Interdisciplinary Applied Mathematics

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m s(x, t) = Apv ■ n,


where A is the area and n the unit normal vector. Here s(x,t) is the number of particles that need to be added to (if s is positive) or removed (if s is negative) the top bin of the overlap domain. Similarly, the momentum flux continuity is enforced by the equation


ms(x,t)(u’) + Y^Fi = АП ■ n,


i


where u’ is the velocity of the added or removed particles, and Fi is an external force acting on particle i in the region of flux-exchange from continuum to particles (P ^ C). Flekkoy et al. employed the compressible Navier-Stokes equations in the coupling, and thus the momentum flux tensor П has the form


П = pvv + p — p(Vv + VvT — V^ v) — (p/3)V ■ v.


Combining the two equations, we observe that the momentum equation is satisfied if the mass equation is satisfied, but in addition we need to enforce


(u’) = v and Fi = А (П — pvv) ■ n.


In order to avoid drifting of particles, a weight function g(x) was introduced. In    particular,    this    function    obeys    g(x)    =    g'(x)    = 0    for    x < 0 and


diverges as


Ф — i/2) * -щ—


at    the    edge    of    the region    (P    ^ C). The coordinate    x    runs    parallel    to n,


and x = 0 is in the middle of the region (P ^ C) where flux-exchange from continuum to particles take place. Also, L is the size of the bins in the n direction. In addition, in order to maintain thermal equilibrium it was found necessary to thermalize the particles in the subdomain P ^ C using Langevin dynamics. Specifically, a force of the form

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