Navier-Stokes-Boussinsq equations in a tube; estimates when the data is front-like

TL;DR

This paper analyzes the stability and evolution of front-like solutions to the Navier-Stokes-Boussinesq equations in a thin 3D tube, providing uniform estimates and demonstrating the persistence of front-like behavior over time.

Contribution

It establishes uniform estimates for solutions with front-like initial data and proves the existence and stability of front-like solutions in a three-dimensional thin tube.

Findings

Derived uniform estimates on burning rate and flow velocity.

Proved solutions remain front-like over time.

Demonstrated stability of laminar front solutions.

Abstract

For the solutions of Navier-Stokes-Boussinesq equations in a three-dimensional thin tube with front like initial data, we derive some uniform estimates on the burning rate and the flow velocity, which can be interpreted as stability results for the laminar front. We also show that the front-like datum admits a solution which will stay front-like in time. We consider no-slip (Dirichlet) boundary condition for the flow, and no-flux (Neumann) boundary condition for the reactant(temperature).