Next: Electrohydrodynamic Coupling
Up: Hydrodynamic Interactions
Previous: Hydrodynamic coupling to a
Pair diffusion near a wall
Figure 7:
Measured inplane pair diffusion coefficients for two
1
diameter silica spheres
in water at
at height
above a glass wall.
Dashed curves result from linear superposition of drag
coefficients, while solid curves result from
stokeslet analysis.

In his 1927 treatise, Oseen suggested that Faxén's
results might be applied to more complicated systems
even if his methods could not [63].
Oseen pointed out that
the drag on a sphere moving in direction near a wall can be
factored into the sphere's drag in an unbounded
system and an additional contribution
due to the wall:

(40) 
He proposed that the drag coefficient for motion
in a given direction
might be approximated by linearly superposing
individual contributions
from all bounding surfaces
and neighboring particles:

(41) 
Oseen emphasized that this
cannot be rigorously correct because it violates
boundary conditions on all surfaces.
Even so, if the surfaces are wellseparated, the
errors may be acceptably small.
Given this hope,
Oseen's linear superposition approximation has
been widely adopted.
The data in Fig. 7 show measured inplane
pair diffusion coefficients for 1
diameter
spheres positioned by
optical tweezers at
above a
glass wall.
Naively adding the drag coefficients [58]
due to spheresphere
and spherewall interactions yields
,
whose predictions appear as
dashed curves in Fig. 7 and
agree poorly with measured diffusivities.
Figure 8:
Left: Hydrodynamic interactions for two spheres near
a planar boundary. Right: Normal modes of motion obtained
from stokeslet analysis of this system [64].

A more complete treatment not only resolves these quantitative
discrepancies but also reveals an additional
influence of the bounding surface: the
highly symmetric and experimentally accessible modes parallel to
the wall are no longer independent.
As shown in Fig. 8,
each sphere interacts with its own image, its neighbor, and its
neighbor's image.
These influences contribute
to the mobility of sphere in the direction.
Eigenvectors of the corresponding diffusivity tensor appear in
Fig. 8.
The independent modes of motion are rotated with respect to the
bounding wall by an amount which depends strongly on both
and .
Even though the experimentally measured inplane motions
are not independent, they still
satisfy Eq. (32)
with pairdiffusion coefficients
,
where the positive sign corresponds to collective motion, the negative to
relative motion, and indicates directions either perpendicular
or parallel to the line connecting the spheres' centers.
Explicitly, we obtain [64]
up to
and
, where
.
These results appear as solid curves in Figs. 6
and 7.
Not only does the stokeslet approximation perform better than
linear superposition for spheres close to a wall, it performs
equally well at all separations we have examined [64].
Next: Electrohydrodynamic Coupling
Up: Hydrodynamic Interactions
Previous: Hydrodynamic coupling to a
David G. Grier
20010116