Marangoni immobilization of a bubble in straining flow

For an almost spherical inviscid bubble in exterior Stokes flow $\mathbf{Ex}$, let $\gamma ={\gamma }_0-{\gamma }_1(C-C_0)$ and $K=C_0{a}^2/(2D_s)$. The quadrupolar surfactant distribution on a spherical interface makes the tangential velocity ${\mathbf{u}}_s=\alpha {\mathbf{I}}_s\mathbf{Ex}$ satisfyThe Marangoni stress approaches the stress needed to suppress surface motion as $M$ grows. The quadrupolar shape perturbation $r=a(1+\mathbf{n}\cdot \mathbf{Dn})$ isThese expressions require small deformation and small surface Péclet number; large $M$ alone does not justify a small concentration perturbation.