Dynamic Transitions of Surface Tension Driven Convection

TL;DR

This paper investigates the mathematical behavior and transition types of surface tension driven convection in a 3D box, identifying conditions for different transition scenarios and analyzing the system's stability and attractors.

Contribution

It provides a detailed analysis of the dynamic transitions and stability criteria for surface tension driven convection, including numerical computation of transition types.

Findings

Transition occurs at critical Marangoni number crossing.

Type-I transition is numerically favored in single-mode cases.

Multiple modes can lead to Type-II or Type-III transitions.

Abstract

We study the well-posedness and dynamic transitions of the surface tension driven convection in a three-dimensional (3D) rectangular box with non-deformable upper surface and with free-slip boundary conditions. It is shown that as the Marangoni number crosses the critical threshold, the system always undergoes a dynamic transition. In particular, two different scenarios are studied. In the first scenario, a single mode losing its stability at the critical parameter gives rise to either a Type-I (continuous) or a Type-II (jump) transition. The type of transitions is dictated by the sign of a computable non-dimensional parameter, and the numerical computation of this parameter suggests that a Type-I transition is favorable. The second scenario deals with the case where the geometry of the domain allows two critical modes which possibly characterize a hexagonal pattern. In this case we show that the transition can only be either a Type-II or a Type-III (mixed) transition depending on another computable non-dimensional parameter. We only encountered Type-III transition in our numerical calculations. The second part of the paper deals with the well-posedness and existence of global attractors for the problem.