Interdisciplinary Applied Mathematics

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Fi



1



m    mNj



Д/2


tMD


Nj


EFi —


i=1



1



Nj



AtMD



Nj



i=1


The motion equations employed in the relaxation method in (O’Connell and Thompson, 1995), are similar to the equations (16.27) but with the last two terms on the right-hand side premultiplied by the relaxation parameter £ ^ 1. This means that the solution relaxes to a converged solution after time on    the    order    of    AtMD/£,.    This    delay may    be necessary    in    order to


prevent the constraint from canceling intrinsic thermal fluctuations on time scales less than the autocorrelation time tvv. However, as arugued in (Nie et al., 2004), it has the undesirable effect that the particle velocities always lag the continuum solution, which is incorrect in an accelerating flow.


A weight function similar to the flux-exchange method was also introduced in (Nie et al., 2004), in order to prevent drifting of molecules away from the MD domain but also to minimize density oscillations. In order to ensure mass continuity at the MD-continuum interface, the number of particles in each    cell    is    modified    by    the    net    flux    over    a    time    step    of the


Navier-Stokes equations,


n’ = -AAtpv ■ n/m.


In general, this method has features of the relaxation method and the flux-exchange method, but momentum flux continuity is not directly imposed. It has been used successfully in simulating accelerating Couette flow, cavity flow, and flow over an obstacle. Typical values were 15.5a for the overlap region in a domain 52.1a wide. The Navier-Stokes time step was At = 50AtMD with AtMD = 0.005т, and r = yTna2 jt the characteristic time of the Lennard-Jones potential.

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