# Interdisciplinary Applied Mathematics

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lim fe (h) ^ 1, lim fe(h) ^ 1.55.

The relative motion of two spheres with the velocity ±V perpendicular to their    line of    centers    was    solved    by    the    multiple    reflection    method ana

lytically (O’Neill and Majumdar, 1970). A rational function approximation of the drag force for this case is given by

Fd

f (h)

6npVa

~1W

МЩк/а)2+ 4815 (h/a) + 67 3433(/i/a)2 + 2591 (h/a) + 31 ’ where h is the separation distance between the two spheres. In the asymptotic limits of h ^ 0 and h ^<x>, the above relation gives

lim fe(h) ^ 1,    lim fe(h) ^ 2.17.

h——ж    h—— 0

Collective motion of two spheres with the same velocity V perpendicular to their line of centers was obtained by (O’Neill and Majumdar, 1970). A rational function approximation of drag force for this case is given by

Fd

f(h)

6npVa

f(h) ’

-Щк/а)3 + 8275(h/a)2 + 14720 (h/a) + 45 -31 (h/a)3 + 8252 (h/a)2 + 20843(/i/a) + 62’ where h is the separation distance between the two spheres. In the asymptotic limits the above relation gives

lim fe(h) ^ 1,    lim fe(h) ^ 0.725.

h—— ж    h——0

The rational function approximations for the drag force, presented above, are valid for very specific particle/surface configurations. In 1976, Batchelor combined all of these two-body hydrodynamic problems and specified particle interactions in a tensor form. This related the velocity and force field vectors via the mobility tensor (Batchelor, 1976). This treatment enabled development of Stokesian dynamics algorithms (Brady and Bossis, 1988).

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