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 (creeping flow) conditions.
First law Faxén's first law was introduced in 1922 by Swedish physicist Hilding Faxén, who at the time was active at Uppsala University, and is given byDurlofsky, Louis, John F. Brady, and Georges Bossis. "Dynamic simulation of hydrodynamically interacting particles." Journal of fluid mechanics 180.1 (1987): 21–49 , equations (2.15a, b, c). Note sign change.
where is the force exerted by the fluid on the sphere is the Newtonian viscosity of the solvent in which the sphere is placed is the sphere's radius is the (translational) velocity of the sphere is the disturbance velocity caused by the other spheres in suspension (not by the background impressed flow), evaluated at the sphere centre is the background impressed flow, evaluated at the sphere centre (set to zero in some references).
It can also be written in the form
where is the hydrodynamic 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 Faxén's second law is given by
where is the torque exerted by the fluid on the sphere is the angular velocity of the sphere is the angular velocity of the background flow, evaluated at the sphere centre (set to zero in some references).
'Third law' George Batchelor and J. T. Green derived an equation for the stresslet, given by
where is the stresslet (symmetric part of the first moment of force) exerted by the fluid on the sphere, is the velocity gradient tensor; represents transpose; and so is the rate of strain, or deformation, tensor. is the rate of strain of the background flow, evaluated at the sphere centre (set to zero in some references).
Note there is no rate of strain on the sphere (no ) 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.Single molecule measurements and biological motors - Glossary , accessed on May 12, 2009
See also Immersed boundary method
Notes
References