Viscous fingering under elastic membranes

这里提出的研究是由最近惊人和意想不到的观察到的对弹性壁Hele-Shaw单元中粘性指进不稳定性的抑制而引起的。当两个刚性板之间的狭窄缝隙中包含的薄液体层被粘性较低的流体置换时，容易形成的树枝状图案是前沿传播和图案形成现象的原型。我们已经观察到，约束流体的板的弹性变形可以对这种不稳定性的开始和非线性发展产生巨大的影响。具体地说，在弹性壁Hele-Shaw池中，其中一个包围板被乳胶膜取代，对于刚性系统已经表现出强烈非线性界面生长的注入速率值，界面保持轴对称。弹性壁系统轴对称膨胀界面失稳的临界注入量约为刚性系统的1000倍。壁弹性不仅影响不稳定的开始，而且对随后发展的手指的结构也有很大的影响。此外，如果沿包围板的流体压力变化足够大(相对于板的刚度)，则流体加载可能会导致板弯曲(或起皱)，从而导致两种基于流体和基于固体的不稳定性之间的强烈相互作用。我们建议采用实验、理论和计算相结合的方法来提供对这些新的弹性流体动力学现象的机制的全面理解。

The research proposed here is motivated by recent striking and unexpected observations of the suppression of a viscous fingering instability in elastic-walled Hele-Shaw cells. The dendritic patterns that readily develop when a thin fluid layer contained in a narrow gap between two rigid plates is displaced by a less viscous fluid, is an archetype for front-propagating, pattern-forming phenomena. We have observed that elastic deformations of the plates that bound the fluid can have a dramatic effect on the onset and nonlinear development of this instability. Specifically, in an elastic-walled Hele-Shaw cell where one of the bounding plates is replaced by a latex membrane, the interface remains axisymmetric for values of the injection rate at which the rigid system already exhibits strongly nonlinear interfacial growth. The critical injection rate beyond which the axisymmetrically expanding interface becomes unstable in the elastic-walled system is approximately 1000 times larger than the corresponding value in the rigid system. Wall elasticity not only affects the onset of the instability but also has a strong effect on the structure of the fingers that develop subsequently. Moreover, if the variations in fluid pressure along the bounding plate are sufficiently large (relative to the plate's stiffness), the fluid loading can cause the plate to buckle (or wrinkle), leading to a strong interaction between two fluid- and solid-based instabilities. We propose to employ a combination of experimental, theoretical and computational approaches to provide a comprehensive understanding of the mechanisms responsible for these novel elasto-hydrodynamic phenomena.