Interdisciplinary Applied Mathematics

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ЦPn — MiVi = 0,


where Ni is the total number of particles and Mi is the mass of the continuum fluid element in the 23-bin, and pn is the momentum of the nth particle in the v-direction. This constraint can be integrated into standard Lagrange’s equations governing the motion of the rest of the molecules in the MD region. This approach was successfully implemented by (O’Connell and Thompson, 1995), who used an overlap region of 14a. A free parameter in this approach is the strength of the constraint £ in relation to the extent of the overlap region. Specifically, the equations of motion in the 23-bin are


x23 =—h £

P i


<9x ’




1 vA pn

Ni 3-^ rri




FIGURE 16.4. Schematic of one Schwarz iteration cycle (Courtesy of N.G. Had-jiconstantinou.)

where m is the particle mass. Small values of the parameter will provide an inadequate coupling between MD and Navier-Stokes, while large values will lead to excessive damping of particle fluctuations, which in turn will lead to divergence in the solution. In (O’Connell and Thompson, 1995), a value of £ = 0.01 was used for simulating a slow-startup Couette flow. It was shown that the best choice of £ is to have AtMD greater than the autocorrelation time tvv, where AMD is the time step in the MD simulation. This relaxation approach does not handle correctly the mass flux at the MD-continuum interface, and this is an important limitation of the method.

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