In the simplest sense, mixing acts on an initially heterogeneous system, transforming it to a homogeneous state through the actions of stirring and diffusion. The theory of dynamical systems has been successful in improving understanding of underlying features in fluid mixing, and how smooth stirring fields, coherent structures and boundaries affect mixing rates. The main stirring mechanism in fluids at low Reynolds number is the stretching and folding of fluid elements, although this is not the only mechanism to achieve complicated dynamics.
Mixing by cutting and shuffling occurs in many situations, for example in micro–fluidic split and recombine flows, through the closing and re-orientation of values in sink–source flows, and within the bulk flow of
granular material. The dynamics of this mixing mechanism are subtle and not well understood. Here, mixing rates arising from fundamental models capturing the essence of discontinuous, chaotic stirring with diffusion are investigated.
In purely cutting and shuffling flows it is found that the number of cuts introduced iteratively is the most important mechanism driving the approach to uniformity. A balance between cutting, shuffling and diffusion achieves a long-time exponential mixing rate, but similar mechanisms dominate the finite time mixing observed through
the interaction of many slowly decaying eigenfunctions. The time to achieve a mixed condition varies polynomially with diffusivity rate
κ, obeying t ∝ κ^{−η} . For the transformations meeting good stirring criteria, η < 1. Considering the time to achieve a mixed condition
to be governed by a balance between cutting, shuffling, and diffusion derives η ∼ 1/2, which shows good agreement with numerical results.
In stirring fields which are predominantly chaotic and exponentially mixing, it is observed that the addition of discontinuous transformations contaminates mixing when the stretching rates are uniform, or close to uniform. The contamination comes from an increase in scales of the concentration field by the reassembly of striations when cut
and shuffled. Mixing stemming from this process is unpredictable, and the discontinuities destroy the possibility to approximate early mixing rates from stretching histories. A speed up in mixing rate can
be achieved if the discontinuity aids particle transport into islands of the original transformation, or chops and rearranges large striations generated from highly non-uniform stretching. The long-time mixing rates and time to achieve a mixed condition are shown to behave counter-intuitively when varying the diffusivity rate. A deceleration of mixing with increasing diffusion coefficient is observed, sometimes
overshooting analytically derived bounds.
