Navier-Stokes equations under Marangoni boundary conditions generate all hyperbolic dynamics

TL;DR

This paper demonstrates that the Navier-Stokes equations with Marangoni boundary conditions in 2D can produce any structurally stable dynamics on compact manifolds by tuning parameters like Prandtl number and heat source.

Contribution

It shows that these fluid dynamics equations are capable of generating all possible structurally stable dynamics through parameter adjustments.

Findings

Local semiflows can generate all structurally stable dynamics.

Adjusting Prandtl number and heat source suffices to prescribe desired dynamics.

The model includes convection, diffusion, and capillary effects.

Abstract

The dynamics defined by the Navier-Stokes equations under the Marangoni boundary conditions in a two dimensional domain is considered. This model of fluid dynamics involve fundamental physical effects: convection, diffusion and capillary forces. The main result is as follows: local semiflows, defined by the corresponding initial boundary value problem, can generate all possible structurally stable dynamics defined by $C^1$ smooth vector fields on compact smooth manifolds (up to an orbital topological equivalence). To generate a prescribed dynamics, it is sufficient to adjust some parameters in the equations, namely, the Prandtl number and an external heat source.