# Exact energy stability of B\'enard-Marangoni convection at infinite Prandtl number

## TL;DR

This paper uses the energy method to precisely determine the conditions under which pure conduction remains stable in Bénard-Marangoni convection at infinite Prandtl number, revealing that energy stability aligns with linear instability thresholds only for insulating boundaries.

## Contribution

The study provides an exact solution to the energy stability problem for Bénard-Marangoni convection, extending previous results and clarifying boundary condition effects on stability.

## Key findings

- Energy stability proven up to linear instability threshold for insulating boundaries.
- Energy method does not exclude subcritical instabilities in non-insulating cases.
- Exact stability conditions depend on boundary thermal properties.

## Abstract

Using the energy method we investigate the stability of pure conduction in Pearson's model for B\'enard-Marangoni convection in a layer of fluid at infinite Prandtl number. Upon extending the space of admissible perturbations to the conductive state, we find an exact solution to the energy stability variational problem for a range of thermal boundary conditions describing perfectly conducting, imperfectly conducting, and insulating boundaries. Our analysis extends and improves previous results, and shows that with the energy method global stability can be proven up to the linear instability threshold only when the top and bottom boundaries of the fluid layer are insulating. Contrary to the well-known Rayleigh-B\'enard convection setup, therefore, energy stability theory does not exclude the possibility of subcritical instabilities against finite-amplitude perturbations.