Stokes used this approach in 1851 to obtain the force needed to translate a sphere of radius through an otherwise quiescent fluid of viscosity at a constant velocity . The flow past the sphere has the form

where is the distance from the sphere's center and the displacement along the direction of motion. The flow's contribution to the pressure is

Substituting these into Eqs. (16) and (17) yields

for the drag, where the sphere's drag coefficient is

The same drag coefficient parameterizes the force needed to hold the sphere stationary in a uniform fluid flow . Conversely, a constant force applied to the sphere causes it to attain a steady-state velocity

where is the sphere's mobility.

Similar calculations for more complicated systems can be acutely difficult; few have analytic solutions. For this reason, a great many highly specialized approximation schemes have been devised for hydrodynamic problems. Progress for many-body systems such as colloidal suspensions has been steady, but slow.