Faxen's Law Proof Been working on this for a couple weeks and have had no dice. Does anyone know of a nice way to prove Faxen's law (force on a sphere in a fluid flow)? Particularly using the reciprocal theorem? Here is the link to what Faxen's law is: https://en.wikipedia.org/wiki/Fax%C3%A9n%27s_law
In particular the first law:
$$\mathbf{F} = 6 \pi \mu a \left[ \left( 1 + \frac{a^2}{6} \nabla^2\right) \mathbf{u}' - (\mathbf{U} - \mathbf{u}^\infty) \right]$$
So far everything I have researched seems very complicated involving an extremely long employment of stokeslets and stresslets. Any guidance would be great.
 A: The most straightforward development I've seen of the Faxén laws, at least for the first one you mention, is in Kim and Karrila's book which precisely uses the Lorentz reciprocal theorem.
Although I won't reproduce it exactly, I'll mention the key conceptual aspects and you can take it from there:


*

*Consider the general form of the Lorentz reciprocal theorem: $$\oint_S \vec{v}_1\cdot\left(\bar{\bar{\sigma}}_2\cdot\vec{n}\right)dS\,-\int_V \vec{v}_1\cdot\left(\nabla \cdot\bar{\bar{\sigma}}_2\right)dV = \oint_S \vec{v}_2\cdot\left(\bar{\bar{\sigma}}_1\cdot\vec{n}\right)dS\,-\int_V \vec{v}_2\cdot\left(\nabla \cdot\bar{\bar{\sigma}}_1\right)dV$$ where $\vec{v}_i$ represents the fluid velocities and $\bar{\bar{\sigma}}_i$ represents the fluid stress tensor for the $i$th solution of a given Stokes flow problem.

*Take your $\vec{v}_1$ and $\bar{\bar{\sigma}}_1$ to represent the solution for a spherical particle translating with constant velocity $\vec{U}$ in a quiescent infinite fluid. Note $\nabla\cdot\bar{\bar{\sigma}}_1 = \vec{0}$ everywhere in the fluid.

*Take your $\vec{v}_2$ and $\bar{\bar{\sigma}}_2$ to represent the solution for a stationary spherical particle embedded in a fluid with a Stokeslet present at some position $\vec{y}$ not inside the spherical particle. Note $\vec{v}_2 = \vec{0}$ everywhere on $S$.

*Simplify the expression above by using the singularity solutions for a sphere, in particular for $\vec{v}_1$, in order to pull out the first Faxén law for a sphere.


The derivation of the other Faxén laws follow from similar constructions when one considers the singularity solutions for the object under consideration (in your case, a sphere).
Hope this helps!
