垂直管内趋旋微游动体悬浮液中的一系列跨临界分岔
A sequence of transcritical bifurcations in a suspension of gyrotactic microswimmers in vertical pipe
AI总结：
本研究拓展了垂直管内趋旋微游动体悬浮液的稳态解研究，发现随Richardson数升高会出现可数无穷个跨临界分岔解，多数解不稳定，仅在瞬态动力学中起作用。
摘要：
Kessler（《Nature》，第313卷，1985年，第218-220页）首次表明，垂直管内的趋旋微游动体静态悬浮液与流动悬浮液中都会自发出现羽流状结构。近期研究证实，这类系统存在多个稳态轴对称、轴向均匀的解（Bees, M. A. 与 Croze, O. A.，《Proc. R. Soc. A.》，第466卷，2010年，第2057-2077页）。本研究拓展了这一发现，报告称随着Richardson数（理查森数）升高，会出现可数无穷个此类解。我们开展了线性稳定性分析、弱非线性分析与完全非线性分析，结果表明每个解均源于均匀悬浮液的失稳。解的离散数量由有限流动域导致，而分岔的跨临界特性则源于圆柱几何结构打破了系统的水平对称性。研究进一步表明，若要保持流动稳态，每个解的可达到向下流量存在最大阈值——此时改变压力梯度无法再提升该解对应的流量。除了在最低Richardson数下产生的那个解之外，所有找到的解都是不稳定的，这意味着它们会在均匀悬浮液向充分发展趋旋模式演化的路径中，对瞬态动力学过程发挥作用。
英文摘要：
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.
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