# Interdisciplinary Applied Mathematics

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(r x + dx)

mod Lx, ry ^ Ly,

0    ^ ry < Ly ,

rX = S rx    mod Lx,

_ (rx — dx)    mod Lx, ry < 0,

r’y = ry    mod Ly,

r’z = rz    mod Lz,

v

x

v

y

(vx + Ux), ry ^ Ly,

* vx,    0 ^ ry ^ Ly,

(vx Ux^ ry < 0,

vy and v’z = vz.

Here dx = UxA is the time-dependent offeset. Therefore, we avoid providing explicitly any new boundary conditions in this case.

Next, we examine how to incorporate reflections at the wall. We have already described the aforementioned types of reflection earlier in this book, but we briefly review them here as well. In specular reflections the velocity component that is tangential to the wall does not change, while the normal component is reversed. In the bounce-back reflection both components are reversed. A Maxwelian reflection involves particles that are introduced back into the flow with a velocity following a Maxwelian distribution centered around the wall velocity.

In the continuum limit, it is interesting to investigate which one of these boundary conditions honors the no-slip condition. We focus the discussion on the    third    boundary    condition    from    the    above list,    since it is    the    most

general approach. In order to parameterize the DPD flow system, following the work of Espanol and his group (Revenga et al., 1999), we identify the key nondimensional parameter that affects the slip velocity. Specifically, there are five governing parameters in the DPD fluid system: m (the mass of particles); 7    (the    friction    coefficient);    rc    (the    cutoff    radius);    kBT    (tem

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