# Interdisciplinary Applied Mathematics

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We now discuss hydrodynamic interactions between two particles in Stokes flow, which generally scale as the inverse of the distance between their centers. First, we consider a particle interacting with a wall, and subsequently we will present results for particle-particle interactions.

An analytical solution for drag force on a sphere moving perpendicular to a    plane    wall    was    obtained    by    (Goldman    et    al.,    1967)    in    the    form    of

an infinite series solution. Using regression techniques, (Bevan and Prieve, 2000) developed the following rational function approximation:

Fd

f (h)

6npV a

~JW’

6(h/a)2 + 2 (h/a)

6 (h/a)2 + 9 (h/a) + 2’

where a is the particle radius, h is the particle-surface separation distance, and V is the    particle    velocity.    In    the    asymptotic    limits    the    above    relation

gives

lim fe(h) ^ 1,    lim fe(h) ^ h/a.

h——ж    h——0

An analytical solution for drag force on a sphere moving parallel to a plane wall was obtained in (O’Neill and Majumdar, 1970). A rational function approximation for this solution is given by

Fd

f(h)

6npV a

~1W

368 (h/a)3 + 559 (h/a)2 + 81 (h/a) 368(h/a)3+ 779(h/a)2+ 250(h/a)’

In the asymptotic limits the above relation gives

lim fe(h) ^ 1,    lim fe(h) ^ 0.324.

h——ж    h——0

Hydrodynamic interaction of a particle near a flat surface also affects its diffusion characteristics. The particle diffusion in the translational direction

FIGURE 14.9. Distribution of random points and typical clouds in the meshless method.

(parallel) to the wall is given as

D(h)

g(h)