Fluid elasticity and the transition to chaos in thermal convection.

Abstract

The influence of fluid elasticity on the onset of aperiodic or chaotic motion of an upper-convected Maxwellian fluid is examined in the context of the Rayleigh-B\'enard thermal convection problem. A truncated Fourier representation of the flow and temperature fields leads to a four-dimensional dynamical system that constitutes a generalization of the classical Lorenz system for Newtonian fluids. It is found that, to the order of the present truncation and above a critical value of the Deborah number ${\mathrm{De}}^{\mathit{c}}$, steady convection cannot set in, with the fluid becoming overstable instead. For De${\mathrm{De}}^{\mathit{c}}$, and even close to the Newtonian limit, the presence of fluid elasticity appears to alter significantly the circumstances leading to the onset of chaotic motion. Depending on the value of the Prandtl number, chaos is found to set in through the quasiperiodic route or period doubling. In general, fluid elasticity tends to destabilize the convective cell structure, precipitating the onset of chaos, at a Rayleigh number that may be well below that corresponding to Newtonian fluids.