Past exam of the advanced fluid dynamics course of the University of Oxford / 2018 / 2 / b

Write the supplied velocity in the radial formAt $r=a$, $f=1$ and $g=0$, giving the no-slip boundary condition $\mathbf{u}=\mathbf{U}$. At infinity $\mathbf{u}\to 0$ and the supplied gauge-fixed pressure tends to zero. These are the laboratory-frame conditions for a sphere translating through otherwise quiescent fluid.

The exterior Stokes equations can also be checked directly:andTo compute surface traction in translating-sphere Stokes flow, let $\mathbf{n}=\mathbf{x}/a$ point from the sphere into the fluid. At the surface, $f^{\prime }(a)=-3/(2a)$, $g^{\prime }(a)=3/(2a^3)$ and $g(a)=0$. HenceSubstitute into the Newtonian fluid stress tensor. Its viscous contribution to the surface traction is $3\mu [-\mathbf{U}+(\mathbf{U}\cdot \mathbf{n})\mathbf{n}]/(2a)$, so the normal part cancels the pressure contribution:The vector is constant over the entire sphere in the stated pressure gauge. Integrating it gives the fluid's drag force on the sphere,This is Stokes's law. On an exterior fluid domain the outward domain normal at its inner boundary is $-\mathbf{n}$; that distinction will matter in the reciprocal calculation.