Interdisciplinary Applied Mathematics

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equals 43.8 x 6nap. It is much stronger when particles start going into each other. Figure 14.20 shows the magnitude of the repulsion force versus the distance between two particles for n = 2 and n = 3.


A collision model of the same nature is developed for particles bumping

FIGURE 14.20. Normalized elastic collision force versus normalized distance.


into a wall at an angle. Here the force is no longer pairwise, but instead it acts on the particle from the wall. This force is computed as


Fi = C



(1.05a)2 — r2

CW




where Y is    the    coordinate    of    the    contact    point    at    a    wall,    and r is    the


distance from the center of particle j to the contact point. The same cutoff distance is    used here.    The    contact    point is    found    by    a loop    over    the


grid points on    the wall    and picking    the    point    closest to    the    center    of    the


particle.


Nonspherical Particles


The FCM as described above is valid for spherical nondeformed particles. However, extensions to ellipsoid particles are straightforward by setting the two axes of the ellipsoid thus:


at = рк(7 and a-i =


In this case the Gaussian envelope is ellipsoidal with extent defined by a1 and <72. The expressions above are obtained by matching the settling velocities in FCM with the analytically obtained settling velocities for ellipsoid particles. The additional computational complexity is that equations of angular momentum have to be solved for each particle in order to obtain the required orientation information. Moreover, arbitrary shapes, e.g., particle chains or clusters, can be modeled as an assembly of ellipsoids with appropriate linkages and constraints in their motion. The important point is that the same governing equations for the microflow are valid, i.e., equation (14.15), and only the particle dynamics have to be modeled properly. Details of the implementation and validation results are presented in (Liu, 2004).

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