Complex bifurcations in Bénard–Marangoni convection

Abstract

Abstract. We study the dynamics of a system defined by the Navier-Stokes equations for a non-compressible fluid with Marangoni boundary conditions in the two dimensional case. We show that more complicated bifurcations can appear in this system for a certain nonlinear temperature profile as compared to bifurcations in the classical Rayleigh-Bénard and Bénard-Marangoni systems with simple linear vertical temperature profiles. In terms of the Bénard-Marangoni convection, the obtained mathematical results lead to our understanding of complex spatial patterns at a free liquid surface, which can be induced by a complicated profile of temperature or a chemical concentration at that surface. In addition, we discuss some possible applications of the results to turbulence theory and climate science.

Strange attractors for Overbeck -Boussinesq model

“…The relations, obtained in the paper, give an analytical description of spatio-temporal patterns induced by strange attractors. The patterns are similar to found in [24] for Marangoni flows, they are quasiperiodic in x, and have the boundary layer form, i.e., located at the top boundary y = 0.…”

“…When we vary C U , all λ k with k = k j pass through 0 simultaneously. In this case there is a bifurcation, which involves a number of the unstable modes e k (this effect was found in [24] for the Marangoni problem).…”

“…We omit an elementary proof. In coming sections we investigate system (8.181) mainly following works [24] and [25,23].…”

“…This difficulty is not too hard and it can be overcome by the methods of the invariant manifold theory [10, 2, 1] that allows us to reduce systems (1.1) of large dimension to analogous systems of smaller dimension, where the coefficients, which define K(X) can be considered as free parameters (for more detail, see Sect. 10 and [24]).…”

“…To find the operator L having such properties is not easy, and the construction of L is a main technical part of the paper. The further application of the RVF method to (1.1) follows works [24,25,23] with small modifications.…”