Using experiments and a depth-averaged numerical model, we study instabilities of two-phase flows in a Hele-Shaw channel with an elastic upper boundary and a non-uniform cross-section prescribed by initial collapse. Experimentally, we find increasingly complex and unsteady modes of air-finger propagation as the dimensionless bubble speed $Ca$ and level of collapse are increased, including pointed fingers, indented fingers and the feathered modes first identified by Cuttle et al. (J. Fluid Mech., vol. 886, 2020, A20). By introducing a measure of the viscous contribution to finger propagation, we identify a $Ca$ threshold beyond which viscous forces are superseded by elastic effects. Quantitative prediction of this transition between ‘viscous’ and ‘elastic’ reopening regimes across levels of collapse establishes the fidelity of the numerical model. In the viscous regime, we recover the non-monotonic dependence on $Ca$ of the finger pressure, which is characteristic of benchtop models of airway reopening. To explore the elastic regime numerically, we extend the depth-averaged model introduced by Fontana et al. (J. Fluid Mech., vol. 916, 2021, A27) to include an artificial disjoining pressure that prevents the unphysical self-intersection of the interface. Using time simulations, we capture for the first time the majority of experimental finger dynamics, including feathered modes. We show that these disordered states evolve continually, with no evidence of convergence to steady or periodic states. We find that the steady bifurcation structure satisfactorily predicts the bubble pressure as a function of $Ca$ , but that it does not provide sufficient information to predict the transition to unsteady dynamics that appears strongly nonlinear.

中文翻译：

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手指在弹性 Hele-Shaw 通道中脱皮

通过实验和深度平均数值模型，我们研究了 Hele-Shaw 河道中两相流的不稳定性，该河道具有弹性上边界和由初始塌陷规定的非均匀横截面。通过实验，我们发现随着无量纲气泡速度的增加，气指传播模式变得越来越复杂和不稳定 加元$ 塌陷程度增加，包括尖指、锯齿状指和 Cuttle 首先识别的羽化模式等人。（J.流体机械。，卷。 886, 2020, A20）。通过引入对手指传播的粘性贡献的测量，我们确定了 加元$ 超过该阈值，粘性力将被弹性效应取代。对跨塌陷级别的“粘性”和“弹性”重新开放状态之间的转变的定量预测建立了数值模型的保真度。在粘性状态下，我们恢复了对 加元$ 手指压力，这是气道重新开放的台式模型的特征。为了从数值上探索弹性状态，我们扩展了 Fontana 引入的深度平均模型等人。（J.流体机械。，卷。 916, 2021, A27) 包括人工分离压力，以防止界面非物理自相交。通过时间模拟，我们首次捕获了大部分实验手指动态，包括羽化模式。我们证明这些无序状态不断演化，没有证据表明它们会收敛到稳定或周期性状态。我们发现稳定分叉结构令人满意地预测了气泡压力作为以下函数的函数 加元$ ，但它没有提供足够的信息来预测向呈现强烈非线性的非稳态动力学的转变。