Interdisciplinary Applied Mathematics

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For a dense particulate microflow where the size of the particle becomes comparable with the characteristic length of the domain, the above description may be of very limited use. On the other hand, computing in great detail all the pair collisions is a computationally formidable task (Johnson and Tezduyar, 1996). To this end, we describe next the force coupling method (FCM), which was developed by Maxey and his students (Maxey et al., 1997; Maxey and Dent, 1998; Maxey and Patel, 2001). It was used and validated for particulate microflows with great success in (Lomholt, 2000; Liu et al., 2002). In FCM, the same stationary mesh is used throughout the simulation, and in combination with the spectral element method we described earlier it gives very accurate results (Liu, 2004). A comparison of the overall accuracy of FCM for an array of spheres is shown in Figure 14.16. The comparison is against a direct numerical simulation using the spectral element method in computations performed by (Dent, 1999). A very good agreement is achieved even at relatively high Rep and this holds for quite dense particulate flows, e.g., up to 20% concentration. Typically, in particulate microflows the particle Reynolds number is small, e.g.,

Rep < 10.

The basic idea of the FCM is to model the disturbance flow via a calibrated multipole expansion modifying the Navier-Stokes equation. The particles are    virtual,    and    the    slip    or    no-slip    boundary    conditions    on    their

surface is only approximately satisfied. This alleviates the severe numerical stiffness of the problem associated with exact boundary condition enforcement on many moving surfaces. The modified Navier-Stokes equations are then

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