# User:Eugene M. Izhikevich/Proposed/Chaotic advection

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## Advection by fluid flow

In fluid mechanics the term advection means the transport of material by the flow. Examples abound: smoke carried by the wind, pollutants carried by a river, cream stirred into coffee, and so on. Sometimes, to emphasize that the material being transported follows the flow exactly, the term passive advection is used. Advection implies that the transported material is so light and inert that it follows the flow transporting it, at each instant adjusting to the prevailing flow velocity. Not all material transport by a flow is of this type, of course, e.g., the transport of sediments or the motion of the bug that hits the windshield of a vehicle. Nevertheless, the notion of (passive) advection is often a very useful approximation, and it describes the kinematics of the fluid itself exactly.

Advection, then, is the statement that the velocity of a particle of the transported material at any time and any point in space equals the velocity of the flow field. If the position of the particle as a function of time is given in cartesian coordinates as \((x(t), y(t), z(t))\ ,\) and if the cartesian components of the velocity field are \((u,v,w)\ ,\) all three of which depend on position \((x,y,z)\) and time, \(t\ ,\) then the *advection equations* are

\(\frac{dx}{dt} = u(x,y,z,t),\) \(\frac{dy}{dt} = v(x,y,z,t),\) \(\frac{dz}{dt} = w(z,y,z,t).\)

The components of the velocity field \(u\ ,\) \(v\) and \(w\) are assumed to be known from the solution of one of the standard dynamical equations governing fluid flow, typically the Navier-Stokes equation in one of its many forms.

## Advection as a dynamical system

With the flow given, the advection equations constitute a three-dimensional dynamical system. The only approximation inherent in this system is the assumption that the particle follows the flow. It is well known from dynamical systems theory that a system of three coupled, ordinary differential equations may be non-integrable and, thus, have chaotic solutions. When the advection equations are non-integrable, and particle trajectories in the flow can be chaotic, we have the phenomenon of *chaotic advection*.

Advection by a two-dimensional steady flow, given by

\(\frac{dx}{dt} = u(x,y),\) \(\frac{dy}{dt} = v(x,y),\)

is always regular, i.e., non-chaotic. Time-dependent two-dimensional flows, however, may lead to chaotic advection. Three-dimensional flows, both steady and unsteady, will often produce chaotic advection. Two-dimensional incompressible flow provides an interesting example. In that case, the velocity components may be given in terms of a streamfunction, \(\psi(x,y,t)\ ,\) by

\( u = \frac{\partial \psi}{\partial y},\ \ \ \ \) \( v = -\frac{\partial \psi}{\partial x}.\)

In this case the advection equations are

\(\frac{dx}{dt} = \frac{\partial \psi}{\partial y},\) \(\frac{dy}{dt} = -\frac{\partial \psi}{\partial x}.\)

These equations are in the form of Hamilton's canonical equations, with the streamfunction \(\psi\) in the role of the Hamiltonian, for a system with one degree of freedom. The coordinates \(x\) and \(y\) play the role of generalized coordinate and generalized momentum, respectively. Hence, the configuration space of an advected particle is the phase space of the Hamiltonian system. The rich spatial structure that can emerge when the Hamiltonian system is time-dependent and the advection is chaotic is, thus, a manifestation of the phase space structure in a non-integrable Hamiltonian system with one degree of freedom.

## History and applications

The term *chaotic advection* and an elucidation of the phenomenon as described above is due to Aref (1984). For a historical review see also Aref (2002). The concept has now found application in many areas of fluid dynamics, recently in the subject of microfluidics. Chaotic advection is used as a classifying keyword in journals and conferences dealing with fluid dynamics.</math>

## References

- H. Aref, 1984 Stirring by chaotic advection.
*Journal of Fluid Mechanics***143**, 1-21. - H. Aref, 2002 The development of chaotic advection.
*Physics of Fluids***14**, 1315-1325.