A fluid flow can be considered as a dynamical system, that is a set of ordinary differential equations that determines the evolution of a Lagrangian trajectory. These equations are called advection equations:
where  are the components of the velocity field, which are assumed to be known from the solution of the equations governing fluid flow, such as the Navier-Stokes equations,
and  is the physical position. If the dynamical system governing trajectories is chaotic, the integration of a trajectory is extremely sensitive to initial conditions, and neighboring points separate exponentially with time. This phenomenon is called chaotic advection.
Dynamical systems and chaos theory state that at least 3 degrees of freedom are necessary for a dynamic system to be chaotic. Three-dimensional flows have three degrees of freedom corresponding to the three coordinates, and usually result in chaotic advection, except when the flow has symmetries that reduce the number of degrees of freedom. In flows with less than 3 degrees of freedom, Lagrangian trajectories are confined to closed tubes, and shear-induced mixing can only proceed within these tubes.
This is the case for 2-D stationary flows in which there are only two degrees of freedom  and . For stationary (time-independent) flows, Lagrangian trajectories of fluid particles coincide with the streamlines of the flow, that are isolines of the stream function. In 2-D, streamlines are concentric closed curves that cross only at stagnation points. Thus, a spot of dyed fluid to be mixed can only explore the region bounded by the most external and internal streamline, on which it is lying at the initial time. Regarding practical applications, this configuration is not very satisfying.
For 2-D unstationary (time-dependent) flows, instantaneous closed streamlines and Lagrangian trajectories do not coincide any more. Hence, Lagrangian trajectories explore a larger volume of the volume, resulting in better mixing. Chaotic advection is observed for most 2-D unstationary flows. A famous example is the blinking vortex flow introduced by Aref, where two fixed rod-like agitators are alternately rotated inside the fluid. Switching periodically the active (rotating) agitator introduces a time-dependency in the flow, which results in chaotic advection. Lagrangian trajectories can therefore escape from closed streamlines, and visit a large fraction of the fluid domain.