Surfactant transport with exchange relaxation ($D_{t}C=-C{\nabla }_s\cdot {\mathbf{u}}_s-C{u}_n\kappa +D_s{\Delta }_{s}C-k(C-C_0)$)

The material derivative of surface concentration combines tangential and normal area change, surface diffusion, and relaxation towards a reservoir concentration $C_0$. The rate $k$ represents exchange; setting $k=0$ recovers transport without exchange. The concentration is measured per interfacial area, so stretching dilutes it.

A spherical inviscid bubble moving slowly through a viscous liquid has a dipolar surfactant perturbation. The associated Marangoni stress reduces the tangential velocity from its clean-bubble value towards zero. In the linear regime, with ${\mathbf{u}}_s=A(I-\mathbf{nn})\mathbf{U}$, the consistent Newtonian fluid stress tensor gives $A=-1/[2(1+2\lambda )]$.

The dipolar normal traction is balanced by buoyant hydrostatic pressure and the concentration-dependent normal capillary stress, in addition to the constant gas pressure and equilibrium Laplace pressure. A degree-one shape perturbation translates a sphere rather than deforming it; the balance fixes the terminal speed. A uniform gas pressure alone cannot balance this mode.

Linear steady transport on a translating sphere gives $B=2A{C}_{0}a/(k{a}^2+2D_s)$. The components of its unit normal vector are degree-one spherical harmonics. The condition $2|A|Ua/(k{a}^2+2D_s)\ll 1$ ensures the concentration perturbation is small relative to $C_0$.