Interdisciplinary Applied Mathematics

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A more convenient measure for quantifying chaotic mixing is the finitetime Lyapunov exponent (FTLE). It is given by a similar definition to the one in equation (9.5) but without the limit t ^ ж, i.e.,


1    d(t + T)

T Пd(t)


It clearly depends on the time t and also on the Lagrangian position £, and it converges to LE as T ^ж. In addition, as shown in (Tang and Boozer, 1996), the FTLE satisfies the following equation:

Af(£, t) = m/t + /(£, t)/Vi + Aoc,    (9.7)

where A(£) is a smooth function of geometry only, and the function f (£, t) satisfies

hm f{^t)/Vt = 0.


Although the distribution of FTLE is strongly space- and time-dependent, it has been shown in (Tang and Boozer, 1996), that the mean FTLE converges very rapidly to the actual value of LE. Its effectiveness in micromixers has been demonstrated in (Niu et al., 2003), for both active and passive mixers.

The dimensionless advection time of the system, within which complete mixing can be achieved, can be written in terms of the FTLE as



ln(2Q) — 1 2Л p

FIGURE 9.11.    Poincare    section    of    the    twisted    pipe    (left)    and FTLE contour    at

X = n/2; c = 100. (Courtesy of Y.K. Lee.)

where H = XpL2/D ^ 1 for micromixing. A criterion for complete mixing was derived in (Niu et al., 2003), as follows:




ln(2AF(t)L2/D) — 1 «

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