Viscous fingering on soft substrates

这里提出的研究的动机是最近引人注目的观察软基板上的粘性指进不稳定性。如果粘性流体在刚性板之间的窄间隙中被粘性较小的流体置换，则这种不稳定性容易发展，并且通常导致在两种流体之间的界面上的复杂的高度分支的指状图案。粘性指触一直是人们长期以来的研究兴趣，因为它是前方传播模式形成现象的原型，与细菌菌落的生长和火焰前锋的传播一样多种多样。我们发现，这种不稳定性可以通过用软基板代替底部边界板来操纵。首先，我们观察到的不稳定性可以延迟，使接口保持轴对称的注入率在相应的刚性系统中的图案已经表现出非线性增长。这类似于其中流体流与柔性边界（例如，与薄弹性膜）相互作用的其他系统，尽管软基板与那些基板相比变形不同。其次，可以调整软衬底的厚度和刚度以改变指状物的结构，一旦圆形界面变得对非轴对称扰动不稳定，指状物就会发展。由此产生的模式范围从在刚性系统中观察到的高度分支的指进到在以相同速度共同旋转的两个圆柱体之间的弯月面上发展的双指进。因此，不稳定性强烈地受到软基底的变形的影响，软基底又响应于两相流而变形，并且可能具有粘弹性。我们建议采用实验，理论和计算方法相结合，以充分解释复杂的流体-结构相互作用，导致这些新的现象。

The research proposed here is motivated by recent striking observations of the viscous fingering instability on soft substrates. This instability readily develops if a viscous fluid is displaced by a less viscous fluid in a narrow gap between rigid plates, and typically results in complex highly branched patterns of fingers on the interface between the two fluids. Viscous fingering has been of long-standing research interest because it serves as an archetype for front-propagating pattern-forming phenomena, as diverse as the growth of bacterial colonies and the propagation of flame fronts. We find that this instability can be manipulated by replacing the bottom bounding plate with a soft substrate. Firstly, we observe that the instability can be delayed, so that the interface remains axisymmetric for injection rates at which the pattern in the corresponding rigid system already exhibits nonlinear growth. This is similar to other systems in which the fluid flow interacts with a compliant boundary, for example, with thin elastic membranes, although soft substrates deform differently compared to those. Secondly, the thickness and the stiffness of the soft substrate can be tuned to change the structure of the fingers that do develop once the circular interface becomes unstable to non-axisymmetric perturbations. The resulting patterns range from the highly branched fingering observed in rigid systems to the stubby fingers that develop on the meniscus between two cylinders co-rotating with the same speed. Thus, the instability is strongly influenced by the deformation of the soft substrate, which in turn deforms in response to the two-phase flow, and is likely to have viscoelastic properties. We propose to employ a combination of experimental, theoretical and computational approaches to fully-characterise the complex fluid-structure interactions that lead to these novel phenomena.