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Existence of orthogonal domain walls in B{é}nard-Rayleigh convection

Gérard Iooss

TL;DR

This work establishes the persistence of orthogonal domain walls in Bénard-Rayleigh convection by embedding the steady Navier–Stokes–Boussinesq system into an eight-dimensional reversible framework and performing a Lyapunov–Schmidt reduction around the known heteroclinic orbit of the reduced amplitude system. The analysis decouples a dominant real part from oscillatory complex components, derives precise estimates for nonlinear terms, and reduces the problem to a scalar bifurcation equation with a positive discriminant. Consequently, for small amplitudes and with a symmetry in y, there exists a one-parameter family of heteroclinic connections linking orthogonal roll sets, with the end states characterized by wave numbers connected via a wall-parallel shift. The results generalize symmetric-wall cases by allowing differing infinity wave numbers and quantify how the shift and wave-number mismatch depend on the cubic normal-form coefficients and the Prandtl number.

Abstract

In B{é}nard-Rayleigh convection we consider the pattern defect in orthogonal domain walls connecting a set of convective rolls with another set of rolls orthogonal to the first set. This is understood as an heteroclinic orbit of a reversible system where the x-coordinate plays the role of time. This appears as a perturbation of the heteroclinic orbit proved to exist in a reduced 6-dimensional system studied by a variational method in [3], and studied analytically in [10]. We then prove for a given amplitude, and an imposed symmetry in coordinate y, the existence of a one parameter family of heteroclinic connections between orthogonal sets of rolls, with wave numbers (different in general) which are linked to an adapted ''shift'' of rolls parallel to the wall.

Existence of orthogonal domain walls in B{é}nard-Rayleigh convection

TL;DR

This work establishes the persistence of orthogonal domain walls in Bénard-Rayleigh convection by embedding the steady Navier–Stokes–Boussinesq system into an eight-dimensional reversible framework and performing a Lyapunov–Schmidt reduction around the known heteroclinic orbit of the reduced amplitude system. The analysis decouples a dominant real part from oscillatory complex components, derives precise estimates for nonlinear terms, and reduces the problem to a scalar bifurcation equation with a positive discriminant. Consequently, for small amplitudes and with a symmetry in y, there exists a one-parameter family of heteroclinic connections linking orthogonal roll sets, with the end states characterized by wave numbers connected via a wall-parallel shift. The results generalize symmetric-wall cases by allowing differing infinity wave numbers and quantify how the shift and wave-number mismatch depend on the cubic normal-form coefficients and the Prandtl number.

Abstract

In B{é}nard-Rayleigh convection we consider the pattern defect in orthogonal domain walls connecting a set of convective rolls with another set of rolls orthogonal to the first set. This is understood as an heteroclinic orbit of a reversible system where the x-coordinate plays the role of time. This appears as a perturbation of the heteroclinic orbit proved to exist in a reduced 6-dimensional system studied by a variational method in [3], and studied analytically in [10]. We then prove for a given amplitude, and an imposed symmetry in coordinate y, the existence of a one parameter family of heteroclinic connections between orthogonal sets of rolls, with wave numbers (different in general) which are linked to an adapted ''shift'' of rolls parallel to the wall.

Paper Structure

This paper contains 21 sections, 7 theorems, 198 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The reduced system
  3. Setting of the perturbed system
  4. Solutions at infinities
  5. First change of variable
  6. Second change of variables
  7. Properties of linear operators $\mathcal{M}_{g}$ and $\mathcal{L}_{g}$ (defined in (\ref{['def Mg']},\ref{['def Lg']}))
  8. Estimates for the right hand sides of $\mathcal{M}_{g}(\widehat{A_{0}},\widehat{C_{0}})$ and $\mathcal{L}_{g}\widehat{D_{0}}$
  9. First component of $\mathcal{M}_{g}(\widehat{A_{0}},\widehat{C_{0}})$
  10. Second component of $\mathcal{M}_{g}(\widehat{A_{0}},\widehat{C_{0}})$
  11. Component $\mathcal{L}_{g}\widehat{D_{0}}$
  12. Bifurcation equation
  13. Resolution with respect to $\widetilde{{ \if@compatibility \mathchar"0121 {} \mathchar"0121 } }_{+}$ and $w$
  14. Resolution with respect to $(u,v)$
  15. Final bifurcation equation
  16. ...and 6 more sections

Key Result

Theorem 1

Let us choose $\frac{1}{3}\leq { \if@compatibility \mathchar"010E {} \mathchar"010E } \leq 1,$ and admit a certain conjecture on a 4th order differential equation with boundary conditions on a bounded interval, all being independent of ${ \if@compatibility \mathchar"0122 {} \mathchar"0122

Figures (1)

  • Figure 1: Orthogonal domain wall

Theorems & Definitions (28)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 1
  • Corollary 2
  • Corollary 3
  • Theorem 4
  • Remark 5
  • Remark 6
  • ...and 18 more