# Interdisciplinary Applied Mathematics

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Ct

33.932

1.236

5.064

33.929

1.236

5.066

33.929

1.236

5.066

35.122

1.274

4.872

35.175

1.274

5.251

35.306

1.276

5.251

33.475

1.172

5.236

33.533

1.211

5.157

33.724

1.210

5.175

TABLE 14.4. Drag coefficient (Cd = i Рг,аД ■, where Um is the maximum velocity

2PfUmd

at the    center    line    of    the    channel    and d    is the    cylinder    diameter),    lift    coefficient

(C’l = j_^h/f2 rf), and the torque coefficient (Ct = i    ) on the sphere near

one wall in a 3D channel.

FIGURE 14.19. Relative distance between the centers of two particles in a microchannel. The circles denote experimental results, the solid line FCM predictions with monopole and dipole terms, and the dotted line FCM predictions with monopole terms only. (Courtesy of S. Lomholt.)

t

vertical grid lines, the torque will be overpredicted on the grid 96 x 84 x 96 (with a value 5.37).

In Section 14.3.1 we have presented several semianalytical results that can be used to analyze the hydrodynamic interactions between the particles as they approach each other, i.e., in the lubrication limit. The effect of lubrication is not explicitly included in FCM, and it has to be added via separate models (Dance and Maxey, 2003). For exact representation of both the lubrication effect and the far-field interaction, an infinite number of multipoles is required, which will make the method inefficient. However, it is interesting to note the effect of the dipole term in this context. Lomholt (2000) has studied this effect by considering the dynamics of the two particles when they touch, using FCM simulations with both the monopole and the dipole terms. A typical result is shown in Figure 14.19, where we see    that    the inclusion    of    the    dipole    contribution    improves    the    results.

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