具有内部毛细作用的黏性流体中Rayleigh-Taylor不稳定性的数学理论

Rayleigh-Taylor(RT)不稳定性是流体力学中著名的不稳定性之一，其数学理论的研究一直是应用数学和工程技术领域中的热点问题之一。.本项目旨在研究具有内部毛细作用的黏性流体RT不稳定性。首先，我们分别在有界域与条形区域内研究具有内部毛细作用的非均匀不可压黏性流体RT不稳定性，分析毛细作用是否可抑制RT不稳定性及区域形状的改变是否有助于毛细作用抑制RT不稳定性。其次，我们研究分层具有内部毛细作用的等熵可压缩黏性流体RT不稳定性，分析毛细作用是否可抑制RT不稳定性。最后，我们分别在有界域与条形区域内研究具有内部毛细作用的非等熵可压缩无热传导黏性流体RT不稳定性，分析毛细作用是否可抑制RT不稳定性及可压缩性、温度变化与区域形状的改变是否有助于毛细作用抑制RT不稳定性。.上述研究内容不仅是应用数学中的前沿课题，有极其重要的理论意义，而且紧密联系应用科学和工程技术实际，有广阔的应用前景。

The Rayleigh-Taylor (RT) instability is one of the famous instability in fluid dynamics, and the research of its mathematical theory has always been a hot topic in the field of applied mathematics and engineering technology. .In this project, we will focus on the RT instability in viscous fluids with internal capillarity. First, we study the RT instability in inhomogeneous incompressible viscous fluids with internal capillarity in bounded and strip domains respectively, and analyze whether the capillarity can inhibit the RT instability or not, and whether the change of the domain shape can help the capillarity to inhibit the RT instability or not. Second, we consider the RT instability in stratified isentropic compressible viscous fluids with internal capillarity, and analyze whether the capillarity can inhibit the RT instability or not. Finally, we investigate the RT instability in non-isentropic compressible viscous fluids with internal capillarity and without heat conductivity in bounded and strip domains respectively, and analyze whether the capillarity can inhibit the RT instability or not, and whether the compressibility, and the changes of the temperature and the domain shape can help the capillarity to inhibit the RT instability or not..The above research contents are not only the frontier topics in applied mathematics, and has extremely important theoretical significance, but also closely connected with applied science and engineering technology, and has wide prospective application.