\subsection{Surfactant-covered bubble or drop in steady state}
\label{sec:validsteady}
\noindent For single surfactant-covered bubbles, analytical solutions to interface position $z$ and surfactant concentration $\rho$ exist in certain cases.
The case regarded here is that of a bubble in an extensional flow, $\mathbf{u}_\infty = Q(x,-y)$, at \textit{steady state}, for which exact solutions were originally obtained by \citeauthor{siegelvalid} \cite{siegelvalid}. A steady state is defined as the time $T$ when $\mathbf{u}\cdot\mathbf{n}=0$. These solutions exist for bubbles deforming from an initially circular shape with uniform surfactant concentration and where there is no diffusion of surfactants along the interface, i.e. $Pe_\Gamma = \infty$. To couple surfactant concentration to surface tension, the linear equation of state \eqref{eq:form_eqstate_linear_nd} is used. For details and derivations on how to obtain these solutions, the reader is referred to the original reference. Here, it will be explained how to compute these solutions to use as a validation case. Given a deformation at steady state, i.e. $z$ at time $T$, these solutions are used to analytically compute $\rho$ at time $T$ as well as the Capillary number $Q$ that is needed to obtain the set deformation at steady state. Using a numerical method to simulate the deformation and surfactant evolution under this Capillary number, the solution at steady state can then be compared to the analytical solution to determine the accuracy.

To describe the bubble interface, a conformal mapping is used,
\begin{align}
	z(\zeta,t) = \dfrac{a(t)}{\zeta} + b(t)\zeta,
	\label{eq:siegel_confmap}
\end{align}
where $a(t)<0$ and $b(t)$ is real. Furthermore, the condition of constant area gives $a^2-b^2 = 1$. At the bubble interface, $|\zeta|=1$, i.e. $\zeta=e^{i\nu}$ for $\nu\in[0,2\pi]$. At steady state, it is required that
\begin{align}
	Q  = -\dfrac{Ab}{(1+b^2)^{1/2}},
	\label{eq:siegel_Q}
\end{align}
where
\begin{align}
	A = \dfrac{\int_0^{2\pi}B(b,\nu)^{1/2}d\nu-2\pi E}{\int_0^{2\pi}B(b,\nu)d\nu},
	\label{eq:siegel_A}
\end{align}
with $E$ the elasticity number in \eqref{eq:form_eqstate_linear_nd}, and
\begin{align}
	B(b,\nu) = 1 + 2b^2-2(1+b^2)^{1/2}b\cos(2\nu).
	\label{eq:siegel_B}
\end{align}
The deformation of a bubble is measured as
\begin{align}
	D=\dfrac{R_{max}-R_{min}}{R_{max}+R_{min}},
	\label{eq:siegel_D}
\end{align}
where $R_{max}$, $R_{min}$ are the maximum and minimum radial distance of a point on the interface to the centre respectively.

To compute the surfactant concentration at steady state, \eqref{eq:form_surfacteq_nondim} is used. In steady state, this can be simplified to
\begin{align}
	\rho \Re\left[ \dfrac{u(z)+\overline{z}_\nu}{|z_\nu|}\right] = \dfrac{1}{Pe_\Gamma}\left(\dfrac{\rho_\nu}{|z_\nu|}\right) = 0,
\end{align}
where the second equality comes from having no diffusion, $Pe_\Gamma=\infty$. This means, that on any part of the interface at steady state, either $\rho = 0$ or $u(z) = 0$ (no slip condition). Assuming the extensional flow is sufficiently low such that the concentration of surfactants is non-zero everywhere, it fulfils
\begin{align}
	\rho(\nu) = \dfrac{1-A|z_\nu|}{E}.
	\label{eq:siegel_rho}
\end{align}
An algorithm to compute the steady state surfactant concentration $\rho(\nu)$ and Capillary number $Q$ given the steady state deformation can be found in Algorithm~\ref{alg:steady}. The deformation at steady state is determined through the conformal mapping parameters $a$ and $b$ in \eqref{eq:siegel_confmap} which are given as input to the algorithm. How the deformation $D$ depends on the Capillary number $Q$ for different elasticity numbers is shown in Figure~\ref{fig:plot4}.

To validate the results of a numerical method against these exact solutions, the results at steady state are compared when simulating an initially circular bubble with uniform surfactant concentration $\rho_0 = 1$ in an extensional flow with Capillary number $Q$ and $Pe_\Gamma=\infty$, i.e. no diffusion of surfactants.


\IncMargin{1em}
\RestyleAlgo{boxruled}
\SetCommentSty{emph}
\SetKwComment{Comment}{}{}
\DontPrintSemicolon
\LinesNumbered

\begin{algorithm}[H]
\SetKwInOut{Input}{Input}\SetKwInOut{Output}{Output}

\Input{Conformal mapping parameters $a$ and $b$ at steady state.}

\Output{Deformation $D$. Capillary number $Q$ for which this deformation is obtained at steady state. Interface position $z(\nu_j)$ and surfactant concentration $\rho(\nu_j)$ at steady state, for $\nu_j=\frac{2\pi j}{M}$, $j\in[0,M-1]$.}
\vspace{2mm}

Discretise $\nu_j = \dfrac{2\pi j}{M}$, $j\in[0,M-1].$ \;
Compute $z(\nu_j)$ through \eqref{eq:siegel_confmap}.\;
Find $R_{max}$, $R_{min}$ and compute deformation $D$ in \eqref{eq:siegel_D}. \;
Compute $B(b,\nu_j)$ in \eqref{eq:siegel_B}.\;
Compute $A$ in \eqref{eq:siegel_A} using trapezoidal rule. \;
Compute $\rho(\nu_j)$ with \eqref{eq:siegel_rho}. \;
Compute $Q$ from \eqref{eq:siegel_Q}. \;

\caption{Steady state computation}
\label{alg:steady}
\end{algorithm}\DecMargin{1em}


As was discussed by \citeauthor{Milliken1993a} in \cite{Milliken1993a}, in the case of no diffusion along the interface the viscosity ratio does not affect steady state deformation. When $Pe_\Gamma = \infty$, the interfacial velocity is dominated by the high Marangoni stresses and the effect of viscosity ratio is therefore negligible.  Thus, the results above can be used to validate the steady state deformation and surfactant concentration also for drops, where $\lambda \neq 0$.
{"config": "arxiv", "file": "1806.10371/srcfiles/valid_steady.tex"}