Faxén's law
In fluid dynamics, Faxén's laws relate a sphere's velocity U {\displaystyle \mathbf {U} } and angular velocity Ω {\displaystyle \mathbf {\Omega } } 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 by F = 6 π μ a [ ( 1 + a 2 6 ∇ 2 ) u ′ − ( U − u ∞ ) ] , {\displaystyle \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 F {\displaystyle \mathbf {F} } is the force exerted by the fluid on the sphere μ {\displaystyle \mu } is the Newtonian viscosity of the solvent in which the sphere is placed a {\displaystyle a} is the sphere's radius U {\displaystyle \mathbf {U} } is the (translational) velocity of the sphere u ′ {\displaystyle \mathbf {u} '} is the disturbance velocity caused by the other spheres in suspension (not by the background impressed flow), evaluated at the sphere centre u ∞ {\displaystyle \mathbf {u} ^{\infty }} is the background impressed flow, evaluated at the sphere centre (set to zero in some references).