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A sphere in an unbounded fluid

Stokes used this approach in 1851 to obtain the force needed to translate a sphere of radius $ a$ through an otherwise quiescent fluid of viscosity $ \eta$ at a constant velocity $ \vec v = v \hat z$. The flow past the sphere has the form

$\displaystyle \frac{u_\alpha (\vec r)}{v} = \frac{3}{4} \, a \, \left( \frac{\d...
... a^3 \, \left( \frac{\delta_{\alpha z}}{r^3} - \frac{3 z r_\alpha}{r^5}\right),$ (18)

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

$\displaystyle p (\vec r) = \frac{3}{2} \, \eta a \frac{\vec r \cdot \vec v}{r^3}.$ (19)

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

$\displaystyle \vec F = \gamma_0 \, \vec v,$ (20)

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

$\displaystyle \gamma_0 = 6 \pi \eta a.$ (21)

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

$\displaystyle \vec v = b_0 \, \vec F,$ (22)

where $ b_0 = 1/\gamma_0$ 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.


next up previous
Next: Faxén's Law Up: Stokeslet Analysis Previous: Stokeslet Analysis
David G. Grier 2001-01-16