX]{x1D716})$ expansion of the concentration field, in contrast to other slender body theories (Johnson Reference Johnson1979; Götz Reference Götz2000; Yariv Reference Yariv2008; Schnitzer & Yariv Reference Schnitzer and Yariv2015; Ibrahim et al. Reference Ibrahim, Golestanian and Liverpool2017; Koens & Lauga Reference Koens and Lauga2018; Yariv Reference Yariv2019).

For the derivation of the following theory, we will assume that the activity ${\mathcal{A}}(s,\unicode[STIX]{x1D703})$ and curvature $\unicode[STIX]{x1D705}(s)$ of the filament are slowly varying with $s$. We will also assume that either the filament as prolate spheroidal cross-section $\unicode[STIX]{x1D70C}(s)=\sqrt{(1-s^{2})}$, or the activity is zero near the filament ends. From a practical calculation standpoint, our validation against boundary element simulation will demonstrate that these assumptions can often be relaxed. Finally, to allow a convenient decomposition of the azimuthal slip velocity into modes, for simple use in the slender body theory of Koens & Lauga (Reference Koens and Lauga2018), we will also assume axisymmetric mobility ${\mathcal{M}}(s)$, though this assumption does not come into the derivation of the slender phoretic theory itself.

2.5 Asymptotic expansion of the boundary integral kernels

The surface element from (2.11), now in its non-dimensionalised form,

(2.19)$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\boldsymbol{S}}{\unicode[STIX]{x2202}\tilde{s}}\times \frac{\unicode[STIX]{x2202}\boldsymbol{S}}{\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D703}}}=\unicode[STIX]{x1D716}\unicode[STIX]{x1D70C}\left(-\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(\tilde{s},\tilde{\unicode[STIX]{x1D703}})+\unicode[STIX]{x1D716}\left[\frac{\text{d}\unicode[STIX]{x1D70C}(\tilde{s})}{\text{d}\tilde{s}}\hat{\boldsymbol{t}}(\tilde{s})+\unicode[STIX]{x1D70C}(\tilde{s})\unicode[STIX]{x1D705}(\tilde{s})\cos \unicode[STIX]{x1D703}_{m}(\tilde{s},\tilde{\unicode[STIX]{x1D703}})\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(\tilde{s},\tilde{\unicode[STIX]{x1D703}})\right]\right),\end{eqnarray}$$

has magnitude

(2.20)$$\begin{eqnarray}\displaystyle \left|\frac{\unicode[STIX]{x2202}\boldsymbol{S}}{\unicode[STIX]{x2202}\tilde{s}}\times \frac{\unicode[STIX]{x2202}\boldsymbol{S}}{\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D703}}}\right| & = & \displaystyle \unicode[STIX]{x1D716}\unicode[STIX]{x1D70C}(\tilde{s})\sqrt{\unicode[STIX]{x1D716}^{2}\left(\frac{\text{d}\unicode[STIX]{x1D70C}(\tilde{s})}{\text{d}\tilde{s}}\right)^{2}+[1-\unicode[STIX]{x1D716}\unicode[STIX]{x1D70C}(\tilde{s})\unicode[STIX]{x1D705}(\tilde{s})\cos \unicode[STIX]{x1D703}_{m}(\tilde{s},\tilde{\unicode[STIX]{x1D703}})]^{2}}\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D716}\unicode[STIX]{x1D70C}(\tilde{s})[1-\unicode[STIX]{x1D716}\unicode[STIX]{x1D70C}(\tilde{s})\unicode[STIX]{x1D705}(\tilde{s})\cos \unicode[STIX]{x1D703}_{m}(\tilde{s},\tilde{\unicode[STIX]{x1D703}})+O(\unicode[STIX]{x1D716}^{2})].\end{eqnarray}$$

The non-dimensionalised version of the boundary integral expression of (2.18) is

(2.21)$$\begin{eqnarray}2\unicode[STIX]{x03C0}c(s,\unicode[STIX]{x1D703})=\int _{-1}^{1}\int _{-\unicode[STIX]{x03C0}}^{\unicode[STIX]{x03C0}}[K_{1}(s,\unicode[STIX]{x1D703},\tilde{s},\tilde{\unicode[STIX]{x1D703}})+K_{2}(s,\unicode[STIX]{x1D703},\tilde{s},\tilde{\unicode[STIX]{x1D703}})]\,\text{d}\tilde{\unicode[STIX]{x1D703}}\,\text{d}\tilde{s},\end{eqnarray}$$

with the two kernels $K_{1}(s,\tilde{s},\unicode[STIX]{x1D703},\tilde{\unicode[STIX]{x1D703}}),K_{2}(s,\tilde{s},\unicode[STIX]{x1D703},\tilde{\unicode[STIX]{x1D703}})$ defined as

(2.22a)$$\begin{eqnarray}\displaystyle & \displaystyle K_{1}=\frac{\unicode[STIX]{x1D70C}(\tilde{s}){\mathcal{A}}(\tilde{s},\tilde{\unicode[STIX]{x1D703}})}{|\boldsymbol{R}|}\quad [1-\unicode[STIX]{x1D716}\unicode[STIX]{x1D70C}(\tilde{s})\unicode[STIX]{x1D705}(\tilde{s})\cos \unicode[STIX]{x1D703}_{m}(\tilde{s},\tilde{\unicode[STIX]{x1D703}})+O(\unicode[STIX]{x1D716}^{2})], & \displaystyle\end{eqnarray}$$

(2.22b)$$\begin{eqnarray}\displaystyle & \displaystyle K_{2}=\unicode[STIX]{x1D716}\frac{\unicode[STIX]{x1D70C}(\tilde{s})c(\tilde{s},\tilde{\unicode[STIX]{x1D703}})}{|\boldsymbol{R}|^{3}}\boldsymbol{R}\boldsymbol{\cdot }\left(\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(\tilde{s},\tilde{\unicode[STIX]{x1D703}})-\unicode[STIX]{x1D716}\left[\frac{\text{d}\unicode[STIX]{x1D70C}(\tilde{s})}{\text{d}\tilde{s}}\hat{\boldsymbol{t}}(\tilde{s})+\unicode[STIX]{x1D70C}(\tilde{s})\unicode[STIX]{x1D705}(\tilde{s})\cos \unicode[STIX]{x1D703}_{m}(\tilde{s},\tilde{\unicode[STIX]{x1D703}})\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(\tilde{s},\tilde{\unicode[STIX]{x1D703}})\right]\right). & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

In the slender limit where $\unicode[STIX]{x1D716}$ is small, for each evaluation parameter $s$ two regions of the integration variable $\tilde{s}$ arise, according to how $\tilde{s}-s$ compares to $\unicode[STIX]{x1D716}$. In the outer region, $\tilde{s}-s=O(1)$, the integration variable is far away from the evaluation arclength parameter. In the inner region, $\tilde{s}-s=O(\unicode[STIX]{x1D716})$, the integration variable is within $\unicode[STIX]{x1D716}$ arclength distance from the evaluation arclength. We proceed with a matched-asymptotic expansion of the integral kernels prior to their integration.

2.5.1 Outer region

In the outer region, $s-\tilde{s}=O(1)$, hence $\boldsymbol{R}_{0}(s,\tilde{s})=O(1)$, so we can approximate $\boldsymbol{R}\approx \boldsymbol{R}_{0}$. To first order,

(2.23)$$\begin{eqnarray}\boldsymbol{R}=\boldsymbol{R}_{0}(s,\tilde{s})+\unicode[STIX]{x1D716}\boldsymbol{{\mathcal{D}}}_{(\tilde{s},\tilde{\unicode[STIX]{x1D703}})}^{(s,\unicode[STIX]{x1D703})},\end{eqnarray}$$

where

(2.24)$$\begin{eqnarray}\boldsymbol{{\mathcal{D}}}_{(\tilde{s},\tilde{\unicode[STIX]{x1D703}})}^{(s,\unicode[STIX]{x1D703})}\equiv \unicode[STIX]{x1D70C}(s)\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(s,\unicode[STIX]{x1D703})-\unicode[STIX]{x1D70C}(\tilde{s})\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(\tilde{s},\tilde{\unicode[STIX]{x1D703}}).\end{eqnarray}$$

Expanding and collecting the orders of $\unicode[STIX]{x1D716}$ (see § A.1 for details), gives the outer expansions ${K_{1}}^{(o)},{K_{2}}^{(o)}$ of the kernels $K_{1},K_{2}$, defined in (2.22a)–(2.22b), as

(2.25)$$\begin{eqnarray}\displaystyle & \displaystyle {K_{1}}^{(o)}=\frac{\unicode[STIX]{x1D70C}(\tilde{s}){\mathcal{A}}(\tilde{s},\tilde{\unicode[STIX]{x1D703}})}{|\boldsymbol{R}_{0}(s,\tilde{s})|}\left(1-\unicode[STIX]{x1D716}\left[\unicode[STIX]{x1D70C}(\tilde{s})\unicode[STIX]{x1D705}(\tilde{s})\cos \unicode[STIX]{x1D703}_{m}(\tilde{s},\tilde{\unicode[STIX]{x1D703}})+\frac{\boldsymbol{R}_{0}}{|\boldsymbol{R}_{0}|^{2}}\boldsymbol{\cdot }\boldsymbol{{\mathcal{D}}}_{(\tilde{s},\tilde{\unicode[STIX]{x1D703}})}^{(s,\unicode[STIX]{x1D703})}\right]+O(\unicode[STIX]{x1D716}^{2})\right),\qquad & \displaystyle\end{eqnarray}$$

(2.26)$$\begin{eqnarray}\displaystyle {K_{2}}^{(o)} & = & \displaystyle \unicode[STIX]{x1D716}\frac{\unicode[STIX]{x1D70C}(\tilde{s})c(\tilde{s},\tilde{\unicode[STIX]{x1D703}})}{|\boldsymbol{R}_{0}|^{3}}\left(\vphantom{{\displaystyle \frac{3\boldsymbol{R}_{0}\boldsymbol{\cdot }\boldsymbol{{\mathcal{D}}}_{(\tilde{s},\tilde{\unicode[STIX]{x1D703}})}^{(s,\unicode[STIX]{x1D703})}}{|\boldsymbol{R}_{0}|^{2}}}}\boldsymbol{R}_{0}\boldsymbol{\cdot }\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(\tilde{s},\tilde{\unicode[STIX]{x1D703}})+\unicode[STIX]{x1D716}(\boldsymbol{{\mathcal{D}}}_{(\tilde{s},\tilde{\unicode[STIX]{x1D703}})}^{(s,\unicode[STIX]{x1D703})})\boldsymbol{\cdot }\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(\tilde{s},\tilde{\unicode[STIX]{x1D703}})\right.\nonumber\\ \displaystyle & & \displaystyle -\,\unicode[STIX]{x1D716}\boldsymbol{R}_{0}\boldsymbol{\cdot }\left[\frac{\text{d}\unicode[STIX]{x1D70C}(\tilde{s})}{\text{d}\tilde{s}}\hat{\boldsymbol{t}}(\tilde{s})+\unicode[STIX]{x1D70C}(\tilde{s})\unicode[STIX]{x1D705}(\tilde{s})\cos \unicode[STIX]{x1D703}_{m}(\tilde{s},\tilde{\unicode[STIX]{x1D703}})\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(\tilde{s},\tilde{\unicode[STIX]{x1D703}})\right]\nonumber\\ \displaystyle & & \displaystyle -\left.\unicode[STIX]{x1D716}(\boldsymbol{R}_{0}\boldsymbol{\cdot }\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(\tilde{s},\tilde{\unicode[STIX]{x1D703}}))\frac{3\boldsymbol{R}_{0}\boldsymbol{\cdot }\boldsymbol{{\mathcal{D}}}_{(\tilde{s},\tilde{\unicode[STIX]{x1D703}})}^{(s,\unicode[STIX]{x1D703})}}{|\boldsymbol{R}_{0}|^{2}}+O(\unicode[STIX]{x1D716}^{2})\right).\end{eqnarray}$$

2.5.2 Inner region

In the inner region, $s-\tilde{s}=O(\unicode[STIX]{x1D716})$, and we let $\tilde{s}=s+\unicode[STIX]{x1D716}\unicode[STIX]{x1D712}$, where $\unicode[STIX]{x1D712}$ is $O(1)$. The expansion and calculation, which we will now summarise, are given in detail in § A.2. Taylor expanding functions of $\tilde{s}$ around $s$ (for example, $\unicode[STIX]{x1D70C}(\tilde{s})=\unicode[STIX]{x1D70C}(s)+\unicode[STIX]{x1D716}\unicode[STIX]{x1D712}\,\text{d}\unicode[STIX]{x1D70C}(s)/\text{d}s+O(\unicode[STIX]{x1D716}^{2})$) gives the inner approximation for (2.17) as

(2.27)$$\begin{eqnarray}\boldsymbol{R}=\unicode[STIX]{x1D716}[\hat{\boldsymbol{R}}_{(i)}^{(1)}+\unicode[STIX]{x1D716}\hat{\boldsymbol{R}}_{(i)}^{(2)}]+O(\unicode[STIX]{x1D716}^{3}),\end{eqnarray}$$

where

(2.28a)$$\begin{eqnarray}\displaystyle & \hat{\boldsymbol{R}}_{(i)}^{(1)}=-\unicode[STIX]{x1D712}\hat{\boldsymbol{t}}(s)+\unicode[STIX]{x1D70C}(s)[\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(s,\unicode[STIX]{x1D703})-\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(s,\tilde{\unicode[STIX]{x1D703}})], & \displaystyle\end{eqnarray}$$

(2.28b)$$\begin{eqnarray}\displaystyle & \displaystyle \hat{\boldsymbol{R}}_{(i)}^{(2)}=-\left[\frac{1}{2}\unicode[STIX]{x1D712}^{2}\unicode[STIX]{x1D705}\hat{\boldsymbol{n}}(s)+\unicode[STIX]{x1D712}\frac{\text{d}\unicode[STIX]{x1D70C}(s)}{\text{d}s}\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(s,\tilde{\unicode[STIX]{x1D703}})-\unicode[STIX]{x1D712}\unicode[STIX]{x1D70C}(s)\unicode[STIX]{x1D705}(s)\cos \unicode[STIX]{x1D703}_{m}(s,\tilde{\unicode[STIX]{x1D703}})\hat{\boldsymbol{t}}(s)\right], & \displaystyle\end{eqnarray}$$

(2.28c)$$\begin{eqnarray}\displaystyle & |\hat{\boldsymbol{R}}_{(i)}^{(1)}+\unicode[STIX]{x1D716}\hat{\boldsymbol{R}}_{(i)}^{(2)}|=|\hat{\boldsymbol{R}}_{(i)}^{(1)}|[1+\unicode[STIX]{x1D716}\hat{\boldsymbol{R}}_{(i)}^{(1)}\boldsymbol{\cdot }\hat{\boldsymbol{R}}_{(i)}^{(2)}/|\hat{\boldsymbol{R}}_{(i)}^{(1)}|^{2}+O(\unicode[STIX]{x1D716}^{2})]. & \displaystyle\end{eqnarray}$$

Noting that

(2.29)$$\begin{eqnarray}|\hat{\boldsymbol{R}}_{(i)}^{(1)}|=\sqrt{\unicode[STIX]{x1D712}^{2}+\unicode[STIX]{x1D6FE}^{2}},\end{eqnarray}$$

with

(2.30)$$\begin{eqnarray}\unicode[STIX]{x1D6FE}^{2}=2\unicode[STIX]{x1D70C}^{2}(s)[1-\cos (\unicode[STIX]{x1D703}-\tilde{\unicode[STIX]{x1D703}})],\end{eqnarray}$$

the expansion of $1/|\boldsymbol{R}|$ will give rise to a factor $1/\sqrt{\unicode[STIX]{x1D712}^{2}+\unicode[STIX]{x1D6FE}^{2}}$. As a result, powers of $1/\sqrt{\unicode[STIX]{x1D712}^{2}+\unicode[STIX]{x1D6FE}^{2}}$ with different exponents appear in different terms of the expression of ${K_{1}}^{(i)},{K_{2}}^{(i)}$, given in § A.2.2.

Performing the integration in $\tilde{s}$, we treat the integrals of the form $I_{j}^{i}=\int _{-1}^{1}(\unicode[STIX]{x1D712}^{i}/\unicode[STIX]{x1D716}\sqrt{\unicode[STIX]{x1D712}^{2}+\unicode[STIX]{x1D6FE}^{2}}^{j})\,\text{d}\tilde{s}$, where $i,j$ positive constants, as in Koens & Lauga (Reference Koens and Lauga2018), see § A.2.3. Some of these integrals give rise to logarithmic terms, and we arrive at

(2.31)$$\begin{eqnarray}\displaystyle \int _{-1}^{1}{K_{1}}^{(i)}\,\text{d}\tilde{s} & = & \displaystyle \unicode[STIX]{x1D70C}(s){\mathcal{A}}(s,\tilde{\unicode[STIX]{x1D703}})\log \left(\frac{2(1-s^{2})}{\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D70C}^{2}(s)[1-\cos (\unicode[STIX]{x1D703}-\tilde{\unicode[STIX]{x1D703}})]}\right)-2s\unicode[STIX]{x2202}_{s}[\unicode[STIX]{x1D70C}(s){\mathcal{A}}(s,\tilde{\unicode[STIX]{x1D703}})]\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D716}\left\{-{\mathcal{A}}(s,\tilde{\unicode[STIX]{x1D703}})\unicode[STIX]{x1D70C}^{2}(s)\unicode[STIX]{x1D705}(s)\cos \unicode[STIX]{x1D703}_{m}(s,\tilde{\unicode[STIX]{x1D703}})\log \left(\frac{2(1-s^{2})}{\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D70C}^{2}(s)[1-\cos (\unicode[STIX]{x1D703}-\tilde{\unicode[STIX]{x1D703}})]}\right)\right.\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D70C}^{2}(s){\mathcal{A}}(s,\tilde{\unicode[STIX]{x1D703}})\frac{1}{2}\unicode[STIX]{x1D705}(s)[\cos \unicode[STIX]{x1D703}_{m}(s,\tilde{\unicode[STIX]{x1D703}})+\cos \unicode[STIX]{x1D703}_{m}(s,\unicode[STIX]{x1D703})]\nonumber\\ \displaystyle & & \displaystyle \times \left.\!\left[\log \left(\frac{2(1-s^{2})}{\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D70C}^{2}(s)[1-\cos (\unicode[STIX]{x1D703}-\tilde{\unicode[STIX]{x1D703}})]}\right)-2\right]\right\}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{2s\unicode[STIX]{x1D716}^{2}}{s^{2}-1}\unicode[STIX]{x1D70C}^{2}(s){\mathcal{A}}(s,\tilde{\unicode[STIX]{x1D703}})\frac{\text{d}\unicode[STIX]{x1D70C}(s)}{\text{d}s}[\cos (\unicode[STIX]{x1D703}-\tilde{\unicode[STIX]{x1D703}})-1]+O(\unicode[STIX]{x1D716}^{2}),\end{eqnarray}$$

(2.32)$$\begin{eqnarray}\displaystyle \int _{-1}^{1}{K_{2}}^{(i)}\,\text{d}\tilde{s} & = & \displaystyle -c(s,\tilde{\unicode[STIX]{x1D703}})\nonumber\\ \displaystyle & & \displaystyle +\,\frac{1}{2}\unicode[STIX]{x1D716}c(s,\tilde{\unicode[STIX]{x1D703}})\left[\log \left(\frac{2(1-s^{2})}{\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D70C}^{2}(s)[1-\cos (\unicode[STIX]{x1D703}-\tilde{\unicode[STIX]{x1D703}})]}\right)\right]\unicode[STIX]{x1D705}(s)\unicode[STIX]{x1D70C}(s)\cos \unicode[STIX]{x1D703}_{m}(s,\tilde{\unicode[STIX]{x1D703}})\nonumber\\ \displaystyle & & \displaystyle -\,\unicode[STIX]{x1D716}c(s,\tilde{\unicode[STIX]{x1D703}})\frac{1}{2}\unicode[STIX]{x1D70C}(s)\unicode[STIX]{x1D705}(s)[\cos \unicode[STIX]{x1D703}_{m}(s,\tilde{\unicode[STIX]{x1D703}})+\cos \unicode[STIX]{x1D703}_{m}(s,\unicode[STIX]{x1D703})]\nonumber\\ \displaystyle & & \displaystyle +\,\frac{2s\unicode[STIX]{x1D716}^{2}}{s^{2}-1}\unicode[STIX]{x1D70C}(s)[\cos (\unicode[STIX]{x1D703}-\tilde{\unicode[STIX]{x1D703}})-1]\unicode[STIX]{x2202}_{s}[\unicode[STIX]{x1D70C}(s)c(s,\tilde{\unicode[STIX]{x1D703}})]+O(\unicode[STIX]{x1D716}^{2}c).\end{eqnarray}$$

While terms incorporating the fraction $2\unicode[STIX]{x1D716}^{2}(s\unicode[STIX]{x1D70C}^{2}(s)/(s^{2}-1))(\text{d}\unicode[STIX]{x1D70C}(s)/\text{d}s)$ are $O(\unicode[STIX]{x1D716}^{2})$, we have written them explicitly above as they generally diverge as $s\rightarrow \pm 1$, and can thus become leading order in a very small region $\unicode[STIX]{x1D716}^{2}$ from the ends. This divergence can be circumvented by assuming a prolate spheroidal shape filament $\unicode[STIX]{x1D70C}(s)\sim \sqrt{1-s^{2}}$, or ensuring the activity ${\mathcal{A}}$ decays to zero at either end. For a more detailed scaling argument, see appendix C.

2.5.3 Matching: common part

We follow the van Dyke matching method, and use $\unicode[STIX]{x1D716}\unicode[STIX]{x1D712}=\tilde{s}-s$ expanding the inner region in terms of the outer region variable $\tilde{s}$, and the outer region in terms of the inner variable $\unicode[STIX]{x1D712}$, and finding the common part, expected to be the same, by expanding in $\unicode[STIX]{x1D716}$. We use the superscripts $(i)\in (o)$ for the expansion of the inner region kernel in terms of the outer variable, and $(o)\in (i)$ for the expansion of the outer region kernel in terms of the inner variable.

In order to obtain ${K_{1}}^{(o)\in (i)}$ and ${K_{2}}^{(o)\in (i)}$, we substitute $\tilde{s}=s+\unicode[STIX]{x1D716}\unicode[STIX]{x1D712}$ in the expressions for ${K_{1}}^{(o)}$ and ${K_{2}}^{(o)}$ and expand. The resulting expressions for ${K_{1}}^{(o)\in (i)}$ and ${K_{2}}^{(o)\in (i)}$ are given in § A.3 (we note that these are the same as for ${K_{1}}^{(i)\in (o)},{K_{2}}^{(i)\in (o)}$, as expected). Following integration,

(2.33)$$\begin{eqnarray}\displaystyle \int _{-1}^{1}\int _{-\unicode[STIX]{x03C0}}^{\unicode[STIX]{x03C0}}{K_{1}}^{(o)\in (i)}\,\text{d}\tilde{s}\,\text{d}\tilde{\unicode[STIX]{x1D703}} & = & \displaystyle \int _{-1}^{1}\frac{\unicode[STIX]{x1D70C}(s)}{|\tilde{s}-s|}\int _{-\unicode[STIX]{x03C0}}^{\unicode[STIX]{x03C0}}{\mathcal{A}}(s,\tilde{\unicode[STIX]{x1D703}})\,\text{d}\tilde{\unicode[STIX]{x1D703}}\,\text{d}\tilde{s}-2s\unicode[STIX]{x2202}_{s}\int _{-\unicode[STIX]{x03C0}}^{\unicode[STIX]{x03C0}}[\unicode[STIX]{x1D70C}(s){\mathcal{A}}(s,\tilde{\unicode[STIX]{x1D703}})]\,\text{d}\tilde{\unicode[STIX]{x1D703}}\nonumber\\ \displaystyle & & \displaystyle -\,\frac{1}{2}\unicode[STIX]{x1D716}\unicode[STIX]{x1D70C}^{2}(s)\unicode[STIX]{x1D705}(s)\int _{-1}^{1}\frac{1}{|\tilde{s}-s|}\int _{-\unicode[STIX]{x03C0}}^{\unicode[STIX]{x03C0}}{\mathcal{A}}(s,\tilde{\unicode[STIX]{x1D703}})\nonumber\\ \displaystyle & & \displaystyle \times \,[\cos \unicode[STIX]{x1D703}_{m}(s,\tilde{\unicode[STIX]{x1D703}})-\cos \unicode[STIX]{x1D703}_{m}(s,\unicode[STIX]{x1D703})]\,\text{d}\tilde{\unicode[STIX]{x1D703}}\,\text{d}\tilde{s}+O(\unicode[STIX]{x1D716}^{2}),\end{eqnarray}$$

(2.34)$$\begin{eqnarray}\displaystyle & \displaystyle \int _{-1}^{1}\int _{-\unicode[STIX]{x03C0}}^{\unicode[STIX]{x03C0}}{K_{2}}^{(o)\in (i)}\,\text{d}\tilde{s}\,\text{d}\tilde{\unicode[STIX]{x1D703}}=O(\unicode[STIX]{x1D716}^{2}). & \displaystyle\end{eqnarray}$$

2.5.4 Composite solution

The boundary integral (2.18) is approximated by adding the outer and inner expansions and subtracting from each the common part,

(2.35)$$\begin{eqnarray}2\unicode[STIX]{x03C0}c(s,\unicode[STIX]{x1D703})\approx \int _{-1}^{1}\int _{-\unicode[STIX]{x03C0}}^{\unicode[STIX]{x03C0}}({K_{1}}^{(o)}+{K_{1}}^{(i)}-{K_{1}}^{(i)\in (o)}+{K_{2}}^{(o)}+{K_{2}}^{(i)}-{K_{2}}^{(i)\in (o)})\,\text{d}\tilde{\unicode[STIX]{x1D703}}\,\text{d}\tilde{s}.\end{eqnarray}$$

The full expansion for the concentration field is given in § B.1. We now use the notation $c^{(0)}$ and $c^{(1)}$ for the leading- and first-order algebraic corrections of $c$,

(2.36)$$\begin{eqnarray}c=c^{(0)}+\unicode[STIX]{x1D716}c^{(1)}+O(\unicode[STIX]{x1D716}^{2}),\end{eqnarray}$$

and consider each in turn. The leading order includes all terms of $O(1)$ and $O(\log \unicode[STIX]{x1D716})$. The correction to the leading order, $\unicode[STIX]{x1D716}c^{(1)}$, vanishes as $\unicode[STIX]{x1D716}\rightarrow 0$, and includes both $O(\unicode[STIX]{x1D716})$ and $O(\unicode[STIX]{x1D716}\log \unicode[STIX]{x1D716})$. The details of simplifications are given in §§ B.2 and B.3. We now present the resulting expressions.

2.6 Leading-order concentration field

Writing $\langle f(s)\rangle \equiv \int _{-\unicode[STIX]{x03C0}}^{\unicode[STIX]{x03C0}}f(s,\tilde{\unicode[STIX]{x1D703}})\,\text{d}\tilde{\unicode[STIX]{x1D703}}$, the leading-order expression for the concentration field (originating from (B 2) in § B.2) is given by

(2.37)$$\begin{eqnarray}\displaystyle 2\unicode[STIX]{x03C0}c^{(0)}(s,\unicode[STIX]{x1D703})+\langle c^{(0)}(s)\rangle & = & \displaystyle +\int _{-1}^{1}\left[\frac{\unicode[STIX]{x1D70C}(\tilde{s})\langle {\mathcal{A}}(\tilde{s})\rangle }{|\boldsymbol{R}_{0}(s,\tilde{s})|}-\frac{\unicode[STIX]{x1D70C}(s)\langle {\mathcal{A}}(s)\rangle }{|\tilde{s}-s|}\right]\text{d}\tilde{s}\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D70C}(s)\langle {\mathcal{A}}(s)\rangle \log \left(\frac{2(1-s^{2})}{\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D70C}^{2}(s)}\right)\nonumber\\ \displaystyle & & \displaystyle -\,\unicode[STIX]{x1D70C}(s)\int _{-\unicode[STIX]{x03C0}}^{\unicode[STIX]{x03C0}}{\mathcal{A}}(s,\tilde{\unicode[STIX]{x1D703}})\log [1-\cos (\unicode[STIX]{x1D703}-\tilde{\unicode[STIX]{x1D703}})]\,\text{d}\tilde{\unicode[STIX]{x1D703}}.\qquad\end{eqnarray}$$

Integrating this again over $\unicode[STIX]{x1D703}$ allows us to evaluate $\langle c^{(0)}(s)\rangle$ and then subtract it from (2.37), the details of which are given in § B.2. We thus arrive at the leading-order slender boundary integral equation

(2.38)$$\begin{eqnarray}\displaystyle 4\unicode[STIX]{x03C0}c^{(0)}(s,\unicode[STIX]{x1D703}) & = & \displaystyle \int _{-1}^{1}\left[\frac{\unicode[STIX]{x1D70C}(\tilde{s})\langle {\mathcal{A}}(\tilde{s})\rangle }{|\boldsymbol{R}_{0}(s,\tilde{s})|}-\frac{\unicode[STIX]{x1D70C}(s)\langle {\mathcal{A}}(s)\rangle }{|\tilde{s}-s|}\right]\text{d}\tilde{s}+\unicode[STIX]{x1D70C}(s)\langle {\mathcal{A}}(s)\rangle \log \left(\frac{(1-s^{2})}{\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D70C}^{2}(s)}\right)\nonumber\\ \displaystyle & & \displaystyle -\,2\unicode[STIX]{x1D70C}(s)\int _{-\unicode[STIX]{x03C0}}^{\unicode[STIX]{x03C0}}{\mathcal{A}}(s,\tilde{\unicode[STIX]{x1D703}})\log [1-\cos (\unicode[STIX]{x1D703}-\tilde{\unicode[STIX]{x1D703}})]\,\text{d}\tilde{\unicode[STIX]{x1D703}}.\end{eqnarray}$$

We thus see that for non-axisymmetric filaments, there is a leading-order contribution to the concentration that varies with $\unicode[STIX]{x1D703}$. As such, we expect the slip velocity, and resulting dynamics, of such filaments, to be dominated by the final term in (2.38).

For filaments with axisymmetric activity, ${\mathcal{A}}(s,\unicode[STIX]{x1D703})\equiv {\mathcal{A}}(s)$, and $\langle {\mathcal{A}}(s,\unicode[STIX]{x1D703})\rangle =2\unicode[STIX]{x03C0}{\mathcal{A}}(s)$, and using $\int _{-\unicode[STIX]{x03C0}}^{\unicode[STIX]{x03C0}}\log [1-\cos (\unicode[STIX]{x1D703}-\tilde{\unicode[STIX]{x1D703}})]\,\text{d}\tilde{\unicode[STIX]{x1D703}}=-2\unicode[STIX]{x03C0}\log (2)$, we arrive at

(2.39)$$\begin{eqnarray}2c^{(0)}(s,\unicode[STIX]{x1D703})=+\int _{-1}^{1}\left[\frac{\unicode[STIX]{x1D70C}(\tilde{s}){\mathcal{A}}(\tilde{s})}{|\boldsymbol{R}_{0}(s,\tilde{s})|}-\frac{\unicode[STIX]{x1D70C}(s){\mathcal{A}}(s)}{|\tilde{s}-s|}\right]\text{d}\tilde{s}+\unicode[STIX]{x1D70C}(s){\mathcal{A}}(s)\log \left(\frac{4(1-s^{2})}{\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D70C}^{2}(s)}\right).\end{eqnarray}$$

The two terms inside the integrand of (2.39) both include non-local effects, while the logarithmic term, and the $\tilde{\unicode[STIX]{x1D703}}$ integral in the non-axisymmetric case (2.38), are local. Physically, equation (2.39) represents a line distribution of point sources located on the filament centreline, weighted by the filament activity and radius, with a local correction arising because we are evaluating the concentration on the filament.

Note that the terms of the integrand inside the square brackets both diverge when the integration variable $\tilde{s}$ passes through the evaluation arclength parameter $s$, however, the two singularities cancel each other, such that the integrand is regular. As explained in detail in Koens & Lauga (Reference Koens and Lauga2018), the fraction inside the logarithmic term of (2.39) means that close to the endpoints, the cross-sectional radius $\unicode[STIX]{x1D70C}(s)$ must decrease as $\sqrt{(1-s^{2})}$ (or less abruptly) for the logarithmic term not to diverge, i.e. the analysis is valid for prolate spheroidal ends.

2.7 Next-order correction

Equation (B 6) in § B.3 gives the full expression for $c^{(1)}(s,\unicode[STIX]{x1D703})$ for a general activity ${\mathcal{A}}(s,\unicode[STIX]{x1D703})$. Simplifying for the axisymmetric case ${\mathcal{A}}(s,\unicode[STIX]{x1D703})\equiv {\mathcal{A}}(s)$, after some algebra (see § B.3) we finally arrive at

(2.40)$$\begin{eqnarray}\displaystyle 2\unicode[STIX]{x03C0}c^{(1)}(s,\unicode[STIX]{x1D703})+\langle c^{(1)}(s)\rangle & = & \displaystyle +\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}^{2}(s)\unicode[STIX]{x1D705}(s){\mathcal{A}}(s)\cos \unicode[STIX]{x1D703}_{m}(s,\unicode[STIX]{x1D703})\left[\log \left(\frac{4(1-s^{2})}{\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D70C}^{2}(s)}\right)-3\right]\nonumber\\ \displaystyle & & \displaystyle -\,2\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}(s)\int _{-1}^{1}\unicode[STIX]{x1D70C}(\tilde{s}){\mathcal{A}}(\tilde{s})\frac{\boldsymbol{R}_{0}}{|\boldsymbol{R}_{0}|^{3}}\,\text{d}\tilde{s}\boldsymbol{\cdot }\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(s,\unicode[STIX]{x1D703})\nonumber\\ \displaystyle & & \displaystyle -\,\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}^{2}(s)\unicode[STIX]{x1D705}(s)\int _{-1}^{1}\frac{1}{|\tilde{s}-s|}{\mathcal{A}}(s)\cos \unicode[STIX]{x1D703}_{m}(s,\unicode[STIX]{x1D703})\,\text{d}\tilde{s}.\end{eqnarray}$$

Integrating over $\unicode[STIX]{x1D703}$ we see that $\langle c^{(1)}(s)\rangle =0$, and the first-order correction to the slender boundary integral expression in the case of axisymmetric activity becomes

(2.41)$$\begin{eqnarray}\displaystyle 2c^{(1)}(s,\unicode[STIX]{x1D703}) & = & \displaystyle \left[\log \left(\frac{4(1-s^{2})}{\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D70C}^{2}(s)}\right)-3\right]\unicode[STIX]{x1D70C}^{2}(s)\unicode[STIX]{x1D705}(s){\mathcal{A}}(s)\cos [\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{i}(s)]\nonumber\\ \displaystyle & & \displaystyle -\,\unicode[STIX]{x1D70C}(s)\int _{-1}^{1}\left[\frac{2\unicode[STIX]{x1D70C}(\tilde{s}){\mathcal{A}}(\tilde{s})\boldsymbol{R}_{0}\boldsymbol{\cdot }\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(s,\unicode[STIX]{x1D703})}{|\boldsymbol{R}_{0}|^{3}}+\frac{\unicode[STIX]{x1D70C}(s)\unicode[STIX]{x1D705}(s){\mathcal{A}}(s)\cos [\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{i}(s)]}{|\tilde{s}-s|}\right]\text{d}\tilde{s}.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Importantly (though perhaps not obviously), the integral converges. This can be seen by substituting

(2.42)$$\begin{eqnarray}\boldsymbol{R}_{0}=-(\tilde{s}-s)\left[\hat{\boldsymbol{t}}(s)+{\textstyle \frac{1}{2}}(\tilde{s}-s)\unicode[STIX]{x1D705}(s)\hat{\boldsymbol{n}}(s)+O((\tilde{s}-s)^{2})\right],\end{eqnarray}$$

in the first fraction of the integrand, whereupon the leading-order term of the expansion of the first fraction of the integrand cancels the second term of the integrand, $\unicode[STIX]{x1D70C}(s)\unicode[STIX]{x1D705}(s){\mathcal{A}}(s)\cos [\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{i}(s)]/|\tilde{s}-s|$, regularising the singularity. Physically, equation (2.41) represents a line distribution of source dipoles located on the filament centreline, weighted by the filament activity, radius and curvature, with a local correction.

Thus, for axisymmetric activity the full surface concentration is given (up to $O(\unicode[STIX]{x1D716}^{2})$ corrections) by

(2.43)$$\begin{eqnarray}\displaystyle 2c(s,\unicode[STIX]{x1D703}) & = & \displaystyle \int _{-1}^{1}\left[\frac{\unicode[STIX]{x1D70C}(\tilde{s}){\mathcal{A}}(\tilde{s})}{|\boldsymbol{R}_{0}(s,\tilde{s})|}-\frac{\unicode[STIX]{x1D70C}(s){\mathcal{A}}(s)}{|\tilde{s}-s|}\right]\text{d}\tilde{s}+\unicode[STIX]{x1D70C}(s){\mathcal{A}}(s)\log \left(\frac{4(1-s^{2})}{\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D70C}^{2}(s)}\right)\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D716}\left[\left[\log \left(\frac{4(1-s^{2})}{\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D70C}^{2}(s)}\right)-3\right]\unicode[STIX]{x1D70C}^{2}(s)\unicode[STIX]{x1D705}(s){\mathcal{A}}(s)\cos [\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{i}(s)]\right.\nonumber\\ \displaystyle & & \displaystyle -\left.\unicode[STIX]{x1D70C}(s)\int _{-1}^{1}\left[\frac{2\unicode[STIX]{x1D70C}(\tilde{s}){\mathcal{A}}(\tilde{s})\boldsymbol{R}_{0}\boldsymbol{\cdot }\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(s,\unicode[STIX]{x1D703})}{|\boldsymbol{R}_{0}|^{3}}+\frac{\unicode[STIX]{x1D70C}(s)\unicode[STIX]{x1D705}(s){\mathcal{A}}(s)\cos [\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{i}(s)]}{|\tilde{s}-s|}\right]\right]\text{d}\tilde{s}\nonumber\\ \displaystyle & & \displaystyle +\,O(\unicode[STIX]{x1D716}^{2}).\end{eqnarray}$$

In contrast to slender body equations for viscous flows, equation (2.43) is explicit; given the filament activity and geometry, one can directly calculate the concentration field by simply evaluating a line integral, rather than having to solve an integral equation (cf. (2.22b), that has the concentration in the kernel $K_{2}$). Note further that, since we have non-dimensionalised with respect to the filament radius for asymptotic convenience, in order to make comparisons with existing results (non-dimensionalised with respect to the filament semiaxis), the entire solution must be premultiplied by $\unicode[STIX]{x1D716}$.

We can now clearly see that there is azimuthal variation in the concentration field for curved axisymmetric filaments that is $O(\unicode[STIX]{x1D716})$. As such, these azimuthal variations will in general have a leading-order effect on the filament swimming velocity. For a straight filament, all the $O(\unicode[STIX]{x1D716})$ terms vanish in (2.43), which means that the leading-order expression for the concentration of a straight filament with axisymmetric activity is correct to $O(\unicode[STIX]{x1D716}^{2})$.

This azimuthal dependence has a natural modal form in terms of $\sin [\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{i}(s)]$ and $\cos [\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{i}(s)]$, which will carry through as a modal expression for the phoretic slip velocity, and will naturally lead to a Fourier modes approach for the kinematics which we now describe.

2.8 Slip velocity – azimuthal modes

We now proceed to calculate the slip velocity for slender curved filaments with axisymmetric activity. In experimental systems, activity arises from deposition of catalyst at the surface, and so axisymmetric activity implies no azimuthal variation in surface chemistry. As such, we may also assume axisymmetric mobility ${\mathcal{M}}(s)$.

As in the above, the following analysis is not valid in a very small region $O(\unicode[STIX]{x1D716}^{2})$ from the filament ends (typically 0.01 % of the total filament length for the examples we consider). The contribution to the dynamics from this region is discussed, and shown to be negligible, in appendix C. The leading-order slip velocity (2.3) for axisymmetric activity is given by

(2.44)$$\begin{eqnarray}\boldsymbol{v}_{slip}(s,\unicode[STIX]{x1D703})={\mathcal{M}}(s)\left[\hat{\boldsymbol{t}}(s)\frac{\unicode[STIX]{x2202}c^{(0)}}{\unicode[STIX]{x2202}s}+\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D703}}(s,\unicode[STIX]{x1D703})\frac{1}{\unicode[STIX]{x1D70C}(s)}\frac{\unicode[STIX]{x2202}c^{(1)}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}\right].\end{eqnarray}$$

Taking the $\unicode[STIX]{x1D703}$-derivative of (2.43), we see that

(2.45)$$\begin{eqnarray}\displaystyle \frac{2}{\unicode[STIX]{x1D70C}(s)}\frac{\unicode[STIX]{x2202}c^{(1)}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}} & = & \displaystyle -\left\{\int _{-1}^{1}\left[\frac{2\unicode[STIX]{x1D70C}(\tilde{s}){\mathcal{A}}(\tilde{s})\boldsymbol{R}_{0}\boldsymbol{\cdot }\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D703}}(s,\unicode[STIX]{x1D703})}{|\boldsymbol{R}_{0}|^{3}}-\frac{\unicode[STIX]{x1D70C}(s)\unicode[STIX]{x1D705}(s){\mathcal{A}}(s)\sin [\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{i}(s)]}{|\tilde{s}-s|}\right]\text{d}\tilde{s}\right.\nonumber\\ \displaystyle & & \displaystyle +\left.\!\left[\log \left(\frac{4(1-s^{2})}{\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D70C}^{2}(s)}\right)-3\right]\unicode[STIX]{x1D70C}(s)\unicode[STIX]{x1D705}(s){\mathcal{A}}(s)\sin [\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{i}(s)]\right\},\end{eqnarray}$$

where $\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D703}}(s,\unicode[STIX]{x1D703})=-\hat{\boldsymbol{n}}(s)\sin [\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{i}(s)]+\hat{\boldsymbol{b}}(s)\cos [\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{i}(s)]$. Using the notation $\unicode[STIX]{x1D703}_{m}(s,\unicode[STIX]{x1D703})\equiv \unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{i}(s)$, we can rearrange (2.45) as

(2.46)$$\begin{eqnarray}\frac{1}{\unicode[STIX]{x1D70C}(s)}\frac{\unicode[STIX]{x2202}c^{(1)}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}=+A_{s}(s)\sin \unicode[STIX]{x1D703}_{m}(s,\unicode[STIX]{x1D703})+A_{c}(s)\cos \unicode[STIX]{x1D703}_{m}(s,\unicode[STIX]{x1D703}),\end{eqnarray}$$

where

(2.47)$$\begin{eqnarray}\displaystyle A_{s}(s) & = & \displaystyle \int _{-1}^{1}\left[\frac{\unicode[STIX]{x1D70C}(\tilde{s}){\mathcal{A}}(\tilde{s})[\boldsymbol{R}_{0}\boldsymbol{\cdot }\hat{\boldsymbol{n}}(s)]}{|\boldsymbol{R}_{0}|^{3}}+\frac{\unicode[STIX]{x1D70C}(s)\unicode[STIX]{x1D705}(s){\mathcal{A}}(s)}{2|\tilde{s}-s|}\right]\text{d}\tilde{s}\nonumber\\ \displaystyle & & \displaystyle -\,\frac{1}{2}\left[\log \left(\frac{4(1-s^{2})}{\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D70C}^{2}(s)}\right)-3\right]\unicode[STIX]{x1D70C}(s)\unicode[STIX]{x1D705}(s){\mathcal{A}}(s),\end{eqnarray}$$

(2.48)$$\begin{eqnarray}\displaystyle A_{c}(s)=-\int _{-1}^{1}\frac{\unicode[STIX]{x1D70C}(\tilde{s}){\mathcal{A}}(\tilde{s})\boldsymbol{R}_{0}\boldsymbol{\cdot }\hat{\boldsymbol{b}}(s)}{|\boldsymbol{R}_{0}|^{3}}\,\text{d}\tilde{s}. & & \displaystyle\end{eqnarray}$$

Substituting (2.46) into (2.44) gives

(2.49)$$\begin{eqnarray}\displaystyle \boldsymbol{v}_{slip}(s,\unicode[STIX]{x1D703}) & = & \displaystyle {\mathcal{M}}(s)\left(\hat{\boldsymbol{t}}(s)\frac{\unicode[STIX]{x2202}c^{(0)}}{\unicode[STIX]{x2202}s}+[A_{s}(s)\sin \unicode[STIX]{x1D703}_{m}(s,\unicode[STIX]{x1D703})+A_{c}(s)\cos \unicode[STIX]{x1D703}_{m}(s,\unicode[STIX]{x1D703})]\right.\nonumber\\ \displaystyle & & \displaystyle \times \left.[-\hat{\boldsymbol{n}}(s)\sin \unicode[STIX]{x1D703}_{m}(s,\unicode[STIX]{x1D703})+\hat{\boldsymbol{b}}(s)\cos \unicode[STIX]{x1D703}_{m}(s,\unicode[STIX]{x1D703})]\vphantom{\frac{\unicode[STIX]{x2202}c^{(0)}}{\unicode[STIX]{x2202}s}}\right),\end{eqnarray}$$

and using the double angle trigonometrical formulae we find that the slip velocity has only zero and second modes,

(2.50)$$\begin{eqnarray}\displaystyle \boldsymbol{v}_{slip}(s,\unicode[STIX]{x1D703}) & = & \displaystyle \frac{1}{2}{\mathcal{M}}(s)\left\{\left[2\hat{\boldsymbol{t}}(s)\frac{\unicode[STIX]{x2202}c^{(0)}}{\unicode[STIX]{x2202}s}-\hat{\boldsymbol{n}}(s)A_{s}(s)+\hat{\boldsymbol{b}}(s)A_{c}(s)\right]\right.\nonumber\\ \displaystyle & & \displaystyle +\,\cos 2\unicode[STIX]{x1D703}_{m}(s,\unicode[STIX]{x1D703})[+\hat{\boldsymbol{n}}(s)A_{s}(s)+\hat{\boldsymbol{b}}(s)A_{c}(s)]\nonumber\\ \displaystyle & & \displaystyle +\left.\sin 2\unicode[STIX]{x1D703}_{m}(s,\unicode[STIX]{x1D703})[-\hat{\boldsymbol{n}}(s)A_{c}(s)+\hat{\boldsymbol{b}}(s)A_{s}(s)]\vphantom{\frac{\unicode[STIX]{x2202}c^{(0)}}{\unicode[STIX]{x2202}s}}\right\}.\end{eqnarray}$$

In the case of planar filaments, $\boldsymbol{R}_{0}(s,\tilde{s})\boldsymbol{\cdot }\hat{\boldsymbol{b}}(s)\equiv 0$ and hence $A_{c}=0$. This reduces (2.50) to

(2.51)$$\begin{eqnarray}\boldsymbol{v}_{slip}(s,\unicode[STIX]{x1D703})=\frac{1}{2}{\mathcal{M}}(s)\left(2\hat{\boldsymbol{t}}(s)\frac{\unicode[STIX]{x2202}c^{(0)}}{\unicode[STIX]{x2202}s}+A_{s}(s)[\hat{\boldsymbol{n}}(s)(\cos 2\unicode[STIX]{x1D703}_{m}(s,\unicode[STIX]{x1D703})-1)+\hat{\boldsymbol{b}}(s)\sin 2\unicode[STIX]{x1D703}_{m}(s,\unicode[STIX]{x1D703})]\right).\end{eqnarray}$$

Given the activity, mobility and geometry of the filament, we can find the concentration, the concentration gradients and then the phoretic slip velocity field on the surface of the filament according to (2.50).

2.9 Phoretic swimming kinematics

We now turn to the problem of finding the leading-order swimming (rigid body) dynamics resulting from the slip velocity forcing defined in (2.50), namely the translational velocity $\boldsymbol{U}_{sw}$ and rotational velocity $\unicode[STIX]{x1D734}_{sw}$.

2.9.1 Fourier modes of surface velocities and tractions

Equations (2.50), (2.51) already have the correct form in order to use the Fourier mode description of surface velocity described by Koens & Lauga (Reference Koens and Lauga2018), who decomposed the surface velocity and traction in Fourier modes

(2.52a)$$\begin{eqnarray}\displaystyle & \displaystyle 2\unicode[STIX]{x03C0}\boldsymbol{U}(s,\unicode[STIX]{x1D703})=\boldsymbol{U}_{0}(s)+\mathop{\sum }_{n=1}^{\infty }[\boldsymbol{U}_{c,n}(s)\cos n\unicode[STIX]{x1D703}_{m}(s,\unicode[STIX]{x1D703})+\boldsymbol{U}_{s,n}(s)\sin n\unicode[STIX]{x1D703}_{m}(s,\unicode[STIX]{x1D703})], & \displaystyle\end{eqnarray}$$

(2.52b)$$\begin{eqnarray}\displaystyle & \displaystyle 2\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}(s)\boldsymbol{f}(s,\unicode[STIX]{x1D703})=\boldsymbol{f}_{0}(s)+\mathop{\sum }_{n=1}^{\infty }[\boldsymbol{f}_{c,n}(s)\cos n\unicode[STIX]{x1D703}_{m}(s,\unicode[STIX]{x1D703})+\boldsymbol{f}_{s,n}(s)\sin n\unicode[STIX]{x1D703}_{m}(s,\unicode[STIX]{x1D703})]. & \displaystyle\end{eqnarray}$$

We write the rigid body motion as

(2.53)$$\begin{eqnarray}\displaystyle \boldsymbol{U}_{sw}+\unicode[STIX]{x1D734}_{sw}\times \boldsymbol{S}(s,\unicode[STIX]{x1D703}) & = & \displaystyle \boldsymbol{U}_{sw}+\unicode[STIX]{x1D734}_{sw}\times \boldsymbol{r}(s)+\unicode[STIX]{x1D716}\unicode[STIX]{x1D70C}(s)\unicode[STIX]{x1D734}_{sw}\times \hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(s,\unicode[STIX]{x1D703})\nonumber\\ \displaystyle & = & \displaystyle \boldsymbol{U}_{sw}+\unicode[STIX]{x1D734}_{sw}\times \boldsymbol{r}(s)\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D716}\unicode[STIX]{x1D70C}(s)\unicode[STIX]{x1D734}_{sw}\times \hat{\boldsymbol{n}}(s)\cos \unicode[STIX]{x1D703}_{m}(s,\unicode[STIX]{x1D703})\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D716}\unicode[STIX]{x1D70C}(s)\unicode[STIX]{x1D734}_{sw}\times \hat{\boldsymbol{b}}(s)\sin \unicode[STIX]{x1D703}_{m}(s,\unicode[STIX]{x1D703}).\end{eqnarray}$$

The contributions of the first sine and cosine modes of the surface velocity due to rigid body motion are $O(\unicode[STIX]{x1D716})$, hence the leading-order Fourier mode decomposition of the total surface velocity (i.e. the sum of the rigid body motion and phoretic slip velocity) is

(2.54a)$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{U}_{0}(s)=2\unicode[STIX]{x03C0}[\boldsymbol{U}_{sw}+\unicode[STIX]{x1D734}_{sw}\times \boldsymbol{r}(s)]+\unicode[STIX]{x03C0}{\mathcal{M}}(s)\left[2\hat{\boldsymbol{t}}(s)\frac{\unicode[STIX]{x2202}c^{(0)}}{\unicode[STIX]{x2202}s}-\hat{\boldsymbol{n}}(s)A_{s}(s)+\hat{\boldsymbol{b}}(s)A_{c}(s)\right]\!,\qquad \quad & \displaystyle\end{eqnarray}$$

(2.54b)$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{U}_{c,2}(s)=\unicode[STIX]{x03C0}{\mathcal{M}}(s)[+\hat{\boldsymbol{n}}(s)A_{s}(s)+\hat{\boldsymbol{b}}(s)A_{c}(s)],\quad \qquad & \displaystyle\end{eqnarray}$$

(2.54c)$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{U}_{s,2}(s)=\unicode[STIX]{x03C0}{\mathcal{M}}(s)[-\hat{\boldsymbol{n}}(s)A_{c}(s)+\hat{\boldsymbol{b}}(s)A_{s}(s)],\quad \qquad & \displaystyle\end{eqnarray}$$

where the rigid body kinematics $(\boldsymbol{U}_{sw},\unicode[STIX]{x1D734}_{sw})$ are to be found by imposing the force and torque balances on the filament. The total force and torque on the filament are

(2.55a)$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{F}_{tot}=\int _{-1}^{1}\int _{-\unicode[STIX]{x03C0}}^{\unicode[STIX]{x03C0}}\boldsymbol{f}(\tilde{s},\tilde{\unicode[STIX]{x1D703}})\left|\frac{\unicode[STIX]{x2202}\boldsymbol{S}}{\unicode[STIX]{x2202}\tilde{s}}\times \frac{\unicode[STIX]{x2202}\boldsymbol{S}}{\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D703}}}\right|\text{d}\tilde{\unicode[STIX]{x1D703}}\,\text{d}\tilde{s}, & \displaystyle\end{eqnarray}$$

(2.55b)$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{T}_{tot}=\int _{-1}^{1}\int _{-\unicode[STIX]{x03C0}}^{\unicode[STIX]{x03C0}}\boldsymbol{S}(\tilde{s},\tilde{\unicode[STIX]{x1D703}})\times \boldsymbol{f}(\tilde{s},\tilde{\unicode[STIX]{x1D703}})\left|\frac{\unicode[STIX]{x2202}\boldsymbol{S}}{\unicode[STIX]{x2202}\tilde{s}}\times \frac{\unicode[STIX]{x2202}\boldsymbol{S}}{\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D703}}}\right|\text{d}\tilde{\unicode[STIX]{x1D703}}\,\text{d}\tilde{s}, & \displaystyle\end{eqnarray}$$

and substituting the surface element form given by (2.20) yields

(2.56a)$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{F}_{tot}=\int _{-1}^{1}\int _{-\unicode[STIX]{x03C0}}^{\unicode[STIX]{x03C0}}\boldsymbol{f}(\tilde{s},\tilde{\unicode[STIX]{x1D703}})\unicode[STIX]{x1D716}\unicode[STIX]{x1D70C}(\tilde{s})[1+O(\unicode[STIX]{x1D716})]\,\text{d}\tilde{\unicode[STIX]{x1D703}}\,\text{d}\tilde{s}, & \displaystyle\end{eqnarray}$$

(2.56b)$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{T}_{tot}=\int _{-1}^{1}\int _{-\unicode[STIX]{x03C0}}^{\unicode[STIX]{x03C0}}[\boldsymbol{r}(\tilde{s})+\unicode[STIX]{x1D716}\unicode[STIX]{x1D70C}(\tilde{s})\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(\tilde{s},\tilde{\unicode[STIX]{x1D703}})]\times \boldsymbol{f}(\tilde{s},\tilde{\unicode[STIX]{x1D703}})\unicode[STIX]{x1D716}\unicode[STIX]{x1D70C}(\tilde{s})[1+O(\unicode[STIX]{x1D716})]\,\text{d}\tilde{\unicode[STIX]{x1D703}}\,\text{d}\tilde{s}. & \displaystyle\end{eqnarray}$$

With $\boldsymbol{f}(s,\unicode[STIX]{x1D703})$ given by (2.52b), due to the periodicity of the cosine and sine modes making vanishing contributions, we see that the force and torque balances to leading order are

(2.57a)$$\begin{eqnarray}\displaystyle & \displaystyle \int _{-1}^{1}\boldsymbol{f}_{0}(\tilde{s})\,\text{d}\tilde{s}=\mathbf{0}, & \displaystyle\end{eqnarray}$$

(2.57b)$$\begin{eqnarray}\displaystyle & \displaystyle \int _{-1}^{1}\boldsymbol{r}(\tilde{s})\times \boldsymbol{f}_{0}(\tilde{s})\,\text{d}\tilde{s}=\mathbf{0}. & \displaystyle\end{eqnarray}$$

2.9.2 Phoretic swimming kinematics from slender body theory of viscous propulsion

After expanding the boundary integral equation for Stokes flow for a 3-D filament of arbitrary geometry, Koens & Lauga (Reference Koens and Lauga2018) derived the following relation between the surface traction modes on the filament and the modes of the surface velocities,

(2.58a)$$\begin{eqnarray}\displaystyle 4\boldsymbol{U}_{0}(s) & = & \displaystyle \int _{-1}^{1}\,\text{d}\tilde{s}\left(\frac{\mathbf{1}+\hat{\boldsymbol{R}}_{0}\hat{\boldsymbol{R}}_{0}}{|\boldsymbol{R}_{0}|}\boldsymbol{\cdot }\boldsymbol{f}_{0}(\tilde{s})-\frac{\mathbf{1}+\hat{\boldsymbol{t}}\hat{\boldsymbol{t}}}{|\tilde{s}-s|}\boldsymbol{\cdot }\boldsymbol{f}_{0}(s)\right)\nonumber\\ \displaystyle & & \displaystyle +\left[\log \left(\frac{4\unicode[STIX]{x1D70C}^{2}(s)}{\unicode[STIX]{x1D716}^{2}(1-s^{2})}\right)(\mathbf{1}+\hat{\boldsymbol{t}}\hat{\boldsymbol{t}})+\mathbf{1}-3\hat{\boldsymbol{t}}\hat{\boldsymbol{t}}\right]\boldsymbol{\cdot }\boldsymbol{f}_{0}(s)+O(\unicode[STIX]{x1D716}),\end{eqnarray}$$

(2.58b)$$\begin{eqnarray}\displaystyle & \displaystyle 2n\boldsymbol{U}_{c,n}(s)=(\mathbf{1}+\hat{\boldsymbol{t}}\hat{\boldsymbol{t}})\boldsymbol{\cdot }\boldsymbol{f}_{c,n}(s)+O(\unicode[STIX]{x1D716}),\quad (n>0) & \displaystyle\end{eqnarray}$$

(2.58c)$$\begin{eqnarray}\displaystyle & \displaystyle 2n\boldsymbol{U}_{s,n}(s)=(\mathbf{1}+\hat{\boldsymbol{t}}\hat{\boldsymbol{t}})\boldsymbol{\cdot }\boldsymbol{f}_{s,n}(s)+O(\unicode[STIX]{x1D716}),\quad (n>0). & \displaystyle\end{eqnarray}$$

Thus in order to find the leading-order rigid body kinematics $(\boldsymbol{U}_{sw},\unicode[STIX]{x1D734}_{sw})$ of the filament, one must solve the following system for the unknown force per unit length $\boldsymbol{f}_{0}(s)$,

(2.59)$$\begin{eqnarray}\displaystyle & & \displaystyle 8\unicode[STIX]{x03C0}[\boldsymbol{U}_{sw}+\unicode[STIX]{x1D734}_{sw}\times \boldsymbol{r}(s)+\boldsymbol{u}_{0}^{phor}(s)]\nonumber\\ \displaystyle & & \displaystyle \quad =\int _{-1}^{1}[\unicode[STIX]{x1D642}(s,\tilde{s})\boldsymbol{\cdot }\boldsymbol{f}_{0}(\tilde{s})-\unicode[STIX]{x1D645}(s,\tilde{s})\boldsymbol{\cdot }\boldsymbol{f}_{0}(s)]\,\text{d}\tilde{s}+\unicode[STIX]{x1D647}(s)\boldsymbol{\cdot }\boldsymbol{f}_{0}(s),\end{eqnarray}$$

where the unknowns $(\boldsymbol{U}_{sw},\unicode[STIX]{x1D734}_{sw})$ are such that zero net force (2.57a) and torque (2.57b) act on the swimmer, and where

(2.60)$$\begin{eqnarray}\boldsymbol{u}_{0}^{phor}(s)=\frac{1}{2}{\mathcal{M}}(s)\left[2\hat{\boldsymbol{t}}(s)\frac{\unicode[STIX]{x2202}c^{(0)}}{\unicode[STIX]{x2202}s}-\hat{\boldsymbol{n}}(s)A_{s}(s)+\hat{\boldsymbol{b}}(s)A_{c}(s)\right],\end{eqnarray}$$

is the average slip velocity (as in (2.50)) over $\unicode[STIX]{x1D703}$, i.e. $\boldsymbol{u}_{0}^{phor}(s)=(1/2\unicode[STIX]{x03C0})\int _{-\unicode[STIX]{x03C0}}^{\unicode[STIX]{x03C0}}\boldsymbol{v}_{slip}(s,\unicode[STIX]{x1D703})\,\text{d}\unicode[STIX]{x1D703}$, that is the zeroth mode of the phoretic slip velocity according to the definition for modes, equation (2.54a) divided by $2\unicode[STIX]{x03C0}$. We also have defined the tensors

(2.61a)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D642}(s,\tilde{s})=\frac{\mathbf{1}+\hat{\boldsymbol{R}}_{0}(s,\tilde{s})\hat{\boldsymbol{R}}_{0}(s,\tilde{s})}{|\boldsymbol{R}_{0}(s,\tilde{s})|}, & \displaystyle\end{eqnarray}$$

(2.61b)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D645}(s,\tilde{s})=\frac{\mathbf{1}+\hat{\boldsymbol{t}}(s)\hat{\boldsymbol{t}}(s)}{|\tilde{s}-s|}, & \displaystyle\end{eqnarray}$$

(2.61c)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D647}(s)=\log \left(\frac{4\unicode[STIX]{x1D70C}^{2}(s)}{\unicode[STIX]{x1D716}^{2}(1-s^{2})}\right)(\mathbf{1}+\hat{\boldsymbol{t}}(s)\hat{\boldsymbol{t}}(s))+\mathbf{1}-3\hat{\boldsymbol{t}}(s)\hat{\boldsymbol{t}}(s), & \displaystyle\end{eqnarray}$$

for a more compact notation. Note that $\unicode[STIX]{x1D642}(s,\tilde{s})=\unicode[STIX]{x1D642}(\tilde{s},s)$ and $\unicode[STIX]{x1D645}(s,\tilde{s})=\unicode[STIX]{x1D645}(\tilde{s},s)$.

A numerical implementation of this system of equations is given in § 3. Following a validation of our theory in §§ 4, 5, we present some simple results in §§ 6, 7.

3.1 Numerical implementation

The filament is first partitioned into $N_{elts}$ segments of equal length $l_{tot}/N_{elts}$. The $n\text{th}$ segment, denoted by $E_{n}$, has contour length $l_{n}$ and parameterised by its contour length parameter $s$ taking values in $(s_{n}-l_{n}/2,s_{n}+l_{n}/2)$ where $s_{n}$ is the midpoint of $E_{n}$ (according to contour length), which we will refer to as the $n\text{th}$ collocation point.

We simply evaluate the concentration field on the filament surface according to (2.43), using Gaussian quadrature over each segment. When the evaluation point lies in the element over which the integration takes place, it is sufficient to simply use a high-order even quadrature rule, so that no quadrature point lies at $\tilde{s}=s$. Although the integrand is non-singular at this point, this regularity arises from two singularities cancelling, and so for numerical purposes the point should be avoided.

Using a bicubic spline interpolant of the concentration field $c(s,\unicode[STIX]{x1D703})$ with periodic boundary conditions in $\unicode[STIX]{x1D703}$ and clamped boundary conditions for $s$, we can use functional derivatives to find the partial derivatives of the concentration along $s,\unicode[STIX]{x1D703}$, and hence evaluate the phoretic slip velocity field on the surface of the filament from (2.44).

For the kinematics, we will only need the zeroth mode of the phoretic slip velocity, $\boldsymbol{u}_{0}^{phor}$, at the collocation points, which we can calculate according to (2.60). For the evaluation of $\unicode[STIX]{x2202}c^{(0)}/\unicode[STIX]{x2202}s$, we use the functional derivative of a cubic spline data interpolation for $c^{(0)}$ based on its values at the collocation points. The coefficients $A_{s},A_{c}$ are evaluated from (2.47)–(2.48) using Gaussian quadrature for the definite integrals.

With the zeroth mode of the phoretic slip velocity at hand, we can now proceed to find the swimming kinematics by numerically implementing slender body theory according to Koens & Lauga (Reference Koens and Lauga2018). The force per unit length $\boldsymbol{f}_{0}(s)$ along $E_{n}$ is approximated as being constant on each segment, taking the value $\boldsymbol{f}_{0}[n]$. Then (2.57a)–(2.57b) become

(3.1a)$$\begin{eqnarray}\displaystyle & \displaystyle \mathop{\sum }_{n=1}^{N_{elts}}l_{n}\boldsymbol{f}_{0}[n]=\mathbf{0}, & \displaystyle\end{eqnarray}$$

(3.1b)$$\begin{eqnarray}\displaystyle & \displaystyle \mathop{\sum }_{n=1}^{N_{elts}}\left(\int _{E_{n}}\boldsymbol{r}(\tilde{s})\,\text{d}\tilde{s}\right)\times \boldsymbol{f}_{0}[n]=\mathbf{0}. & \displaystyle\end{eqnarray}$$

$l_{n}$

$\boldsymbol{r}(\tilde{s})$

$\tilde{s}\in E_{n}$

(3.2)$$\begin{eqnarray}\boldsymbol{r}(\tilde{s})=\boldsymbol{r}(s_{n})+(\tilde{s}-s_{n})\hat{\boldsymbol{t}}(s_{n})+{\textstyle \frac{1}{2}}(\tilde{s}-s_{n})^{2}\unicode[STIX]{x1D705}(s_{n})\hat{\boldsymbol{n}}(s_{n})+O(l_{n}^{3})\end{eqnarray}$$

and use

(3.3)$$\begin{eqnarray}\int _{s_{n}-l_{n}/2}^{s_{n}+l_{n}/2}(\tilde{s}-s_{n})^{m}\,\text{d}\tilde{s}=\frac{l_{n}^{m+1}}{2^{m+1}(m+1)}[1-(-1)^{m+1}]\end{eqnarray}$$

to obtain

(3.4)$$\begin{eqnarray}\int _{E_{n}}\boldsymbol{r}(\tilde{s})\,\text{d}\tilde{s}=\boldsymbol{r}(s_{n})l_{n}+{\textstyle \frac{1}{24}}\unicode[STIX]{x1D705}(s_{n})\hat{\boldsymbol{n}}(s_{n})l_{n}^{3}+O(l_{n}^{5}).\end{eqnarray}$$

We assume that all segments have equal contour lengths $l_{n}=l_{elt}\,\forall n$. Equations (3.1a)–(3.1b) become

(3.5a)$$\begin{eqnarray}\displaystyle & \displaystyle \mathop{\sum }_{n=1}^{N_{elts}}\boldsymbol{f}_{0}[n]=\mathbf{0}, & \displaystyle\end{eqnarray}$$

(3.5b)$$\begin{eqnarray}\displaystyle & \displaystyle \mathop{\sum }_{n=1}^{N_{elts}}\left[\boldsymbol{r}(s_{n})+\frac{l_{elt}^{2}}{24}\unicode[STIX]{x1D705}(s_{n})\hat{\boldsymbol{n}}(s_{n})+O(l_{elt}^{4})\right]\times \boldsymbol{f}_{0}[n]=\mathbf{0}, & \displaystyle\end{eqnarray}$$

$l_{elt}^{2}\unicode[STIX]{x1D705}(s_{n})/24$

$i\text{th}$

$s_{i}$

(3.6)$$\begin{eqnarray}\displaystyle & & \displaystyle 8\unicode[STIX]{x03C0}[\boldsymbol{U}_{sw}+\unicode[STIX]{x1D734}_{sw}\times \boldsymbol{r}(s_{i})+\boldsymbol{u}_{0}^{phor}(s_{i})]\nonumber\\ \displaystyle & & \displaystyle \quad =\int _{-1}^{1}[\unicode[STIX]{x1D642}(s_{i},\tilde{s})\boldsymbol{\cdot }\boldsymbol{f}_{0}(\tilde{s})-\unicode[STIX]{x1D645}(s_{i},\tilde{s})\boldsymbol{\cdot }\boldsymbol{f}_{0}(s_{i})]\,\text{d}\tilde{s}+\unicode[STIX]{x1D647}(s_{i})\boldsymbol{\cdot }\boldsymbol{f}_{0}(s_{i}).\end{eqnarray}$$

For our element-based approach, we break the integral over the entire filament into a sum of integrals over the segments, in each of which $\boldsymbol{f}_{0}(s)$ takes a constant value.

(3.7)$$\begin{eqnarray}\displaystyle & & \displaystyle 8\unicode[STIX]{x03C0}[\boldsymbol{U}_{sw}+\unicode[STIX]{x1D734}_{sw}\times \boldsymbol{r}(s_{i})+\boldsymbol{u}_{0}^{phor}(s_{i})]\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{j=1}^{N_{elts}}\int _{E_{j}}[\unicode[STIX]{x1D642}(s_{i},\tilde{s})\boldsymbol{\cdot }\boldsymbol{f}_{0}[j]-\unicode[STIX]{x1D645}(s_{i},\tilde{s})\boldsymbol{\cdot }\boldsymbol{f}_{0}[i]]\,\text{d}\tilde{s}+\unicode[STIX]{x1D647}(s_{i})\boldsymbol{\cdot }\boldsymbol{f}_{0}[i].\end{eqnarray}$$

We aim to obtain a system of equations of the form

(3.8)$$\begin{eqnarray}8\unicode[STIX]{x03C0}[\boldsymbol{U}_{sw}+\unicode[STIX]{x1D734}_{sw}\times \boldsymbol{r}(s_{i})+\boldsymbol{u}_{0}^{phor}(s_{i})]=\mathop{\sum }_{j=1}^{N_{elts}}\unicode[STIX]{x1D614}_{ij}\boldsymbol{\cdot }\boldsymbol{f}_{0}[j],\end{eqnarray}$$

and solve (3.8), (3.5a) and (3.5b) for the unknowns $\{\boldsymbol{f}_{0}[i],i=1,\ldots ,N_{elts}\},\boldsymbol{U}_{sw},\unicode[STIX]{x1D734}_{sw}$. Each submatrix $\unicode[STIX]{x1D614}_{ij}$ is given by

(3.9)$$\begin{eqnarray}\unicode[STIX]{x1D614}_{ij}=\left\{\begin{array}{@{}ll@{}}\displaystyle \int _{E_{j}}\unicode[STIX]{x1D642}(s_{i},\tilde{s})\,\text{d}\tilde{s}\quad & \text{if}~i\neq j\\ \displaystyle \int _{E_{i}}[\unicode[STIX]{x1D642}(s_{i},\tilde{s})-\unicode[STIX]{x1D645}(s_{i},\tilde{s})]\,\text{d}\tilde{s}-\int _{\unicode[STIX]{x1D6FA}\backslash E_{i}}\unicode[STIX]{x1D645}(s_{i},\tilde{s})\,\text{d}\tilde{s}+\unicode[STIX]{x1D647}(s_{i})\quad & \text{if}~i=j.\end{array}\right.\end{eqnarray}$$

Note for the diagonal submatrices, $\unicode[STIX]{x1D614}_{ij}$ with $i=j$, we avoid any singularities by having the difference $[\unicode[STIX]{x1D642}(s_{i},\tilde{s})-\unicode[STIX]{x1D645}(s_{i},\tilde{s})]$ (which is regular over $E_{i}$ as the singularities have cancelled each other) integrated over $E_{i}$, and $\unicode[STIX]{x1D645}(s_{i})$ (which is singular in $E_{i}$) integrated over the entire filament except $E_{i}$, denoted by $\unicode[STIX]{x1D6FA}\backslash E_{i}$. We formulate the entire system as

(3.10)$$\begin{eqnarray}\left(\begin{array}{@{}ccccc@{}}\unicode[STIX]{x1D614}_{11} & \cdots \, & \unicode[STIX]{x1D614}_{1N_{elts}} & -\mathbf{1}_{3\times 3} & \unicode[STIX]{x1D750}\boldsymbol{\cdot }\boldsymbol{r}(s_{1})\\ \vdots & \ddots & \vdots & \vdots & \vdots \\ \unicode[STIX]{x1D614}_{N_{elts}1} & \cdots \, & \unicode[STIX]{x1D614}_{N_{elts}N_{elts}} & -\mathbf{1}_{3\times 3} & \unicode[STIX]{x1D750}\boldsymbol{\cdot }\boldsymbol{r}(s_{N_{elts}})\\ \boldsymbol{ 1}_{3\times 3} & \cdots \, & \mathbf{1}_{3\times 3} & \mathbf{0}_{3\times 3} & \mathbf{0}_{3\times 3}\\ \unicode[STIX]{x1D750}\boldsymbol{\cdot }\boldsymbol{r}(s_{1}) & \cdots \, & \unicode[STIX]{x1D750}\boldsymbol{\cdot }\boldsymbol{r}(s_{N_{elts}}) & \mathbf{0}_{3\times 3} & \mathbf{0}_{3\times 3}\end{array}\right)\left(\begin{array}{@{}c@{}}\boldsymbol{f}_{0}[1]\\ \vdots \\ \boldsymbol{f}_{0}[N_{elts}]\\ \boldsymbol{U}_{sw}\\ \unicode[STIX]{x1D734}_{sw}\end{array}\right)=\left(\begin{array}{@{}c@{}}\boldsymbol{u}_{0}^{phor}(s_{1})\\ \vdots \\ \boldsymbol{u}_{0}^{phor}(s_{N_{elts}})\\ \mathbf{0}_{3\times 1}\\ \mathbf{0}_{3\times 1}\end{array}\right),\end{eqnarray}$$

where $\unicode[STIX]{x1D750}$ is the Levi-Civita tensor such that $(\unicode[STIX]{x1D750}\boldsymbol{\cdot }\boldsymbol{r})\boldsymbol{\cdot }\unicode[STIX]{x1D734}_{sw}=\boldsymbol{r}\times \unicode[STIX]{x1D734}_{sw}$. We use the notation $\mathbf{0}_{3\times 1}$ for the $3\times 1$ zero vector, the notation $\mathbf{0}_{3\times 3}$ for the $3\times 3$ zero matrix and similarly $\mathbf{1}_{3\times 3}$ for the $3\times 3$ identity matrix. The last two block lines of the matrix on the left-hand side of (3.10) enforce force and torque balances. Note that it is also possible to calculate the swimming velocity using a version of the reciprocal theorem that is appropriate for filaments, which may be used to give more detailed insight into the total contributions of azimuthal slip flows.

3.2 Computational cost

Before we proceed with the method validation and results, it is worth briefly discussing the computational efficiency gains in employing SPT over the boundary element method (BEM), Montenegro-Johnson (Reference Montenegro-Johnson2018) and providing some benchmarks. In doing so, we will focus on the gains in calculating the chemical solute concentration, which represents our novel contribution.

The computational gains from employing SPT over the BEM arise in two parts of the code, (a) generating the filament geometry, in other words computational mesh and quadrature points, and (b) solving for the surface solute concentration. Once $c(s,\unicode[STIX]{x1D703})$ at a cloud of points the surface is available, the same techniques may be used to evaluate the concentration gradient for the slip velocity, and so this process is comparable (and takes a negligible time). Similarly, once this slip velocity is available, either SBT or the BEM may be used to solve for the hydrodynamics (with SBT significantly faster). The benchmark results are given in table 1.

Table 1.

An indication of the computational savings that can be made by employing SPT over the boundary element method. Simulations run on a Macbook Pro 2.9 GHz Intel Core i7 (2017) with 16 GB 2133 MHz LPDDR3 RAM.

Simulations were performed on a 2017 Macbook Pro using Matlab. It should be noted that the code is efficient, but not precompiled. The BEM code uses Fekete quadrature over quadrilateral triangles, with a high-order rule (190 points) when integrating a triangle where the evaluation point lies on a vertex, and a lower-order rule (28 points) for other triangles. Note that while these values seem much higher than the SPT quadrature, the fact that these points are spread over two dimensions makes the high-order rule comparable, and the low-order rule significantly coarser. Calculating the slip velocity via spline interpolation took $O(0.01)$ seconds for both methods.

A key reason for this speed up is not only the reduction in the dimension of the integral equation from 2 (BEM) to 1 (SPT), but also the fact that SPT gives the surface concentration by evaluating an integral, whereas BEM requires one to solve an integral equation (which entails setting up and solving a matrix system). As a crude estimate, evaluation requires $O(N_{s}^{2})$, whereas the solution for BEM requires $O(N_{eval}^{3})$ steps ($N_{eval}\gg N_{s}$) to solve for a direct solver, although this could be reduced using iterative solvers such as the generalized minimal residual method (GMRES).

3.3 Layout of validation and results

In the following sections, we validate and apply SPT to various filament geometries, cross-sections and activity patterns of increasing complexity, as summarised in figure 2. Throughout, we set the mobility ${\mathcal{M}}=-1$, so that slip flows go from high surface solute concentrations to low. Firstly, straight rods are used to validate our SPT results against analytical formulae for prolate spheroids. Secondly, planar curved rods (with uniform cross-section) are validated against boundary element computations using the authors’ previously published regularised singularity code (Montenegro-Johnson, Michelin & Lauga Reference Montenegro-Johnson, Michelin and Lauga2015; Varma, Montenegro-Johnson & Michelin Reference Varma, Montenegro-Johnson and Michelin2018), paying particular attention to the important azimuthal variation. Next, planar curved rods are analysed with SPT – a circular arc and sinusoidal S-shaped centreline. The impact of azimuthal phoretic effects on the resulting kinematics is calculated. Finally, we consider 3-D helical phoretic filaments, focusing on the Janus helix, an autophoretic swimmer with the ability to explore space on a helical trajectory, relevant to sensing and enhanced 3-D mixing applications.

Figure 2.

Summary of the different activity patterns (uniform ${\mathcal{A}}(s)=-1$, smooth decay ${\mathcal{A}}(s)=s^{2}-1$, Saturn – 2 symmetric active caps, Janus – 1 active cap), cross-sectional radius profiles (uniform, hemispherical cap used for BEM, prolate spheroid) and centreline geometries (straight, circular arc, S-shape, helix), used throughout the results section.

6.1 Azimuthal variation of the concentration field

We now use SPT to examine the physics of slender autophoretic filaments of curved centreline. Throughout this section we consider a spheroidal end shape to ensure the regularity of the SPT formulation. Figure 7 shows colour maps of the concentration field (lines of same colour indicate the contours of the concentration field) at different arclength values ($s$) and azimuthal ($\unicode[STIX]{x1D703}$) positions for the three shapes (straight a, circular arc b and S-shaped c) in the case of a uniform catalytic coating, ${\mathcal{A}}(s)=-1$, that depletes its surrounding solute.

Figure 7.

The surface concentration of uniformly active phoretic filaments with a prolate spheroidal end shape and planar centrelines with (a) straight shape which is stationary but pumps fluid, (b) circular arc shape which translates and (c) S-shape which rotates. Small azimuthal variation due to non-local interactions and confinement effects from curvature are visible in the circular arc and S-shaped examples.

For the straight filament (figure 7a), we see axisymmetry (i.e. no variation of colour with different $\unicode[STIX]{x1D703}$ at a given $s$) in the concentration field, as expected. We note that for the straight filament the first-order concentration field vanishes, hence figure 7(a) shows the zeroth-order concentration field which is axisymmetric, but non-uniform in $s$ due to the non-local and endpoint effects already present in (2.39). For example, the closer to the endpoints, the more space there is available for diffusion, hence the solute depletion is less and the solute reactant is at a higher concentration.

For circular arc filaments, (figure 7b), we see a minimum of the concentration field occurring at $s=0$ and $\unicode[STIX]{x1D703}=0(2\unicode[STIX]{x03C0})$, i.e. the part of the surface that is facing the centre of the semicircular centreline. This is a confinement effect arising from the curvature of the centreline, as this inner part of the surface experiences more depletion of the solute from the local surface and the adjacent active parts of the surface. Close to the outer part of the surface, there is more space (due to curvature) for diffusion of solute from the bulk to the filament surface thus the solute that has been depleted due to the reaction can be replenished more easily. Straight and circular arc filament centreline shapes share the symmetry $c(s,\unicode[STIX]{x1D703})=c(-s,\unicode[STIX]{x1D703})$ for the concentration field, arising naturally from the centreline symmetry in $s\rightarrow -s$. This symmetry is broken by S-shaped filaments, for which $c(s,\unicode[STIX]{x1D703})=c(-s,-\unicode[STIX]{x1D703})$, as in figure 7(c). At $s=0$, the concentration is minimised by the azimuthal positions $\unicode[STIX]{x1D703}=0,\unicode[STIX]{x03C0}$ with $c(0,0)=c(0,\unicode[STIX]{x03C0})$, as these lie on the $x,y$ plane and therefore are closer to the nearby active surfaces (by the same amount by symmetry), and $c(0,\unicode[STIX]{x03C0}/2)=c(0,3\unicode[STIX]{x03C0}/2)$.

6.2 Kinematics: leading-order contribution of azimuthal variations of the concentration field

Turning now to the kinematics of these filaments, figure 8 shows the translational and rotational speeds for circular arc and S-shaped filaments when the azimuthal terms are included (‘on’ – solid lines) versus neglected (‘off’ – dashed), for two different activity profiles. The azimuthal effect contribution to the kinematics vanishes as $\unicode[STIX]{x1D6FC}$ decreases to 0 in all the plots of figure 8, as expected for straight filaments that have zero curvature ($\unicode[STIX]{x1D6FC}=0$) and an axisymmetric concentration field. As $\unicode[STIX]{x1D6FC}$ increases, so does the curvature, and hence the azimuthal effects too.

Figure 8.

Leading-order contribution of azimuthal effects to kinematics arising from curvature and confinement. Kinematics of (a) circular arc and (b) S-shaped filaments versus the angle parameter based on SPT, for uniform and Saturn activity patterns, where the Saturn particle has two active end caps and an inert midpiece (see figure 2). Here, $\unicode[STIX]{x1D716}=0.01$. Solid lines include azimuthal slip flows, dashed lines have this contribution removed, i.e. $A_{s}=A_{c}=0$ in (2.60).

The importance of the azimuthal effects, as noted in Montenegro-Johnson (Reference Montenegro-Johnson2018), is quite profound in the case of uniform activity, where neglecting their contribution results in an incorrect prediction in the direction of motion, as shown by the dashed lines in figure 8(a,b). In order to understand this, let us consider the uniformly active, circular arc filament (see also the qualitative schematic figure 9b). If we discount azimuthal effects, the higher solute concentration at the ends drives a longitudinal surface slip flow to the middle, giving rise to translation in the opposite direction, hence the positive velocity shown by the dashed blue line in figure 8(a).

Since the filament consumes solute, confinement effects mean that there is a lower surface concentration in the inner filament surface that faces the centre of the circular arc centreline. This drives an azimuthal tangential surface slip flow from the outer to the inner side of the curve. This means that, locally, the filament is acting as a 2-D squirmer (Blake Reference Blake1971), with symmetry about $\unicode[STIX]{x1D703}=0$, $\unicode[STIX]{x03C0}$. This outer to inner flow locally exerts a flow forcing towards the outer side of the cylinder, propelling the filament in that direction. This forcing is in the opposite direction to that arising from longitudinal concentration gradient, and this contribution is large enough that the overall translational velocity has a negative sign, as shown by the solid blue line in figure 8(a).

For Saturn particles, which have two active caps and an inert midpiece (see figure 2), the active caps ensure that the tangential concentration gradients dominate (due to the fast velocity at the change in activity) and hence the kinematics with the azimuthal terms on/off have the same directionality. In more detail (see also the qualitative schematic figure 9c), let us consider with the case of two catalytic caps of length $\unicode[STIX]{x0394}s=0.2$. The catalytic end caps in the regions $|s|>0.8$ give rise to a lower concentration of solute at the end region, compared to the region $|s|<0.8$, driving a slip flow across the interface at $s=0.8$ directed from the high to the low solute concentration region (i.e. towards the ends). There is of course a small region of higher solute concentration at the very ends due to the end confinement effect (magnified by the presence of the prolate spheroidal ends – see the qualitative schematic figure 9a), driving a small slip flow at the very ends directed to the middle, but overall the slip flow at the interface at $|s|=0.8$ dominates, leading to motion of the filament in the negative direction, as shown by the red, dashed line in figure 8(a).

Now consider the azimuthal effect, focusing on the middle region around $s=0$, which by the confinement effect, has higher solute concentration close to the ‘outer’ surface that looks away from the centre of the semicircle. This drives an azimuthal flow around the filament towards the centre of the semicircle, which contributes to a negative direction, i.e. in the same direction as the kinematics with the azimuthal effects switched ‘off’, hence the overall speed is increased when the azimuthal effects are included, as shown by the red, solid line in figure 8(a).

Similar explanations hold for the kinematics of S-shaped filaments shown in figure 8(b).

Figure 9.

Qualitative schematic: understanding end shape and azimuthal effects. Red text indicates the relative concentration of solute in each figure, with slip flow (blue) moving from higher to lower concentrations, assuming constant, negative mobility. (a) The shape of the filaments affects the geometric confinement at the ends, resulting in different strength of slip flows. (b) A uniformly active prolate spheroid. For our parameter choices, azimuthal flows work in opposition to axial flows and hence the swimming direction is sensitive to which effect is stronger, which in turn depends on geometry and end shape. (c) In a capped filament azimuthal flows and axial flows work together, however, there is an opposing end-effect flow (as with the Janus prolate spheroid) that slows the swimmer.

7.1 Helix geometry, concentration and kinematics

We consider a helical centreline with chirality $h$ (taking the values $\pm 1$ for right-handed and left-handed helices respectively), helical angle $\unicode[STIX]{x1D6FC}$ and $n$ turns,

(7.1)$$\begin{eqnarray}(x,y,z)=\left(R\cos \frac{(s+1)\sin \unicode[STIX]{x1D6FC}}{R},hR\sin \frac{(s+1)\sin \unicode[STIX]{x1D6FC}}{R},s\cos \unicode[STIX]{x1D6FC}\right),\end{eqnarray}$$

where the helical radius $R$, curvature $\unicode[STIX]{x1D705}$ and torsion $\unicode[STIX]{x1D70F}$ are given by

(7.2a-c)$$\begin{eqnarray}R=\frac{\sin \unicode[STIX]{x1D6FC}}{n\unicode[STIX]{x03C0}},\quad \unicode[STIX]{x1D705}=\frac{\sin ^{2}\unicode[STIX]{x1D6FC}}{R},\quad \unicode[STIX]{x1D70F}=h\frac{\sin \unicode[STIX]{x1D6FC}\cos \unicode[STIX]{x1D6FC}}{R},\end{eqnarray}$$

and its pitch by $(2\cos \unicode[STIX]{x1D6FC})/n$. We begin by validating a constant cross-section rod against BEM for a helix with 2.5 turns and $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x03C0}/3$, with regularised activity ${\mathcal{A}}(s)=-(s+1)(1-\tanh (2s))$ and mobility ${\mathcal{M}}=-(s+1)(1-\tanh (4s))$. This regularisation is somewhat similar to a single capped end, but with a smoothly decaying velocity. We achieve approximately 1 % error in the concentration field, and 2 % and 7 % errors in translational and angular velocities respectively, where angular velocity is calculated about the helix centroid. We note that this larger relative error in the swimming velocity arises from the fact that this regularised case only swims relatively slowly.

Continuing with SPT for a filament with prolate spheroidal ends, the surface concentration of a uniformly active helix is shown in figure 10(a) for varying arclengths ($s$) and azimuthal positions, using $\unicode[STIX]{x1D703}+\unicode[STIX]{x1D703}_{i}(s)$ in order to ensure that $\unicode[STIX]{x1D703}$ lies in $[0,2\unicode[STIX]{x03C0})$ (recall that by definition, $\unicode[STIX]{x1D703}_{i}(s)$ follows the torsion of the filament – (2.8)). The apparent shear about $\unicode[STIX]{x1D703}(s)=\unicode[STIX]{x03C0}$ is a result of this choice for $\unicode[STIX]{x1D703}_{i}(s)$. In figure 10(b), we show the surface concentration of a Janus helix capped for $s\leqslant -0.8$, with $\unicode[STIX]{x1D716}=0.01$, showing strong local gradients around the capped end.

7.2 Exploration of space

Many microswimmers make use of helical trajectories for exploring their environment and sensing, and thus also perform chemotaxis towards food resources. Helical trajectories could offer enhanced sensing and 3-D mixing to artificial phoretic microswimmer applications.

As shown schematically in figure 11, Janus helices can explore space on a helical trajectory, with radius and pitch that vary with the geometry of the helical filament. Figure 12(a) shows the trajectories of Janus helices of the same arclength (2), helical angle ($\unicode[STIX]{x03C0}/3$) and length of catalytic cap ($s_{c}=-0.9$), but different number of turns. All simulations run over the same time interval ($t=30$ in dimensionless units). In real space, the axis of the trajectories is aligned with the $\unicode[STIX]{x1D734}$ vector. In figure 12(a) the trajectories have been rotated to be aligned vertically for the purpose of comparison. In order to quantify the ability of the different Janus helices to explore space, in figure 12(b) we plot the explorative area of the helical trajectory, which is equal to $\unicode[STIX]{x03C0}R_{T}^{2}$ for $R_{T}$ the helix radius as in figure 11. Figure 12(c) shows the velocity along the axis of each helical trajectory versus the number of turns.

Figure 11.

Schematic showing a Janus helix, on a helical trajectory of radius $R_{T}$, from a side and top view, demonstrating the ‘exploratory area’ of the swimmer $\unicode[STIX]{x03C0}R_{T}^{2}$.

These plots show interesting nonlinear behaviour, with a number of transition points, the most dramatic of which is close to when the number of turns $n$ increases above a single full turn. There are two limiting behaviours. Firstly, as $n\rightarrow 0$, the helix tends to a straight rod, and the axial velocity in turn tends to that of a straight rod. However, the explorative area does not seem to approach zero in this limit – rather the nearly straight helix may explore a non-zero or even infinite area, but take an infinite time to do it as $n\rightarrow 0$. On the other hand, as $n\rightarrow \infty$, the helix again approaches a straight rod, and we would expect the axial velocity to once more tend to the straight case. However, we cannot analyse this limit within our framework, as it entails divergent curvature $\unicode[STIX]{x1D705}$; our analysis remains valid provided $\unicode[STIX]{x1D705}=n\unicode[STIX]{x03C0}\sin \unicode[STIX]{x1D6FC}\ll 1/\unicode[STIX]{x1D716}$.

We note that between these limits, we expect a decrease in axial velocity (since propulsion force does not lie solely along the axial direction), and hence we expect a minimum in axial velocity – obtained somewhere in $n\in (0.75,1)$ for the particular values of parameters we have chosen here. This behaviour is not monotonic, but displays periodic fluctuations that change depending on whether the helix has an odd or even number of turns, that are not clearly visible in figure 12(a). Finally, we note that smart, stimulus-responsive materials, such as thermoresponsive hydrogel composites, could assist in exploiting both features by changing the number of turns of a Janus helix, as can occur naturally in the polymorphism of bacterial flagella (Spagnolie & Lauga Reference Spagnolie and Lauga2011), so that it can have a slow, explorative mode with less than one turn, and a faster, close to straight swimming mode at higher number of turns.

Figure 12.

Trajectories of helical filaments with one catalytic end of length 0.1, i.e. $s_{c}=-0.9$, helical angle $\unicode[STIX]{x03C0}/3$ and varying number of turns. (a) The different helical trajectories arising from varying the number of turns. All simulations run over the same time interval ($t=30$ in dimensionless units). The axis of the trajectories is aligned with the $\unicode[STIX]{x1D734}$ vector in real space. The trajectories have been rotated to be aligned vertically for the purposes of this plot. (b) Area of helical trajectories, i.e. $\unicode[STIX]{x03C0}R_{T}^{2}$ for helical radius $R_{T}$, versus number of turns of the Janus helix. (c) Velocity along the axis of each helical trajectory versus number of turns of the Janus helix.