Fast-moving pattern interfaces close to a Turing instability in an asymptotic model for the three-dimensional B\'enard-Marangoni problem

TL;DR

This paper analyzes pattern bifurcations and fast-moving interfaces in a long-wave model of the Bénard-Marangoni problem near a Turing instability, deriving amplitude equations and constructing modulating front solutions.

Contribution

It introduces a new asymptotic model for the Bénard-Marangoni problem and develops a detailed bifurcation analysis near the Turing instability, including the construction of modulating front solutions.

Findings

Identification of bifurcation of planar patterns at critical Marangoni number

Derivation of amplitude equations governing pattern modulation

Construction of heteroclinic orbits representing pattern transition fronts

Abstract

We study the bifurcation of planar patterns and fast-moving pattern interfaces in an asymptotic long-wave model for the three-dimensional B\'enard-Marangoni problem, which is close to a Turing instability. We derive the model from the full free-boundary B\'enard-Marangoni problem for a thin liquid film on a heated substrate of low thermal conductivity via a lubrication approximation. This yields a quasilinear, fully coupled, mixed-order degenerate-parabolic system for the film height and temperature. As the Marangoni number $M$ increases beyond a critical value $M^*$, the pure conduction state destabilises via a Turing(-Hopf) instability. Close to this critical value, we formally derive a system of amplitude equations which govern the slow modulation dynamics of square or hexagonal patterns. Using center manifold theory, we then study the bifurcation of square and hexagonal planar patterns. Finally, we construct planar fast-moving modulating travelling front solutions that model the transition between two planar patterns. The proof uses a spatial dynamics formulation and a center manifold reduction to a finite-dimensional invariant manifold, where modulating fronts appear as heteroclinic orbits. These modulating fronts facilitate a possible mechanism for pattern formation, as previously observed in experiments.