Citation
Liñán Martínez, Amable
(1995).
Analytical description of chaotic oscillations in a toroidal thermosyphon.
In:
"Fluid physics : lecture notes of summer schools".
World Scientific series on nonlinear science. Series B, Special theme issues and proceedings
(5).
World Scientific, Singapore, pp. 507-523.
ISBN 9810218990.
Abstract
Multiple time scales and singular perturbation techniques are used to describe the ordinary and the chaotic oscillations due to natural convection in a fluid loop subject to a known external heat flux. The turbulent flow in the loop is modelled using the hydraulic approximation with a quadratic friction law. No steady solutions exist if the heat is added mainly to the top half and extracted from the bottom half of the loop, and two steady convective solutions may exist if one proceeds otherwise; these convective solutions may loose stability when the heat input is shifted from the side toward the bottom. The instability leads, first, to a periodic convective flow and then, after a period doubling Feigenbaum cascade, to a chaotic motion. An intermittent type transition from limit cycles to chaos is also found in the analysis. The transition to chaos can be described in terms of a non-invertible return map, obtained by singular perturbation techniques for loops with long length, when the system becomes strongly dissipative.