Hypermixing in linear shear flow

In this technical note we study mixing in a two-dimensional linear shear flow. We derive analytical expressions for the concentration field for an arbitrary initial condition in an unbounded two-dimensional shear flow. We focus on the solution for a point initial condition and study the evolution of (1) the second centered moments as a measure for the plume dispersion, (2) the dilution index as a measure of the mixing state, and (3) the scalar dissipation rate as a measure for the rate of mixing. It has previously been shown that the solute spreading grows with the cube of time and thus is hyperdispersive. Herein we demonstrate that the dilution index increases quadratically with time in contrast to a homogeneous medium, for which it increases linearly. Similarly, the scalar dissipation rate decays as t(-3), while for a homogeneous medium it decreases more slowly as t(-2). Mixing is much stronger than in a homogeneous medium, and therefore we term the observed behavior hypermixing.