9813

Coupled Lorenz Oscillators

This Demonstration simulates the dynamics of two thermally coupled thermosiphons.
Thermosiphons are tubes filled with fluid that are bent into vertical closed loops that, when subjected to temperature differences, can convey heat due to buoyancy-driven flows; the tubes remove heat from the same source and exchange heat but do not exchange fluid. The equations describing the system are derived in the manner of the Lorenz system [1]:
,
,
,
,
,
,
where , , and , , represent the temperature difference between descending and ascending fluids, the horizontal convective heat transport, and the azimuthal velocities of the two coupled thermosiphons. , , and are the Biot, Rayleigh, and Prandtl numbers and is the ratio of the vertical temperature gradients in the two tubes. If both loops are subject to the same external temperature, then and no heat is transported between the loops; when , the loops decouple and either loop can be used to represent the Lorenz system. In that case, for the second loop the equations are
,
,
.
These equations are the same as the Lorenz equations after a scale transformation and with the geometric factor set to unity; they have steady-state solutions and . More detailed explanation of the behavior of the equations used here can be found in [1] and [2].

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

References
[1] S. H. Davis and M. N. Roppo, "Coupled Lorenz Oscillators," Physica D, 24(13), 1987 pp. 226–242. doi:10.1016/0167-2789(87)90077-7.
[2] C. Sparrow, The Lorenz Equations: Bifurcation, Chaos, and Strange Attractors, New York: Springer, 1982.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+