# Interdisciplinary Applied Mathematics

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+ v — 7в = Ре-172в, dt

with Pe = UL/D is the Peclet number defined as the ratio of the diffusion time (L2/D) to the advection time (L/U), where D is the molecular diffusivity. With regard to numerical discretization of the above equation, algorithms that work well for low Reynolds number but also for high Peclet number are required. To this end, a semi-Lagrangian method can be used to deal with the advection-diffusion equation (Xiu and Karniadakis, 2001). It can effectively bypass the strict time-step restriction imposed in explicit time-stepping integration, and it is stable for very large values of Pe.

Advection dominates for large values of the Peclet number, and a steepening of concentration gradients occurs. However, at much later times molecular diffusion smooths out these steep gradients. Of interest is the characteristic time for which У в is maximum, which we denote by tm. This time can be estimated by equating the diffusion length Id oc f~Dt to the striation thickness (transverse dimension) of the tracer la associated with advection. Stretching in applications achieving chaotic advection is a selfsimilar process at small scales following a lognormal distribution. Here, we refer to the average striation thickness in order to carry out the following order-of-magnitude analysis.

Let us first consider a simple shear flow v = (yy, 0), where у denotes the deformation rate, and regular advection for which we have that laZo//l + (yt)2• Therefore, the characteristic time tbm in this case is

t

s

m

OC