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Influence of constraint on routes to chaotic natural convection in Hele-Shaw slot, loop, and annulus

Influence of constraint on routes to chaotic natural convection in Hele-Shaw slot, loop, and annulus

The Lorenz equations are known to be an explicit `inertial form' for the partial differential equations that govern natural convection in a closed loopof liquid (thermosyphon). What is the fate of the inertial form as the constraint of unidirectional flow imposed by container geometry is relaxed? What happens to the Lorenz-like chaos as secondary parameters are varied? Is there a `break-up' of the inertial form and, if so, how so?

In this computational study, the loop is embedded in a family of container geometries, including the Hele-Shaw slot and annulus, with consistent thermal boundary conditions. Routes to chaotic convection corresponding to deformations within this family and varying Prandtl number are compared. Along each route, branch-tracing is used whenever possible, complemented by solution of initial-value problems. Galerkin and finite-difference discretizations lead to dynamical systems of order 100 equations, typically. Results extend those of [1,2] and evidence points to the Shilnikov-like breakup described in [3].

1. Yorke, Yorke, and Mallet-Paret, Physica D 24, pp 279-91, 1987.

2. Hu and Steen, Phys. Fluids 8(7), 1929-37, 1996.

3. Lyubimov and Zaks, Physica D 9, pp 52-64, 1983.