Interdisciplinary Applied Mathematics

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Velocity Profiles


The velocity    profile    is one    of    the    most    important    measurables    for    fluid


transport, and can be computed in a similar manner as the density profile. Usually, the simulation system is partitioned into n bins, and statistics of the fluid velocity are gathered separately in each bin (Tysanner and Garcia, 2004). Assuming that during an s-step simulation, at each step k, there are nk,i particles in the *th bin, and the velocity of each of these particles (denoted by j) is given by vjk i, then the average fluid velocity ui in the *th


bin can be computed by






Equation (16.20) is used to compute the steady-state velocity profile. If one    is interested in    the    transient    behavior    of    the    velocity    profile,    an


ensemble of simulations will need to be performed. In this case, the velocity profile can still be analyzed using equation (16.20), and the only difference is that the parameter s now denotes the different simulations rather than different time steps.


Diffusion Coefficient


The diffusion coefficient can be calculated using the Einstein relationship D = 1 lim <[»«°+<>—Vo)F


6 t^X    t


where r is the atom position, which can be obtained from the trajectories generated by the MD simulation. See Chapter 10 for a discussion of the calculation of diffusion coefficients of simple fluids.


Stress Tensor


The stress    or    pressure    tensor    of    an    atomic fluid,    denoted    by    P, is    often


defined as the infinitesimal force dF acting on an infinitesimal area dA, which moves    with    the    local    streaming    velocity    u(r, t)    of the    fluid    (Todd

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