On the Rayleigh-Taylor Instability for the Incompressible Viscous Magnetohydrodynamic Equations
On the Rayleigh-Taylor Instability for the Incompressible Viscous Magnetohydrodynamic Equations
复制标题
不可压缩粘性磁流体动力学方程的瑞利-泰勒不稳定性
DOI：
10.1080/03605302.2013.863913
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发表时间：
2015
影响因子：
1.9
通讯作者：
Wang Yanjin
中科院分区：
文献类型：
--
作者：
Jiang Fei;Jiang Song;Wang Yanjin
本文研究了两个不可压缩、不混溶、粘性磁流体动力学（MHD）流的Rayleigh-Taylor不稳定性问题，该流具有零电阻率和表面张力（或无表面张力），在均匀重力场的存在下演化为自由界面。首先，我们将无限大平板中的MHD自由边界问题转化为拉格朗日坐标系中的Navier-Stokes方程组，其中的力项由流体流图引起。然后，我们分析了线性化的问题周围的稳定状态，其中描述了一个更密集的不混溶流体躺在一个轻的一个自由界面分离的两种流体，和两种流体处于（不稳定）平衡。通过研究一个家庭的修改变分问题，我们构造光滑（当限制到每个流体域）的线性化问题的解决方案，增长指数快速的时间在Sobolev空间，从而导致一个全局不稳定的结果线性化问题。最后，利用这些病态解，在适当的意义下证明了相应非线性问题的全局不稳定性。此外，我们估计，所谓的临界数确实等于 分析了粘度和表面张力对不稳定性的影响。
We study the Rayleigh-Taylor instability problem for two incompressible, immiscible, viscous magnetohydrodynamic (MHD) flows with zero resistivity and surface tension (or without surface tension), evolving with a free interface under presence of a uniform gravitational field. First, we reformulate the MHD free boundary problem in an infinite slab as a Navier-Stokes system in Lagrangian coordinates with a force term induced by the fluid flow map. Then, we analyze the linearized problem around the steady state which describes a denser immiscible fluid lying above a light one with a free interface separating the two fluids, and both fluids being in (unstable) equilibrium. By studying a family of modified variational problems, we construct smooth (when restricted to each fluid domain) solutions to the linearized problem that grow exponentially fast in time in Sobolev spaces, thus leading to an global instability result for the linearized problem. Finally, using these pathological solutions, we prove the global instability for the corresponding nonlinear problem in an appropriate sense. Moreover, we evaluate that the so-called critical number indeed is equal to , and analyze the effect of viscosity and surface tension on the instability.