Viscous fingering under elastic membranes

在这里提出的研究是由最近惊人的和意想不到的观察到的粘指不稳定性的抑制弹性壁Hele-Shaw细胞。当包含在两个刚性板之间狭窄间隙中的薄流体层被粘性较低的流体取代时，树枝状图案很容易形成，这是前传播，图案形成现象的原型。我们已经观察到，束缚流体的板块的弹性变形对这种不稳定性的开始和非线性发展具有显著的影响。具体来说，在弹性壁Hele-Shaw细胞中，其中一个边界板被乳胶膜取代，界面保持轴对称，当注入速率达到一定值时，刚性系统已经表现出强烈的非线性界面生长。弹性壁体系轴对称膨胀界面失稳的临界注入速率约为刚性壁体系的1000倍。壁弹性不仅影响不稳定的开始，而且对手指随后发展的结构也有很强的影响。此外，如果沿边界板的流体压力变化足够大（相对于板的刚度），流体载荷可能导致板屈曲（或起皱），从而导致两种基于流体和固体的不稳定性之间的强烈相互作用。我们建议采用实验、理论和计算方法相结合的方法，以提供对这些新型弹性流体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.