Interdisciplinary Applied Mathematics

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FIGURE 16.12. Comparison of boundary conditions for the two cases described in the text: case 1 (high wall density): squares; case 2 (equal wall density): circles. Plotted are velocity profiles (open symbols) and density profiles (solid symbols) across a channel of height 10rc.

16-4-3 Boundary Conditions

The example we presented in the previous section involved only periodic boundaries, and hence there was no need to specify boundary conditions. In a confined geometry, however, the effect of the no-slip or slip boundary condition with the wetted surface has to be modeled carefully. To this end, the boundary conditions that have been used in DPD are based on ideas implemented in both LBM and MD formulations.

There are three main methods to impose boundary conditions in DPD (Revenga et al., 1999):

1. The Lees-Edwards method, also used in LBM, which is a way to avoid directly modeling the physical boundary.

2. Freezing regions of the fluid to create a rigid wall or a rigid body, e.g., in particulate microflows.

3. Combining freezing with proper reflections, namely, specular reflection or bounce-back reflection, or Maxwellian reflection.

Let us first consider how to implement the Lees-Edwards (Lees and Edwards, 1972; Wagner and Pagonabarraga, 2002) boundary conditions for a shear flow    (e.g.,    Couette    flow)    with    the    upper    wall    moving with    velocity

-Ux/2 and    the    lower    wall    moving    at    Ux/2.    Let    us assume that    we    have

a system of particles with positions (rx,ry,rz) and velocities (vx,vy,vz) within a box of dimensions (Lx,Ly ,Lz). We can describe the boundary conditions by providing the new positions r’ and velocities v’ of the particles after the particles have been moved, as follows:

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