Surface traction in translating-sphere Stokes flow

On the sphere, pressure and the normal part of the viscous stress cancel, leaving uniform traction $-3\mu U/(2a)$.

Integrating the translating-sphere traction gives the drag force $F=-6\pi \mu aU$.

A spherical clean bubble with negligible internal dynamic viscosity has zero tangential traction and permits tangential slip. In the creeping-flow, negligible-deformation limit its drag is $-4\pi \mu a\mathbf{U}$, versus $-6\pi \mu a\mathbf{U}$ for a no-slip rigid sphere.

The Stokes–Einstein relation connects translational diffusion and linear drag through $D=k_{B}T/\zeta$. For an unbounded spherical particle, $\zeta =6\pi \mu a$ and $D=k_{B}T/(6\pi \mu a)$.