Transition to disordered front propagation

一个多世纪以来，一个动力系统从一个有序的、不变的状态突然转变为一个不稳定的、无序的动力学状态，这一现象引起了科学家们的兴趣。虽然系统的有序基本状态对于无穷小的扰动是稳定的，但在驱动参数的阈值以上，有限幅度的扰动可以触发复杂的、持续演化的动力学的突然变化。在切变流过渡到湍流的开创性解释中，从稳定状态的漂移对应于对系统弱不稳定状态或所谓的边缘状态的稳定流形的短暂探索，边缘状态决定了将初始条件从增长到湍流的初始条件分离为层流的盆地边界。提出的研究将探讨这种动态情景是否适用于表现出亚临界过渡到无序的更广泛的系统。一个主要的候选者是矩形（Hele-Shaw）通道中的两相位移流，其关键优点是，非线性只产生于两种流体之间的界面，因此仅使用界面形态就可以区分不同的状态。此外，系统的行为可以用一组深度平均方程来描述，这比描述剪切流所需的完整的Navier-Stokes方程更易于分析。当驱动通量足够大时，传播前沿的复杂时间依赖行为可以解释为对边缘状态稳定流形的探索，为了追求这一想法，我们需要确定这些微扰驱动的漂移是否总是短暂的，还是在阈值通量以上成为自维持的。换句话说，是局部的扰动足以引发无序，还是需要空间分布的扰动才能产生长期的无序模式形成？为了回答这个问题，我们建议在EPSRC项目拨款EP/P026044/1资助的大型实验设施中进行系统的实验研究，测试稳定传播的Saffman-Taylor手指对大范围驱动通量的响应，包括局部和空间分布的扰动。实验结果将为模型提供信息，这反过来将加速识别该系统中的过渡情景。

Subcritical transitions whereby a dynamical system abruptly transitions from an ordered, invariant state to unsteady, disordered dynamics have intrigued scientists for over a century. Although the ordered basic state of the system is stable to infinitesimal perturbations, above a threshold value of the driving parameter, finite amplitude perturbations can trigger a sudden change to complex, continuously evolvingdynamics. In the seminal interpretation of the transition to turbulence in shear flows, excursions from the stable state correspond to the transient exploration of the stable manifolds of weakly unstable states of the system or so-called edge states, which determine the basin boundary separating initial conditions decaying to laminar flow from those growing to turbulence.The proposed research will explore whether this dynamical scenario applies to a broader class of systems exhibiting subcritical transitions to disorder. A prime candidate is the two-phase displacement flow in a rectangular (Hele-Shaw) channel which has the key advantage that the only nonlinearity arises due to the presence of the interface between the two fluids so that different states can be distinguished using only the interfacial morphology. Moreover, the behaviour of the system can be described by a depth-averaged set of equations that is more amenable to analysis than the full Navier-Stokes equations required to describe shear flow. In order to pursue the idea that when the driving flux is sufficiently large the complex time-dependent behaviour of the propagating front can be interpreted as an exploration of the stable manifolds of edge states, we need to establish whether these perturbation-driven excursions are always transient or become self-sustaining above a threshold flux. In other words, is a localised perturbation sufficient to trigger disorder or are spatially-distributed perturbations required to generate long-term disordered pattern formation? We propose to answer this question with a systematic experimental investigation in our large-scale experimental facility funded by EPSRC project grant EP/P026044/1 by testing the response of the steadily propagating Saffman-Taylor finger for a wide range of driving fluxes to both localised and spatially-distributed perturbations. The findings from the experiments will inform the model, which will in turn accelerate the identification of the transition scenario in this system.