When a gas bubble rises in an impure liquid, its surface often has an upper spherical cap with negligible shear stress, a lower spherical cap with negligible tangential velocity, and a very small transition region between the two caps.

This paper gives the diffusion boundary-layer theory for the distribution of surfactant around a stagnant-cap bubble, allowing for slowness of both adsorption and diffusion. The resulting singular Volterra integro-differential equations are solved numerically for creeping flow (small Reynolds number). The main result is the relation between the surface pressure of surfactant in the bulk solution, the cap angle and Péclet number of the bubble, and the adsorption depth and adsorption speed of the surfactant. The values of the latter two parameters affect the validity of the approximations much more than the numerical results.