# Interdisciplinary Applied Mathematics

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Specifically, the trajectories of the two particles show a general agreement with the experimental results, but the numerical results, obtained with the monopole term show only erroneous overlap between the two particles. On the other hand, the inclusion of the dipole term leads to particles that touch but do not overlap; similar conclusions were reached in the simulations presented in (Liu et al., 2002). Including force dipoles helps to represent the particle-particle interactions in the flow more accurately, but they are not adequate on their own. Lubrication forces still play a role in low Reynolds number flows. At finite Reynolds numbers collisions do occur, and contact forces need to be explicitly included, e.g., elastic forces that cause particles to bounce. To this end, simple models based on the interparticle distance to mimic the elastic collision effect can be employed, as we show next.

The component of the repulsion force on particle j from particle к in the xi direction is computed as:

(14.16)

x — x*l f (1.05 x 2a)2 — r2 » Cl 2a V C2 x 4a2    )

where Ci = 1.02 and C2 = 0.011 are two constants to adjust the strength of the force; a is the particle radius; xj — xk is the difference in the xcoordinates of particles j and к; and r is the distance between the two particles. The exponent n can take the value 2 or 3 for weak or strong repulsions, respectively. The collision force works in pairs, i.e., Fjk = — Fkj. The cut-off distance is 5% of the diameter away from the particle surface. In other words, when two particles are within distance 1.05 x 2a, the repulsion force will be automatically activated on each particle to push them away. For example, when r > 1.05 x 2a, this repulsive force will remain 0. When r = 1.025    x    2a,    it    is about 6 times    the    Stokes drag    force. When r =    2a,    it

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