论文标题
在垂直管中旋转微武器悬架中的一系列跨临界分叉序列
A sequence of transcritical bifurcations in a suspension of gyrotactic microswimmers in vertical pipe
论文作者
论文摘要
凯斯勒（Nature，第313卷，1985年，第218-220页）首先表明，羽状结构从垂直管中的旋转微晶状体的静止和流动悬浮液自发出现。最近，已经表明，对于这种系统，存在多个稳定的轴对称轴对称轴向均匀的溶液（Bees，M。A.＆Croze，O。A.，Proc。R.Soc。A.，第466卷，2010年，第2057-2077页）。在本研究中，我们通过报告说，随着Richardson数量的增加而出现了大量无限的解决方案来概括这一发现。进行线性稳定性，弱非线性和完全非线性分析，表明每种溶液均来自均匀悬浮液的不稳定。解决方案的离散数是由于有限流量域引起的，而分叉的跨临界性质是由于圆柱形几何形状破坏了系统的水平对称性。进一步表明，如果流动保持稳定，则存在每个溶液可实现的向下流速的最大阈值，因为变化的压力梯度无法再增加溶液的流速。除了以最低的理查森数字出现的那个解决方案，所有发现的解决方案都是不稳定的，这意味着它们将在从统一悬架到完全发达的陀螺模式的路线中的瞬态动力学中发挥作用。
Kessler (Nature, vol. 313, 1985, pp. 218-220) first showed that plume-like structures spontaneously appear from both stationary and flowing suspensions of gyrotactic microswimmers in a vertical pipe. Recently, it has been shown that there exist multiple numbers of steady axisymmetric axially uniform solutions to such a system (Bees, M. A. & Croze, O. A., Proc. R. Soc. A., vol. 466, 2010, pp. 2057-2077). In the present study, we generalise this finding by reporting that a countably infinite number of such solutions emerge as Richardson number increases. Linear stability, weakly nonlinear and fully nonlinear analyses are performed, revealing that each of the solutions arises from the destabilisation of uniform suspension. The discrete number of the solutions is due to the finite flow domain, while the transcritical nature of the bifurcation is because of the cylindrical geometry which breaks the horizontal symmetry of the system. It is further shown that there exists a maximum threshold of achievable downward flow rate for each solution if the flow is to remain steady, as varying the pressure gradient can no longer increase the flow rate from the solution. Except the one arising at the lowest Richardson number, all of the solutions found are unstable, implying that they would play a role in the transient dynamics in the route from a uniform suspension to the fully-developed gyrotactic pattern.
