# Interdisciplinary Applied Mathematics

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s « 2N-1/2.

Once the number of particles (N) is selected, the domain is divided into boxes of size s, and the motion of each particle is calculated for a long time. Then the dispersed particles in each box are counted, and a decision is made regarding the mixing state of each box. For example, tracking 1600 particles requires 400    boxes    in    the    entire    domain.    This results    in a maximum    of    4

particles per box in a homogeneous mixing state (i.e., nmax = 4). For such conditions one can define a Mixing index for each box (Mi) as

if ni ^ nmax if ni ^ nmax:

where ni    is the    number of particles    in    box    i.    There    will    be empty    boxes

in the domain for each overpopulated box (ni> nmax). Therefore, assuming the overpopulated boxes well mixed is reasonable, since impact of this decision is compensated by the low M values of underpopulated boxes in the domain. Another alternative is to characterize particle dispersion by assuming that boxes that contain at least one particle are well mixed (Liu et al.,    1994).    This is    useful,    since    it    is practically    impossible    to    observe    a

homogeneous mixing    state    (ni    =    nmax)    for    all    the    boxes in    the    domain.

Using either definition for Mi, the total mixing index MT is defined as

1N

(98)

FIGURE 9.15. Time variation of the mixing index M for pattern B-C at T = 6 and T = 8, obtained using equation (9.8).

In Figure 9.15, we present the time variation of the mixing index for pattern B-C at T = 6 and T = 8, using the method of Liu et. al (1994). The mixing index for both cases increases exponentially at early times, then reaches the asymptotic limit of unity. The mixing index for T = 6 reaches its asymptotic limit much faster than in the T = 8 case, thus indicating that the case T = 6 corresponds to a better mixer than the T = 8 case. Overall, the behavior of MT for both cases is consistent with the FTLE results, indicating that time variation of the mixing index can be utilized as a consistent measure for monitoring mixing quality.

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