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Two-dimensional Rayleigh–Bénard convection of viscoelastic liquids in Cartesian and cylindrical coordinates: regular and chaotic regimes

Abstract:

Two problems of Rayleigh–Bénard convection (RBC) concerning regular and chaotic motions of viscoelastic liquids in rectangular and cylindrical geometries are investigated. Four viscoelastic liquids and a Newtonian one are considered in the study. The equations for the critical Rayleigh numbers for stationary and oscillatory convection are found to be the same in the two RBC problems considered. However, the critical wave numbers are different due to an additional constraint in the cylindrical problem. In the dynamical system obtained here, the salient feature of it is that unlike the Khayat–Lorenz model, it reduces directly to the classical Lorenz model through a simple limiting procedure. In lieu of the spectrum of Lyapunov exponents, the corresponding spectrum of eigenvalues is considered to conclude that chaos sets in through a Hopf Rayleigh number in Jeffrey and Newtonian liquids while in the other three it is not so. The structure of the modified Khayat–Lorenz model of the paper is the same in the two geometries. The similarity or dissimilarity between the results of the two models in the linear regimes continues into the regular nonlinear regime as well as into the chaotic regime. Chaos is shown to manifest earlier in Maxwell liquids followed by that in a Jeffrey liquid for both the problems. However, the intensity of chaos in the other three liquids does not follow a similar trend. Hyper-chaos when explored is shown to be impossible in the chosen viscoelastic liquids. An in-depth study of the two problems in the two geometries presented a better understanding of the two problems.