Bifurcation of Symmetric Domain Walls for the Bénard–Rayleigh Convection Problem

Abstract

We prove the existence of domain walls for the Bénard-Rayleigh convection problem. Our approach relies upon a spatial dynamics formulation of the hydrodynamic problem, a center manifold reduction, and a normal form analysis of a reduced system. Domain walls are constructed as heteroclinic orbits of this reduced system.

Domain Walls for the Bénard–Rayleigh Convection Problem with “Rigid–Free” Boundary Conditions

“…Domain walls, or grain boundaries, are line defects which may occur between rolls with different orientations (see Figure 1.1), and are steady solutions of the system (1.1)- (1.3). In our previous work [5], we showed that symmetric domain walls bifurcate for the Navier-Stokes-Boussinesq system (1.1)-(1.3) with either "rigid-rigid" or "free-free" boundary conditions for Rayleigh numbers R > R c close to the critical value R c . These domain walls are solutions of the steady system which are periodic in y, symmetric in x and their limits as x → ∓∞ are rolls rotated by opposite angles ±α with α ∈ (0, π/3).…”

“…For the analysis of the bifurcation problem we use the spatial dynamics approach developed for the Swift-Hohenberg equation in [6] and adapted to the Navier-Stokes-Boussinesq system (1.1)-(1.3) in [5]. The steady system (1.1)-(1.3) is written as a dynamical system in which the evolutionary variable is the spatial variable x and a center manifold theorem is used to reduce this infinite dimensional dynamical system to a 12-dimensional system.…”

“…Domain walls are constructed as heteroclinic solutions of this 12dimensional system which connect suitably chosen periodic orbits. In [5], the vertical reflection symmetry has been used to further reduce this system to an 8-dimensional system. The loss of this symmetry in the case of "rigid-free" boundary conditions does not allow for this second reduction of dimension, here we have to analyze the full 12-dimensional system.…”

“…For our reduced system these symmetries are obtained from the invariance of the Navier-Stokes-Boussinesq system under reflections in the horizontal variables x and y and under translations in y, and the non-resonance condition is satisfied for angles α = π/6. Next, similarly to [5], we scale variables in the normal form and identify a leading order system for which we construct a heteroclinic connection. Finally, using the implicit function theorem we prove the persistence of this leading order heteroclinic connection for the full 12-dimensional system.…”

“…The main difference between the case of "rigid-free" boundary conditions considered here and the two cases in [5], is the loss of a vertical reflection symmetry of the governing equations. This symmetry was exploited in [5] to show that bifurcating domain walls lie on an 8-dimensional invariant submanifold of the center manifold. Consequently, the heteroclinic connections were found as solutions of an 8-dimensional, instead of a 12-dimensional, dynamical system.…”