Basset force

Consider an infinitely large plate started impulsively with a step change in velocity—from 0 to u₀—in the direction of the plate–fluid interface plane.

The equation of motion for the fluid—Stokes flow at low Reynolds number—is

${\displaystyle {\frac {\partial u}{\partial t}}=\nu _{c}\,{\frac {\partial ^{2}u}{\partial y^{2}}},}$

where u(y,t) is the velocity of the fluid, at some time t, parallel to the plate, at a distance y from the plate, and v_c is the kinematic viscosity of the fluid (c~continuous phase). The solution to this equation is,[5]

${\displaystyle u=u_{0}-u_{0}\,\operatorname {erf} \left({\frac {y}{2{\sqrt {\nu _{c}t}}}}\right)=u_{0}\operatorname {erfc} \left({\frac {y}{2{\sqrt {\nu _{c}t}}}}\right),}$

where erf and erfc denote the error function and the complementary error function, respectively.

Assuming that an acceleration of the plate can be broken up into a series of such step changes in the velocity, it can be shown that the cumulative effect on the shear stress on the plate is

${\displaystyle \tau ={\sqrt {\frac {\rho _{c}\mu _{c}}{\pi }}}\int \limits _{0}^{t}{\frac {\frac {\partial u_{p}}{\partial t'}}{\sqrt {t-t'}}}\,dt',}$

where u_p(t) is the velocity of the plate, ρ_c is the mass density of the fluid, and μ_c is the viscosity of the fluid.

Boussinesq (1885) and Basset (1888) found that the force F on an accelerating spherical particle in a viscous fluid is[3][4][6][7]

${\displaystyle \mathbf {F} ={\frac {3}{2}}D^{2}{\sqrt {\pi \rho _{c}\mu _{c}}}\int \limits _{0}^{t}{\frac {\mathbf {v} \cdot \mathbf {\nabla } \mathbf {u} +{\frac {\partial \mathbf {u} }{\partial t'}}-{\frac {\partial \mathbf {v} }{\partial t'}}}{\sqrt {t-t'}}}dt',}$

where D is the particle diameter, and u and v are the fluid and particle velocity vectors, respectively.

• Basset–Boussinesq–Oseen equation
• Stokes boundary layer