Interdisciplinary Applied Mathematics

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Fl = -Y(ui — v) + F, (F(t)F(t’)) = 2kBTjS(t — t’)


was added to the force F.j; here 7 is a measure of dissipation.


This scheme was tested for steady Couette and Poiseuille flow using a shifted Lennard-Jones potential, with time steps AtMD = 0.0017т on the MD side and At = 100AMD on the Navier-Stokes side.


In the hybrid    method    of    (Nie    et    al.,    2004)    the    relaxation    procedure    of


O’Connel and Thompson is abandoned, and new motion equations for the MD are derived. In the overall region, the average continuum velocity vj in each cell J is computed by averaging the velocities on the cell’s grid points where the Navier-Stokes equations are discretized. Continuity of the mean velocity is imposed by requiring that the averaged particle velocity in this cell be equal to vJ, i.e.,


^E ui=Vj(t),


i


where NJ is the number of particles in cell J. Taking the Lagrangian derivative of the above, we have


1    ..    _ Dv j(t)


Dt


i


which is a constraint in the equations of motion x = Fi/m. A general solution that satisfies the constraint has the form


x



Dt



+ Co



EZi = °.



By choosing



F


Ci = — ~



1



mNj



Nj


Fi


i=1


we obtain the generalized equations of motion for the *th particle,


x



Fi



Nj



m    mNj



1 EF. + ^U’., £c = 0.    (16.27)


The discretization of the above equation gives x(t + AtMD) — 2x(t) + x(t — AtMD)

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