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Electronic supplementary information for \\ 
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{\Large``Irreversible particle motion in surfactant-laden interfaces due to pressure-dependent surface viscosity''}\\
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Harishankar Manikantan and Todd M. Squires\\
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Department of Chemical Engineering\\
University of California, Santa Barbara\\
Santa Barbara, CA 93101, USA
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Here we list the coefficients in the series solution to the stream function $\Psi$ for the general case of a cylinder within a cylinder enclosing a 2D Newtonian fluid. The translating and/or rotating inner cylinder will be identified by the streamline corresponding to $\alpha_0$, while the stationary outer wall is identified by $\alpha_1$ (see Fig.~1$(b)$). In every case, $\alpha_1=0$ retrieves the limit of a plane wall along $y=0$. We provide the coefficients corresponding to each mode of motion separately: since these are solutions to the Stokes equations, they may be suitably superimposed.

\section{Translation perpendicular to the wall}
For radial motion of the inner cylinder towards the center of the outer cylinder with velocity $V$ (or equivalently, motion along the positive $y$-axis in Fig.~1$(b)$), the only non-zero coefficients are
\begin{subequations}
\begin{align}
D&=\frac{V}{\mathcal{M}}\\
a_1'&=-\frac{1}{2} \frac{V}{\mathcal{M}} \frac{\sinh (\alpha_0+\alpha_1)}{\cosh(\alpha_0-\alpha_1)},\\
b_1'&=\frac{1}{2} \frac{V}{\mathcal{M}} \frac{\cosh (\alpha_0+\alpha_1) }{\cosh(\alpha_0-\alpha_1)},\\
c_1'&=\frac{1}{2} \frac{V}{\mathcal{M}} (\tanh (\alpha_0-\alpha_1)+2 \alpha_1),
\end{align}
\end{subequations}
where
\begin{equation}
\mathcal{M}=\tanh (\alpha_0-\alpha_1)-(\alpha_0-\alpha_1).
\end{equation}

\section{Translation parallel to wall}
For azimuthal translation of the inner cylinder velocity $U$ (or equivalently, motion along the positive $x$-axis in Fig.~1$(b)$), the only non-zero coefficients are
\begin{subequations}
\begin{align}
A&=\frac{U}{\mathcal{L}} \left[\sinh \left(2 \alpha _0\right)+\sinh \left(2 \alpha _1\right)\right],\\
B&=\frac{U}{\mathcal{L}} \left[\alpha _1\left\{\cosh \left(2 \alpha _0\right)+\cosh \left(2 \alpha _1\right)-2\right\}- \cosh \left(\alpha _1\right) \left\{\sinh \left(2 \alpha _0-\alpha _1\right)+3 \sinh \left(\alpha _1\right)\right\}\right],\\
C&=-\frac{U}{\mathcal{L}} \left[\cosh \left(2 \alpha _0\right)+\cosh \left(2 \alpha _1\right)-2\right],\\
a_1&=-\frac{U}{\mathcal{L}} \sinh ^2\left(\alpha _0+\alpha _1\right),\\
b_1&=\frac{1}{2} \frac{U}{\mathcal{L}} \sinh \left(2 \left(\alpha _0+\alpha _1\right)\right),\\
c_1&=\frac{1}{2} \frac{U}{\mathcal{L}} \left[2 \cosh \left(2 \alpha _0\right)+2 \cosh \left(2 \alpha _1\right)-\cosh \left(2 \left(\alpha _0-\alpha _1\right)\right)-3\right],
\end{align}
\end{subequations}
with
\begin{equation}
\mathcal{L}=(\alpha _0-\alpha_1) \left[\cosh \left(2 \alpha _0\right)+\cosh \left(2 \alpha _1\right)-2\right]-4 \sinh \left(\alpha _0\right) \sinh \left(\alpha _1\right) \sinh \left(\alpha _0-\alpha _1\right) .
\end{equation}

\section{Rotation next to wall}
For rotation of inner cylinder towards the center of the outer cylinder with angular velocity $\Omega$ (such that $\Omega$ is positive in the counter-clockwise sense in Fig.~1$(b)$), the only non-zero coefficients are
\begin{subequations}
\begin{align}
A&=-\frac{a\Omega}{\mathcal{N}} \coth \left(\alpha _0-\alpha _1\right) \left[2 \left(\alpha _0-\alpha _1\right) \sinh \left(\alpha _0\right)-\cosh \left(\alpha _0\right)+\cosh \left(\alpha _0-2 \alpha _1\right)\right],\\
B&=-\frac{a\Omega}{\mathcal{N}} \left[\cosh \left(\alpha _0\right)\left\{2 \alpha _1+\sinh \left(2 \alpha _1\right)\right\} \right. \nonumber \\
& \quad \quad \left. +\coth \left(\alpha _0-\alpha _1\right) \left\{\alpha _1 \sinh \left(2 \alpha _1\right) \cosh \left(\alpha _0\right)-2 \alpha _0 \sinh \left(\alpha _0\right) \cosh ^2\left(\alpha _1\right)\right\}\right],\\
C&=2 \frac{a\Omega}{\mathcal{N}} \sinh \left(\alpha _0-\alpha _1\right) \sinh \left(\alpha _1\right),\\
a_1&=-\frac{a\Omega}{\mathcal{N}} \sinh \left(\alpha _0+\alpha _1\right) \left[\sinh \left(\alpha _1\right)+\left(\alpha _1-\alpha _0\right) \sinh \left(\alpha _0\right) \text{csch}\left(\alpha _0-\alpha _1\right)\right],\\
b_1&=-\frac{1}{2} \frac{a\Omega}{\mathcal{N}} \frac{\cosh \left(\alpha _0+\alpha _1\right)}{\sinh\left(\alpha _0-\alpha _1\right)} \left[2 \left(\alpha _0-\alpha _1\right) \sinh \left(\alpha _0\right)-\cosh \left(\alpha _0\right)+\cosh \left(\alpha _0-2 \alpha _1\right)\right],\\
c_1&=-\frac{1}{2} \frac{a\Omega}{\mathcal{N}} \left[2 \left(\alpha _0-\alpha _1\right) \sinh \left(\alpha _0\right) \left\{\sinh \left(2 \alpha _1\right) \coth \left(\alpha _0-\alpha _1\right)+1\right\} \right. \nonumber \\
& \hspace{2.5in} \left. -2 \sinh \left(\alpha _1\right) \sinh \left(\alpha _0+\alpha _1\right)\right],
\end{align}
\end{subequations}
where $\mathcal{L}$ is the same as with parallel translation.

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