Interdisciplinary Applied Mathematics

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In order to demonstrate the relation between the dispersion of particles and the mixing index we present in Figure 9.16 the time evolution of dispersion of 1600 particles using pattern B-C with T = 6 and T = 8. Snapshots of dispersed particles are shown in the figure at the same nondimensional time. Comparisons between the figures show that the T = 6 case is better stirred than the T = 8 case, as also indicated in Figure 9.15.


Finally, we address mixing-efficiency quantifications using numerical solutions of the species transport equation (see Section 14.1). Unlike the previous methods that utilize the Lagrangian motion of pseudo-particles, the species transport equation involves diffusion, and it can better match the experimental mixing results. In both numerical simulations and experiments (that utilize fluorescent dyes), it is possible to define an alternative mixing index by dividing the flow domain into N boxes, and recording the


T = 6


T = 8



t = 0


t = 0



t = 24


t = 24



t = 48


t = 48

t = 72


t = 72


FIGURE 9.16. Dispersion of 1600 particles as a function of time, obtained using pattern B-C at T = 6 (left) and at T = 8 (right).


fluorescence intensity/concentration value in these boxes. Using this, an alternative mixing index can be defined as


M



1


N





N



2



£ Vi-‘



(9.9)



i=1


where 9i    is the    concentration/fluorescence    value in    box    i,    and    9o    is    the

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