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The mobility tensor

This observation immediately suggests a simplified description of colloidal hydrodynamic coupling. Consider a sphere labelled $ j$, located at $ \vec r_j$ within a suspension. A force $ \vec f (\vec r_j)$ applied to this sphere excites a flow $ \vec u(\vec r) = \ensuremath{\mathsf{H}}\xspace (\vec r - \vec r_j) \,
\vec f (\vec r_j) $ at displacement $ \vec r$. This linear relationship is guaranteed by the Stokes equation's linearity. $ \mathsf{H}$ is known as the Oseen tensor and is obtained from Eqs. (18), (20) and (21). At large enough distances, $ \vec u$ appears sufficiently uniform that a sphere at $ \vec r_i$ is simply advected according to Faxén's law:

$\displaystyle v_\alpha(\vec r_i)$ $\displaystyle = u_\alpha(\vec r_i)$ (24)
  $\displaystyle = \ensuremath{\mathsf{b}}\xspace _\ensuremath{{i \alpha, j \beta}}\, f_\beta(\vec r_j),$ (25)

where $ \ensuremath{\mathsf{b}}\xspace _\ensuremath{{i \alpha, j \beta}}$ is a component of a mobility tensor describing sphere $ i$'s motion in the $ \alpha$ direction due to a force applied to sphere $ j$ in the $ \beta$ direction. More to the point, $ \ensuremath{\mathsf{b}}\xspace _\ensuremath{{i \alpha, j \beta}}= \ensuremath{\mathsf{H}}\xspace _\ensuremath{{\alpha\beta}}(\vec r_i - \vec r_j)$. The diagonal elements, $ \ensuremath{\mathsf{b}}\xspace _\ensuremath{{i \alpha, i \alpha}}= b_0$, describe the sphere's own response to an external force.

Any number of sources may contribute to the flow past sphere $ i$, with each contributing linearly in the Stokes approximation. The mobility tensor therefore may be factored into a self-mobility and a mobility, $ \ensuremath{\mathsf{b}}\xspace ^e$, due to all external sources,

$\displaystyle \ensuremath{\mathsf{b}}\xspace _\ensuremath{{i \alpha, j \beta}}=...
...pha\beta}}+ \ensuremath{\mathsf{b}}\xspace ^e_\ensuremath{{i \alpha, j \beta}}.$ (26)

The results in this section all follow from this equation, once the necessary flow fields have been calculated. In particular, we will consider contributions to a sphere's mobility from (1) neighboring spheres and (2) bounding planar surfaces.


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