Faxén's law

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 : \mathbf{F} = 6 \pi \mu a \left[ \left( 1 + \frac{a^2}{6} \nabla^2\right) \mathbf{u}' - (\mathbf{U} - \mathbf{u}^\infty) \right], where

• \mathbf{F} is the force exerted by the fluid on the sphere
• \mu is the Newtonian viscosity of the solvent in which the sphere is placed
• a is the sphere's radius
• \mathbf{U} is the (translational) velocity of the sphere
• \mathbf{u}' is the disturbance velocity caused by the other spheres in suspension (not by the background impressed flow), evaluated at the sphere centre
• \mathbf{u}^\infty is the background impressed flow, evaluated at the sphere centre (set to zero in some references).

It can also be written in the form : \mathbf{U} - \mathbf{u}^\infty = \mathbf{u}' + b_0 \mathbf{F} + \frac{a^2}{6} \nabla^2 \mathbf{u}', where b_0 = -\frac{1}{6 \pi \mu a} 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

Faxen's second law is given by : \mathbf{T} = 8 \pi \mu a^3 \left[ \frac{1}{2} \left( \boldsymbol{\nabla} \times \mathbf{u}' \right) - (\mathbf{\Omega} - \mathbf{\Omega}^\infty) \right], where

• \mathbf{T} is the torque exerted by the fluid on the sphere
• \mathbf{\Omega} is the angular velocity of the sphere
• \mathbf{\Omega}^\infty is the angular velocity of the background flow, evaluated at the sphere centre (set to zero in some references).

'Third law'

Batchelor and Green derived an equation for the stresslet, given by : \boldsymbol{\mathsf{S}} = \frac{10}{3} \pi \mu a^3 \left[2\boldsymbol{\mathsf{E}}^\infty + \left(1 + \frac{1}{10} a^2 \nabla^2\right) \left(\boldsymbol{\nabla} \mathbf{u}' + (\boldsymbol{\nabla} \mathbf{u}')^\mathrm{T}\right)\right], where

• \boldsymbol{\mathsf{S}} is the stresslet (symmetric part of the first moment of force) exerted by the fluid on the sphere,
• \boldsymbol{\nabla}\mathbf{u}' is the velocity gradient tensor; {}^\mathrm{T} represents transpose; and so \frac{1}{2}\left[\boldsymbol{\nabla} \mathbf{u}' + (\boldsymbol{\nabla} \mathbf{u}')^\mathrm{T}\right] is the rate of strain, or deformation, tensor.
• \boldsymbol{\mathsf{E}}^\infty = \frac{1}{2}\left[\boldsymbol{\nabla} \mathbf{u}^\infty + (\boldsymbol{\nabla} \mathbf{u}^\infty)^\mathrm{T}\right] 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 \boldsymbol{\mathsf{E}}) 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.

Notes

References

References

1. (2000). "Faxen's laws of a composite sphere under creeping flow conditions". Journal of Colloid and Interface Science.
2. Durlofsky, Louis, John F. Brady, and Georges Bossis. "Dynamic simulation of hydrodynamically interacting particles." Journal of fluid mechanics 180.1 (1987): 21–49 {{doi. 10.1017/S002211208700171X, equations (2.15a, b, c). Note sign change.
3. (1972). "The hydrodynamic interaction of two small freely-moving spheres in a linear flow field". J. Fluid Mech..
4. [http://www2.bioch.ox.ac.uk/~oubsu/ebjknight/gloss.html Single molecule measurements and biological motors - Glossary] {{webarchive. link. (2007-09-03 , accessed on May 12, 2009)