Viscous fingering in a radial elastic-walled Hele-Shaw cell
Viscous fingering in a radial elastic-walled Hele-Shaw cell
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DOI：
10.1017/jfm.2018.404
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发表时间：
2018-06
影响因子：
3.7
通讯作者：
D. Pihler-Puzović;Gunnar G. Peng;J. Lister;M. Heil;A. Juel
中科院分区：
文献类型：
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作者：
D. Pihler-Puzović;Gunnar G. Peng;J. Lister;M. Heil;A. Juel
本文研究了径向Hele-Shaw盒中的粘性指进不稳定性，其中顶部边界被薄弹性片所代替。壁弹性的引入延迟了指进不稳定性的发生，以更大的值的注射流率。此外，当不稳定性发展时，在膨胀的空气-液体界面上形成的指状物是短的和细长的，与在刚性壁细胞中观察到的高度分支的图案形成对比（Pihler-Puzović等人，物理修订信函：第108卷，2012，074502）。我们报告了这个问题的一个全面的实验研究的结果，并比较实验观察到的预测从一个理论模型，该模型是基于雷诺润滑方程的解决方案，再加上Föppl-von-Kármán方程描述的弹性片材的变形。我们进行了线性稳定性分析，研究小振幅非轴对称扰动的演变的时间演变的基流。然后，我们推导出一个简化的模型，利用观察（i）的非轴对称扰动片是非常小的，（ii）扰动的流动主要发生在一个小的楔形区域前面的空气-液体界面。这使我们能够确定各种物理机制，通过这些机制，粘性指进被壁弹性的存在削弱（甚至抑制）。我们表明，小振幅扰动的增长率的理论预测是在良好的一致性与实验观察的注入流量略大于临界流量所需的发病的不稳定。我们还描述了大幅度指进模式，开发在较大的注入流量。我们表明，这些图案的波数仍然是很好的预测的线性稳定性分析，和手指的长度是由当地的几何形状的顺从细胞。
We study the viscous-fingering instability in a radial Hele-Shaw cell in which the top boundary has been replaced by a thin elastic sheet. The introduction of wall elasticity delays the onset of the fingering instability to much larger values of the injection flow rate. Furthermore, when the instability develops, the fingers that form on the expanding air–liquid interface are short and stubby, in contrast with the highly branched patterns observed in rigid-walled cells (Pihler-Puzović et al., Phys. Rev. Lett., vol. 108, 2012, 074502). We report the outcome of a comprehensive experimental study of this problem and compare the experimental observations to the predictions from a theoretical model that is based on the solution of the Reynolds lubrication equations, coupled to the Föppl–von-Kármán equations which describe the deformation of the elastic sheet. We perform a linear stability analysis to study the evolution of small-amplitude non-axisymmetric perturbations to the time-evolving base flow. We then derive a simplified model by exploiting the observations (i) that the non-axisymmetric perturbations to the sheet are very small and (ii) that perturbations to the flow occur predominantly in a small wedge-shaped region ahead of the air–liquid interface. This allows us to identify the various physical mechanisms by which viscous fingering is weakened (or even suppressed) by the presence of wall elasticity. We show that the theoretical predictions for the growth rate of small-amplitude perturbations are in good agreement with experimental observations for injection flow rates that are slightly larger than the critical flow rate required for the onset of the instability. We also characterize the large-amplitude fingering patterns that develop at larger injection flow rates. We show that the wavenumber of these patterns is still well predicted by the linear stability analysis, and that the length of the fingers is set by the local geometry of the compliant cell.