The stabilizing effect of the temperature on buoyancy-driven fluids
The stabilizing effect of the temperature on buoyancy-driven fluids
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DOI：
10.1512/iumj.2022.71.9070
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发表时间：
2020-05
影响因子：
1.1
通讯作者：
Oussama Ben Said;Uddhaba Raj Pandey;Jiahong Wu
中科院分区：
文献类型：
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作者：
Oussama Ben Said;Uddhaba Raj Pandey;Jiahong Wu
浮力驱动流体的Boussinesq系统将浮力驱动的动量方程与温度的对流扩散方程耦合起来。Boussinesq系统的一个基本问题是在流体静力平衡附近扰动的稳定性问题。当系统没有完全耗散时，这个问题可能非常困难。本文研究了一个只有垂直耗散和水平热扩散的二维Boussinesq系统的稳定性问题。我们建立了非线性系统的稳定性，并推导出精确的线性化系统的大时间行为。本文的结果揭示了浮力驱动流体的一个显著现象。也就是说，温度实际上使流体平滑和稳定。如果温度不存在，则流体由仅具有垂直耗散的2D Navier-Stokes方程控制，并且其稳定性保持开放。正是由于Boussinesq系统中温度和速度之间的耦合和相互作用，使得本文所研究的稳定性问题成为可能。数学上，系统可以简化为退化和阻尼波方程，燃料的稳定。
The Boussinesq system for buoyancy driven fluids couples the momentum equation forced by the buoyancy with the convection-diffusion equation for the temperature. One fundamental issue on the Boussinesq system is the stability problem on perturbations near the hydrostatic balance. This problem can be extremely difficult when the system lacks full dissipation. This paper solves the stability problem for a two-dimensional Boussinesq system with only vertical dissipation and horizontal thermal diffusion. We establish the stability for the nonlinear system and derive precise large-time behavior for the linearized system. The results presented in this paper reveal a remarkable phenomenon for buoyancy driven fluids. That is, the temperature actually smooths and stabilizes the fluids. If the temperature were not present, the fluid is governed by the 2D Navier-Stokes with only vertical dissipation and its stability remains open. It is the coupling and interaction between the temperature and the velocity in the Boussinesq system that makes the stability problem studied here possible. Mathematically the system can be reduced to degenerate and damped wave equations that fuel the stabilization.