Faxén's law


In fluid dynamics, Faxén's laws relate a sphere's velocity and angular velocity to the forces, torque, stresslet and flow it experiences under low Reynolds number conditions.

First law

Faxen's first law was introduced in 1922 by Swedish physicist Hilding Faxén, who at the time was active at Uppsala University, and is given by
where
It can also be written in the form
where is the mobility.
In the case that the pressure gradient is small compared with the length scale of the sphere's diameter, and when there is no external force, the last two terms of this form may be neglected. In this case the external fluid flow simply advects the sphere.

Second law

Faxen's second law is given by
where
Batchelor and Green derived an equation for the stresslet, given by
where
Note there is no rate of strain on the sphere since the spheres are assumed to be rigid.
Faxén's law is a correction to Stokes' law for the friction on spherical objects in a viscous fluid, valid where the object moves close to a wall of the container.