Interdisciplinary Applied Mathematics

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Various methods are available for flow simulations. Specifically, we discuss the continuum-based ALE, DLM, and FCM schemes in Chapter 14. These are flexible methods suitable for particulate flows both in low Reynolds number and zero Reynolds number flows. In addition lattice-Boltzmann

simulations (LBM), see Section 15.5, have been developed as a general simulation approach for dispersed two-phase flows as well as dissipative particle dynamics methods; see Section 16.4. For Stokes flows there are several methods based on multipole expansions; see (Kim and Karrila, 1991). The Stokesian dynamics method of (Brady and Bossis, 1988) uses a low-order multipole representation to construct a mobility matrix for the particles. This is supplemented with localized resistance forces and lubrication forces when particles are close to each other or near a wall. These simulations have been used to investigate dense suspensions and the rheology of particles in shear flows. New developments of the method (Ichiki and Brady, 2001) have extended the range of application to larger systems of particles and to possibly higher multipole representations. These methods provide a range of techniques to determine flow characteristics and the force and torques acting on the particles in response to the flow or their motion through the fluid.

At low Reynolds numbers, special attention is needed to determine the motion of    particles that    are close    to each    other    or    near a    wall.    As    the

separation distance is reduced, very large forces can be generated as fluid is displaced from the narrow gap between particles or from shearing motion across the gap. These are characterized by lubrication forces and have been calculated accurately for a number of standard configurations; see section 14.3.2 and    (Jeffrey,    1982;    Kim and    Karrila,    1991).    Even    if    one    uses high-

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