Bifurcations and dynamics in convection with temperature-dependent viscosity in the presence of the O(2) symmetry

Abstract

We focus on the study of a convection problem in a two-dimensional setup in the presence of the O(2) symmetry. The viscosity in the fluid depends on the temperature as it changes its value abruptly in an interval around a temperature of transition. The influence of the viscosity law on the morphology of the plumes is examined for several parameter settings, and a variety of shapes ranging from spout-to mushroom-shaped are found. We explore the impact of the symmetry on the time evolution of this type of fluid, and we find solutions which are greatly influenced by its presence: at a large aspect ratio and high Rayleigh numbers, traveling waves, heteroclinic connections, and chaotic regimes are found. These solutions, which are due to the presence of symmetry, have not been previously described in the context of temperature-dependent viscosities. However, similarities are found with solutions described in other contexts such as flame propagation problems or convection problems with constant viscosity also in the presence of the O(2) symmetry, thus confirming the determining role of the symmetry in the dynamics.

Symmetry and plate-like convection in fluids with temperature-dependent viscosity

“…There also exist new critical thresholds at which stability is lost, thereby giving rise to new bifurcated structures. These stationary solutions are numerically obtained by using an iterative Newton-Raphson method as reported in [9,8].…”

“…The sign in the real part of the eigenvalue λ determines the stability of the solution: if it is negative, the perturbation decays and the stationary solution is stable, while if it is positive the perturbation grows over time and the stationary solution is unstable. For each unknown field expression, (5) is introduced into the system (1)-(3) and the equations are linearized inỹ, which are assumed to be small (see [9,8] for details). Together with their boundary conditions, the equations define a generalized eigenvalue problem.…”

Spectral numerical schemes for time-dependent convection with viscosity dependent on temperature

“…The time dependent solutions displayed in Figure 7 although confirms the validity of the method do not particularly exhibit the influence of the symmetry, and thus the proposed spectral scheme does not show its power on this respect for the chosen test problem. However for other viscosity dependencies as the ones reported in [11] the current spectral method has successfully described solutions whose existence is related to the presence of the symmetry.…”

“…The time dependent solutions found for the exponential viscosity law do not evidence the influence of the symmetry. However in [11] for different viscosity dependences, at high-moderate viscosity contrasts, it is reported that the proposed scheme is successful to this end. Finite element methods and finite volume methods have proven to be successful to reach extremely large viscosity contrasts up to 10 10 -10 20 and we have not improved these limits with spectral methods.…”

“…The stability of the stationary solutions is approached with the collocation method used for the Newton-Raphson iterative method. Expansions of the fields (28) are replaced in equations (32)- (35), and they and the boundary conditions (11) are evaluated at the collocation nodes following the same rules as before. As a result, the discrete form of the generalized eigenvalue problem is obtained:…”