Chaos and overstability in the thermal convection of viscoelastic fluids

Abstract

The onset of aperiodic or chaotic behaviour in viscoelastic fluids is examined in the context of the Rayleigh-Benard thermal convection setup. A truncated Fourier representation of the conservation and constitutive equations, for an Oldroyd-B fluid, leads to a four-dimensional system that constitutes a generalization of the classical Lorenz system for a Newtonian fluid. It is found that, to the order of the present truncation and below a critical Deborah number De c, the critical Rayleigh number Ra c, for the onset of steady thermal convection does not depend on fluid elasticity or retardation. For De > De c, it is shown that steady convection does not exist, with the fluid becoming overstable instead. Fluid overstability, namely when the convective cell structure is time periodic, and which is attributed to fluid elasticity, is found to set in at a Rayleigh number that depends on the Deborah number and fluid retardation, and may be much smaller than Ra c. It is also found that fluid elasticity tends to destabilize the convective cell structure, precipitating the onset of chaotic motion, at a Rayleigh number that may be well below that corresponding to Newtonian fluids.