= Marangoni immobilization of a bubble in straining flow
{c}
For an almost spherical inviscid bubble in exterior <Stokes flow> $\mathbf E\mathbf x$, let $\gamma=\gamma_0-\gamma_1(C-C_0)$ and $K=C_0a^2/(2D_s)$. The <quadrupolar surfactant distribution on a spherical interface> makes the tangential velocity $\mathbf u_s=\alpha\mathbf I_s\mathbf E\mathbf x$ satisfy
$$
\alpha=\frac5{5+2M},\qquad M=\frac{K\gamma_1}{\mu a}.
$$
The <Marangoni stress> approaches the stress needed to suppress surface motion as $M$ grows. The quadrupolar shape perturbation $r=a(1+\mathbf n\cdot\mathbf D\mathbf n)$ is
$$
\mathbf D=\frac{5\mu a}{\gamma_0}\frac{2+M}{5+2M}\mathbf E.
$$
These expressions require small deformation and small surface <Péclet number>; large $M$ alone does not justify a small concentration perturbation.
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