# Interdisciplinary Applied Mathematics

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order multipole methods or other accurate flow simulation methods, the numerical resolution of the gap becomes insufficient to calculate these forces accurately, and in general, some approximation of these effects must be included with the simulations. The lubrication forces are usually included through a summation of discrete pairwise interaction between particles or a particle and a wall, with the forces determined by the particle positions and relative velocity. This has been done for the force coupling method (FCM) in Stokes flows; see (Dance and Maxey, 2003). There are limitations to the accuracy of summing pairwise interactions, and this is evident, for example, for a particle in a narrow channel, where both walls simultaneously can influence the particle motion. A simple model that can be used to prevent overlapping of particles is a velocity repulsion barrier. It can be activated only when particles are closer than a cutoff distance Rref on the order of 20% of the radius a. This repulsive velocity is given by

V

lj

thef 2 a

r2    _ r2

Ref — 4a2

2

viji

where rij is the distance connecting the particles i and j (see sketch in Figure 13.1). Calibration of the relative motion of two or three particles can help in determining the velocity scale vref.

As the separation distance between particles is reduced further, particle-particle forces    will    act.    In    the    context    of    colloids    the    most    relevant    is    the

short-range electrostatic repulsion between particles due to surface charges.

The effects of these charges that are naturally occurring in most systems is generally screened by ions in the suspending liquid phase, and their influence is confined to a thin Debye layer. Their action on particle motion is effectively approximated by a Derjaguin potential; see (Russel et al., 1989).

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