= Solution
The <material derivative> follows the interfacial <surfactant>. The term $-C\nabla_s\cdot\mathbf u_s$ describes dilution by tangential expansion and concentration by compression. The term $-C(\mathbf u\cdot\mathbf n)\nabla_s\cdot\mathbf n$ accounts for changing <surface area> due to normal motion of a curved interface. Finally, $D_s\nabla_s^2C$ describes <surface diffusion>. Together these give <conservation of insoluble surfactant on a moving interface>.
The relevant surface <Péclet number> is
$$
\mathrm{Pe}_s=\frac{a^2|\mathbf A|}{D_s}\ll1.
$$
Write $S=\mathbf n\cdot\mathbf A\cdot\mathbf n$ and $\mathbf u_s=\mathbf A\mathbf x-S\mathbf x$. Since $\mathbf A$ has zero <trace> and $\nabla_s\mathbf n=\mathbf I_s/a$, the <surface divergence> is
$$
\nabla_s\cdot(\mathbf A\mathbf x)=\mathbf I_s:\mathbf A=-S,\qquad
\nabla_s\cdot(S\mathbf x)=2S,
\qquad\boxed{\nabla_s\cdot\mathbf u_s=-3S.}
$$
There is no normal motion. To first order in $\mathrm{Pe}_s$, the steady <surfactant> balance becomes $D_s\nabla_s^2C'=C_0\nabla_s\cdot\mathbf u_s=-3C_0S$. The traceless quadratic $S$ is a degree-two <spherical harmonic>, so $\nabla_s^2S=-6S/a^2$. The mean of $C'$ is zero by total <surfactant> conservation; consequently
$$
\boxed{C'=KS,\qquad K=\frac{C_0a^2}{2D_s}.}
$$
The discarded advective term $\mathbf u_s\cdot\nabla_sC'$ is smaller by $\mathrm{Pe}_s$. This is the <quadrupolar surfactant distribution on a spherical interface>.
With the unit normal directed from the bubble into the exterior, the <interfacial stress balance with variable surface tension> is
$$
[\boldsymbol\sigma\cdot\mathbf n]_-^+=\gamma\kappa\mathbf n-\nabla_s\gamma.
$$
Using the specified first-order <curvature> and $\gamma=\gamma_0-K\gamma_1S$, together with $\nabla_sS=2\mathbf I_s\mathbf A\mathbf n/a$, gives
$$
\boxed{[\boldsymbol\sigma\cdot\mathbf n]_-^+=\frac{2\gamma_0}{a}\mathbf n+\frac{4\gamma_0}{a}(\mathbf n\cdot\mathbf D\cdot\mathbf n)\mathbf n+\frac{2K\gamma_1}{a}\left[\mathbf I_s\mathbf A\mathbf n-S\mathbf n\right].}
$$
The first term is the spherical <capillary pressure>; the last contains the normal tension correction and the tangential <Marangoni stress>.
The ambient <Stokes flow> has the symmetry of the symmetric traceless <tensor> $\mathbf E$. In the <Unscaled Papkovich–Neuber representation>, $\chi_\infty=\mathbf x\cdot\mathbf E\cdot\mathbf x/2$ is <harmonic> and yields $\mathbf E\mathbf x$. The decaying vector potential must have the form $\mathbf E\nabla(1/r)$, and the scalar disturbance must have the degree-two form $\mathbf E:\nabla\nabla(1/r)$. They are <harmonic> outside the bubble and have precisely the required rotational covariance. Dimensionless amplitudes may therefore be written
$$
\boldsymbol\Phi=\frac{Pa^3}{3}\mathbf E\nabla\frac1r,\qquad
\chi=\frac12\mathbf x\cdot\mathbf E\cdot\mathbf x+\frac{Qa^5}{3}\mathbf E:\nabla\nabla\frac1r.
$$
For a steady bubble, the <no-penetration boundary condition> gives $1+P-3Q=0$. Its tangential velocity gives $\alpha=1+2Q$. The tangential <stress boundary condition> then yields
$$
2\mu(1+P-8Q)=\frac{2K\gamma_1}{a}\alpha,
\qquad
5(1-\alpha)=2M\alpha,
\qquad M=\frac{K\gamma_1}{\mu a}.
$$
Hence $\alpha=5/(5+2M)$, $Q=-M/(5+2M)$ and $P=-(5+5M)/(5+2M)$. The inviscid interior supplies only the constant pressure balancing $2\gamma_0/a$. Matching the remaining normal <stress> gives
$$
2\mu(1-3P+12Q)=\frac{4\gamma_0}{a}\beta-2\mu M\alpha.
$$
Eliminating $P,Q,\alpha$ gives
$$
\boxed{\mathbf D=\frac{5\mu a}{\gamma_0}\frac{2+M}{5+2M}\mathbf E,\qquad\alpha=\frac5{5+2M}.}
$$
As $M\to\infty$, $\alpha\to0$: the <Marangoni stress> suppresses tangential motion and effectively immobilizes the interface. This limit must retain the small surface <Péclet number> and small-deformation assumptions; $M$ can grow through increasing $\gamma_1$ without invalidating the linear concentration approximation.
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