A Spatial Center Manifold Approach to a Hydrodynamical Problem with O(2) Symmetry
A Spatial Center Manifold Approach to a Hydrodynamical Problem with O(2) Symmetry
复制标题
具有 O(2) 对称性的流体动力学问题的空间中心流形方法
DOI：
--
复制
发表时间：
1994
期刊：
影响因子：
--
通讯作者：
A. Mielke
中科院分区：
文献类型：
--
作者：
A. Afendikov;A. Mielke
我们考虑平行板之间的三维Poiffille流的分叉。在“经典”的陈述中，这个问题具有SO（2）× O（2）对称群，从一开始似乎完全类似于库埃特-泰勒问题，但实际上它是完全不同的，因为在物理上最有趣的参数范围内，最危险的是纯二维扰动。与经典研究不同的是，除了分叉解与基本流的一致接近性外，我们对展向的行为没有做任何假设。然而，我们在流向方向上施加时间周期性以及周期为2π/α的空间周期性。这允许应用“空间动力学”方法，将展向变量视为进化变量。对于一定范围的参数α，我们能够减少分歧问题的空间中心流形上的流动是由一个稳定的Ginzburg-Landau方程。所有相关系数都可以从纯2D Poiffuille问题的分析中获得。对于小β的简化问题的研究表明，无论是螺旋和带状，分叉亚临界，相反的库埃特-泰勒问题。这些解在展向方向上是另外2π/β周期的。
We consider bifurcations from the 3D Poiseuille flow between parallel plates. In the “classical” statement the problem possesses SO(2) × O(2) symmetry group and from the beginning seems to be completely analogous the Couette-Taylor problem, but in fact it is quite different as in the most physically interesting range of parameters the most dangerous are pure 2D disturbances. In contrast to the classical studies, we make no assumptions on the behavior in the spanwise direction, except the uniform closeness of the bifurcating solution to the basic flow. However, we impose time periodicity as well as spatial periodicity with period 2π/α in streamwise direction. This allows to apply the “spatial dynamics” approach taking the spanwise variable as an evolutionary one. For a certain range of parameters α, we are able to reduce the bifurcation problem to a spatial center manifold on which the flow is described by a steady Ginzburg-Landau equation. All relevant coefficients can be taken from the analysis of the purely 2D Poiseuille problem. For small β the study of the reduced problem demonstrates that both, spirals and ribbons, bifurcate subcritical, in contrast to the Couette-Taylor problem. These are solutions which are additionally 2π/β periodic in the spanwise direction.