# Interdisciplinary Applied Mathematics

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where Pe = уl2/D and l0 denotes the initial thickness of the layer. On the other hand, for a field with characteristic velocity and length scales U and h that generates chaotic advection, the striation thickness of the tracer is la =    l0    exp(—At).    Here A is    the    stretching    rate    of the    tracer, equal    to    the

asymptotic value of the Lyapunov exponent (Ottino, 1997). In this case, the characteristic time, tm, for long times is estimated as

t

ca

m

oc — In Pe.

Comparison of the above estimates for the mixing time scale implies that the diffusion time increases much faster as Pe increases in linear shear flow compared to chaotic advection. It turns out that this is a general result with the exception of the pure strain flow v = (yx,yy), which although integrable (i.e., nonchaotic) also has mixing times proportional to lnPe. This anomaly is a consequence of an exponential separation associated with this flow and    the    infinite    acceleration    for    x ^    ж;    for    a    detailed

explanation see (Jones, 1991).

The long mixing times imply long mixing lengths, which are impractical in microfluidic applications. To this end, many efforts have been made to design and operate effective mixers that exploit the chaotic advection concept. One of the first theoretical designs suggested for general viscous flows was based on twisted pipes, i.e., pipe segments that are not all in the same plane (Jones et al., 1989). It is well known that in a curved pipe a secondary motion is induced, giving rise to longitudinal vortices. The flow of particles in a twisted pipe with a pitch angle у is represented by a sequence of Dean solutions augmented by a rotation of particles through an angle —у between the successive segments. The specific value of the angle is very important, since it will make the system nonintegrable and generate chaotic advection.

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