= Solution
The <material derivative> follows <surfactant> carried by the moving interface. Tangential <surface divergence> changes the local <surface area> and hence dilutes or concentrates insoluble material. Normal motion changes area through the <curvature>: an expanding <sphere> has positive $u_n\nabla_s\cdot\mathbf n$ and its <concentration> decreases. The last term is <surface diffusion>, which smooths <concentration> <gradients>. There is no exchange with the bulk; this is <conservation of insoluble surfactant on a moving interface>.
The interfacial speed is of order $a|\mathbf F|$. Diffusion dominates redistribution when the surface <Péclet number> satisfies
$$
\boxed{\operatorname{Pe}_s=a^2|\mathbf F|/D_s\ll1.}
$$
Then $C'/C_0=O(\operatorname{Pe}_s)$, and advection of $C'$ is a second-order effect. Write $S=\mathbf n\cdot\mathbf F\mathbf n$. Since $\mathbf u_s=\mathbf F\mathbf x-S\mathbf x$, the projected <divergence> of the first term is $\operatorname{tr}\mathbf F-S=-S$, whereas the second has <divergence> $2S$; its <gradient> contracted with $\mathbf x$ is zero. Hence $\nabla_s\cdot\mathbf u_s=-3S$.
The leading steady transport equation is $D_s\Delta_s C'=C_0\nabla_s\cdot\mathbf u_s=-3C_0S$. Trace-freeness and the supplied <surface Laplacian> identity give $\Delta_s S=-6S/a^2$. Fixing the conserved total <surfactant> removes the uniform null mode and yields
$$
\boxed{C'=AS,\qquad A=\frac{C_0a^2}{2D_s}.}
$$
This is <quadrupolar surfactant distribution on a spherical interface>; its spherical mean is zero because $\langle n_i n_j\rangle=\delta_{ij}/3$.
With $+$ outside, $-$ inside, and the normal directed out of the bubble, the <interfacial stress balance with variable surface tension> is
$$
(\boldsymbol\sigma_+-\boldsymbol\sigma_-)\mathbf n=\gamma\kappa\mathbf n-\nabla_s\gamma.
$$
Now $\nabla_s S=2I_s\mathbf F\mathbf n/a$, so $-\nabla_s\gamma=2A\gamma_1I_s\mathbf F\mathbf n/a$. Multiplying the <curvature> by the tension and retaining linear terms gives
$$
\boxed{[\boldsymbol\sigma\mathbf n]^+_-=\frac{2\gamma_0}{a}\mathbf n+\frac{4\gamma_0}{a}(\mathbf n\cdot\mathbf D\mathbf n)\mathbf n+\frac{2A\gamma_1}{a}\bigl[I_s\mathbf F\mathbf n-(\mathbf n\cdot\mathbf F\mathbf n)\mathbf n\bigr].}
$$
The undeformed isotropic <pressure> jump is kept with its normal; the perturbation terms are evaluated on the reference <sphere>. Products of deformation and <concentration> perturbations are beyond the approximation.
The imposed <tensor> is symmetric and traceless, so $\mathbf x\cdot\mathbf E\mathbf x$ is a <harmonic> <scalar> of degree two. Its decaying partner is $\mathbf E:\nabla\nabla(1/r)$; the corresponding <harmonic> <vector> uses $\mathbf E\nabla(1/r)$. <Linearity> and <rotational symmetry> give precisely these tensorial modes, with the growing <scalar> chosen to reproduce the imposed strain. Thus in the unscaled convention one can take
$$
\boldsymbol\Phi=\frac{Pa^3}{3}\mathbf E\nabla(1/r),\qquad\chi=\frac12\mathbf x\cdot\mathbf E\mathbf x+\frac{Qa^5}{3}\mathbf E:\nabla\nabla(1/r).
$$
The coefficients multiply distinct decaying modes. No other angular <harmonic> is forced at this order.
No penetration in the steady state gives $P=3Q-1$. The given tangential surface <velocity> implies $\alpha=1+2Q$. The tangential <stress> balance, with $M=A\gamma_1/(\mu a)$, gives $1+P-8Q=M\alpha$, or $-5Q=M(1+2Q)$. Therefore
$$
Q=-\frac M{5+2M},\quad P=-\frac{5(1+M)}{5+2M},\quad\boxed{\alpha=\frac5{5+2M}}.
$$
The nonuniform <normal stress> is unaffected by the uniform inviscid interior <pressure>. Its balance is
$$
2\mu(1-3P+12Q)=\frac{4\gamma_0\beta}{a}-\frac{2A\gamma_1\alpha}{a}.
$$
Substitution and rearrangement give
$$
\boxed{\mathbf D=\frac{5\mu a}{\gamma_0}\frac{2+M}{5+2M}\mathbf E.}
$$
As $M\to\infty$, $\alpha\to0$: the concentration-induced <Marangoni stress> suppresses tangential surface motion, making the exterior boundary behave like an immobile surface despite the inviscid interior. The deformation coefficient tends to $5\mu a/(2\gamma_0)$; the clean limit is $2\mu a/\gamma_0$. Both small surface <Péclet number> and small deformation, of order $\mu a|\mathbf E|/\gamma_0$, are required. This is <Marangoni immobilization of a bubble in straining flow>.
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