Concept

Chaotic advection — where it appears

The stretching and folding of material by a flow whose streamlines are chaotic, which needs neither turbulence nor randomness. A steady three-dimensional flow can do it, and the rate at which it stretches material lines is a distribution rather than a number.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

The same patch, 6 periods later, in two flows. A round patch of 848 marked particles, advanced 6 periods by the blinking flow and by a steady flow of the same strength. The steady flow has drawn the patch into a smooth ribbon along a streamline and every particle in it is still on the streamline it started on; the blinking flow has folded the patch through itself repeatedly and its particles are spread across the whole region. Neither flow has any diffusion in it and neither has lost a particle. The difference between them is that one depends on time.

No randomness, and it mixes anyway

A steady two-dimensional flow cannot mix, however fast it is stirred, because its trajectories are its streamlines. Switch two vortices on and off alternately and the same fluid, obeying an exact map with nothing random in it, folds a patch of dye through itself until neighbouring particles separate by a factor of a thousand in six periods.

kinematics · Advection $One rate per moment, and none of them the same. lambda_p = ln<l^p>/(p t) against p. As p goes to zero it is the Lyapunov exponent, the rate of the typical element; at p = 1 it is the rate of the average length, which is nearly twice as large. If ln l were exactly Gaussian this would be a straight line with the Lyapunov exponent as its intercept, and the departure from that line is the same multifractality the velocity increments have.$

The stretching rate that is not one number

A material line in a flow gets longer, and there is a theorem saying its length grows at a definite exponential rate. There is also a rate at which the average length grows, and it is nearly twice as large — and a different rate for every moment of the distribution.

kinematics · Material lines

Named alongside it

The objects these essays reach for when they reach for this one.

The Lyapunov exponent Mixing Model limit ABC flow Averaging Deformation gradient Determinism Eulerian and Lagrangian Kinematics Material line Moments Multifractal

All concepts