Period-Bubbling Transition to Chaos in Thermo-Viscoelastic Fluid Systems

Abstract

We report a 6D nonlinear dynamical system for thermo-viscoelastic fluid by selecting higher modes of infinite Fourier series of flow quantities. This nonlinear system demonstrates overstable convective motion and some organized structures such as period-bubbling and Arnold tongue-like structures. Studies reveal that the stability of the conduction state does not alter for the new 6D system in comparison with the lowest order 4D system of Khayat [1995] . However, the stabilities of the convective state have some differences. The onset of unsteady convection in the 6D system is delayed for weak elasticity of the fluid. There exists a critical range of fluid elasticity where the 4D system exhibits subcritical Hopf bifurcation while the 6D system shows supercritical Hopf bifurcation, which ensures the increase of the domain of stability. In this range, catastrophic route to chaos occurs in the 4D system, whereas the 6D system exhibits intermittent onset of chaos. Comparing the two-parameter dependent dynamics for the two systems, the chaotic zones enclosed by periodic regions are suppressed in the 6D system, so the flow behaviors become more predictable. Owing to interacting thermal buoyancy and fluid elasticity, both the models exhibit period-bubbling transition to chaos, but the period-bubbling cascade in the 6D model occurs at lower Rayleigh number than the 4D model. The convergence rate of the period-bubbling process slows down compared to usual period-doubling and approaches the square root of the Feigenbaum constant asymptotically.