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Stokeslet Analysis

The local velocity $ \vec u$ in a slowly flowing incompressible viscous fluid is described by the Stokes equation

$\displaystyle \eta \nabla^2 \vec u = \vec \nabla p.$ (14)

where $ \eta$ is the fluid's viscosity, and $ p$ is the local pressure. In the absence of sources or sinks,

$\displaystyle \vec \nabla \cdot \vec u = 0$ (15)

completes the flow's description. Eq. (14) is a good approximation for flows at low Reynolds numbers for which viscous damping dominates inertial effects - typical conditions for small collections of diffusing spheres.

Flows vanish on solid surfaces thus setting boundary conditions for solutions to Eqs. (14) and (15). Once these boundary conditions are satisfied, we can calculate the viscous drag on a particle by integrating the pressure tensor

$\displaystyle \Pi_\ensuremath{{\alpha\beta}}= - p \, \delta_\ensuremath{{\alpha\beta}}+ \eta \, \left( \nabla_\alpha u_\beta + \nabla_\beta u_\alpha \right)$ (16)

over its surface:

$\displaystyle F_\alpha = \int_S \Pi_\ensuremath{{\alpha\beta}}\, dS_\beta.$ (17)



Subsections
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Next: A sphere in an Up: Hydrodynamic Interactions Previous: Hydrodynamic Interactions
David G. Grier 2001-01-16