Interdisciplinary Applied Mathematics

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(a)    (b)

FIGURE 14.15. Comparison between the boundary cloud method (BCM) and the finite element method (FEM) for the ж-velocity profiles (a) along line AB; (b) at the outflow.

14.3 Particulate Microflows

Here we consider numerical approaches for simulating particulate microflows in the applications described in chapter 1 (see Figures 1.5, 1.6) and also in self-assembly applications; see Section 13.1. There are a number of approaches in dealing with particulate flows based on direct but computationally expensive methods such as the front tracking technique of (Tryggvason et al., 1998); the ALE (Arbitrary Lagrangian Eulerian) method, see Section 14.1.4 and (Johnson and Tezduyar, 1996); and the fictitious domain method

(DLM) (Glowinsky et al., 1999). A mesoscopic method based on the lattice— Boltzmann equation has been developed by (Ladd, 1994a; Ladd, 1994b); we discuss    this method in    some    detail    in    Section    15.5.    Also,    a    stochas

tic molecular dynamics approach, the dissipative particle dynamics (DPD) method, can be used, especially for dense systems; See section 16.4. At low Reynolds numbers, the Stokesian dynamics approach has been developed by (Brady and Bossis, 1988), which deals effectively with the lubrication forces in particle-particle and particle-wall interactions.

The complexity of dynamics of a particulate flow depends on the volume fraction occupied by the particles relative to the total volume, defined as

_ 12iNiVPti
a V ’

where V is    the    total volume,    Vp>i    is    the    particle    volume, and    Ni,    is    the

number of    particles    present    in volume    Vp>i.    For    a <    10~3    the    particulate

flow is dilute, but for a > 10~3 strong fluid-dynamic interactions and particle collisions occur.

To model particulate microflows accurately, the full Navier-Stokes equations need to be employed; otherwise, important phenomena such as the wake behind the particles are inaccurately represented. The wake influences the history of the particle’s trajectory. Additionally, convective inertia influences the vorticity transport, resulting in the faster decay of the so-called Basset history force. In the following we first review some classical results on hydrodynamic interactions between spheres, and subsequently we focus on the force coupling method, a particularly fast and easy to implement modeling approach for particulate flows.

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