# Interdisciplinary Applied Mathematics

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perature); and A = N-1/d (the average distance between particles, where d is the space dimension). Based on these we can define the dimensionless friction coefficient

_ yA dvT

where vt = /к^Т/то is the thermal velocity. Large values of r mean that    the    particles    move    very    little in the    time scale    associated    with    the velocity decaying due to thermal fluctuations. Also, we define a measure of overlapping between particles by

r

A’

which represents the number of particles that interact within the cutoff radius rc. These two nondimensional parameters define the dynamics of the DPD fluid system. For example, the kinematic viscosity can be computed in terms of these two parameters following a kinetic theory description, i.e.

vT

v

1

— + — s t t 40

Espanol and other researchers have used the plane Couette flow to evaluate the above boundary conditions. The Lees-Edwards boundary conditions work well for this model, as we discussed above, but the objective is to see what type of reflections are appropriate with the freeze-and-reflect approach. In    (Revenga et    al.,    1999),    it    was    shown    that    for    large    values    of

т all three reflections satisfy the no-slip condition. However, for small values of т the specular and Maxwellian reflections produce an excessive slip velocity at the wall, while the bounce-back approach satisfies the no-slip condition. An anomaly, however, has been presented for the temperature profile very close to the wall at small values of t even with the bounce-back boundary conditions. This was explained as large axial velocity dispersion for small values of т. For т « 0 the particles move along almost straight lines, while the bounce-back condition causes a continuous acceleration of the particles after each collision with the wall. However, a closer inspection of the definition employed for the temperature contributions in (Revenga et al., 1999), shows that this behavior for small values of т is not really an anomaly!

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