Transition to disordered front propagation

一个多世纪以来，一个动力学系统突然从有序的、不变的状态转变为不稳定的、无序的动力学的亚临界转变引起了科学家的兴趣。尽管系统的有序基本状态对无限小的扰动是稳定的，但在驱动参数的阈值以上，有限的振幅扰动可以触发突然变化为复杂的、连续演化的动力学。在剪切流向湍流转变的开创性解释中，从稳定状态的漂移对应于对系统的弱不稳定状态或所谓的边缘状态的稳定流形的瞬时探索，这些状态确定了将衰减到层流的初始条件与增长到湍流的初始条件分开的盆地边界。拟议的研究将探索这一动力学情景是否适用于更广泛的表现为亚临界向无序转变的系统。矩形(Hele-Shaw)流道中的两相置换流是一种主要的候选者，它的主要优点是由于两种流体之间界面的存在而产生唯一的非线性，因此仅通过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的方程来描述，这组方程比描述剪切流所需的完整的Navier-Stokes方程更易于分析。为了追求当驱动通量足够大时，传播锋的复杂的随时间变化的行为可以解释为对边缘状态稳定流形的探索的想法，我们需要确定这些由微扰驱动的漂移总是瞬时的，还是在阈值通量以上变得自我维持的。换句话说，局域扰动是否足以引发无序，或者空间分布的扰动是否需要产生长期的无序图案形成？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，通过测试稳定传播的Saffman-Taylor手指对广泛范围的驱动通量对局部和空间分布扰动的响应。实验的结果将为模型提供信息，这反过来将加快该系统中过渡场景的识别。

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.