text: In fluid mixing, one often wishes to homogenize a species, that can be characterized by its concentration field q. Often, the species can be considered as a passive tracer that does not modify the flow. The species can be for example a dye to be mixed.
  The evolution of a concentration field  obeys the advection-diffusion equation, also called convection–diffusion equation:
  Compared to the simple diffusion equation, the term proportional to the velocity field  represents the effect of advection.
  When mixing a spot of tracer, the advection term dominates the evolution of the concentration field at the beginning of the mixing process. Chaotic advection transforms the spot into a bundle of thin filaments. The width of a dye filament decreases exponentially with time, until an equilibrium scale is reached, at which the effect of diffusion starts to be significant. This scale is called the Batchelor scale.  It is defined as the square root of the ratio between the diffusion coefficient and the Lyapunov exponent 
  where  is the Lyapunov exponent and D is the diffusion coefficient.
  This scale measures the balance between stretching and diffusion on the evolution of the concentration field: stretching tends to decrease the width of a filament, while diffusion tends to increase it. The Batchelor scale is the smallest length scale that can be observed in the concentration field, since diffusion smears out quickly any finer detail.
  When most dye filaments reach the Batchelor scale, diffusion begins to decrease significantly the contrast of concentration between the filament and the surrounding domain. The time at which a filament reaches the Batchelor scale is therefore called its mixing time. The resolution of the advection–diffusion equation shows that after the mixing time of a filament, the decrease of the concentration fluctuation due to diffusion is exponential, resulting in fast homogenization with the surrounding fluid.