Modelling the suppression of viscous fingering in elastic-walled Hele-Shaw cells
Modelling the suppression of viscous fingering in elastic-walled Hele-Shaw cells
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DOI：
10.1017/jfm.2013.375
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发表时间：
2013-09-01
影响因子：
3.7
通讯作者：
Heil, Matthias
中科院分区：
文献类型：
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作者：
Pihler-Puzovic, Draga;Perillat, Raphael;Heil, Matthias
Pihler-Puzovic等人最近的实验。启。（vol. 108, 2012, article 074502）的研究表明，在气泡取代粘性流体的圆形Hele-Shaw细胞中，当细胞的顶部边界被弹性膜取代时，粘性指法的发生大大延迟。非轴对称不稳定性仅在更大的流速下观察到，并且发展的大振幅指与刚性壁细胞中高度分支的指有着根本的不同。我们基于一个理论模型，将流体流动的深度平均润滑方程与描述弹性膜变形的Foppl-von Karman方程耦合在一起，利用线性稳定性分析和直接数值模拟相结合的方法解释了抑制不稳定性的机制。我们表明，流固耦合作用主要通过对轴对称基流的两种改变来影响不稳定性：在任何不稳定性发生之前，膜的轴对称膨胀减缓了气泡的膨胀，并迫使气液界面传播到一个收敛的充满流体的间隙。这两种变化都减少了驱动刚性壁细胞指动不稳定性的不稳定粘性效应。相比之下，毛细效应在抑制不稳定性方面只起很小的作用。
Recent experiments by Pihler-Puzovic et al. (Phys. Rev. Lett., vol. 108, 2012, article 074502) have shown that the onset of viscous fingering in circular Hele-Shaw cells in which an air bubble displaces a viscous fluid is delayed considerably when the top boundary of the cell is replaced by an elastic membrane. Non-axisymmetric instabilities are only observed at much larger flow rates, and the large-amplitude fingers that develop are fundamentally different from the highly branched fingers in rigid-walled cells. We explain the mechanism for the suppression of the instability using a combination of linear stability analysis and direct numerical simulations, based on a theoretical model that couples a depth-averaged lubrication equation for the fluid flow to the Foppl-von Karman equations, which describe the deformation of the elastic membrane. We show that fluid-structure interaction affects the instability primarily via two changes to the axisymmetric base flow: the axisymmetric inflation of the membrane prior to the onset of any instabilities slows down the expansion of the air bubble and forces the air-liquid interface to propagate into a converging fluid-filled gap. Both of these changes reduce the destabilizing viscous effects that drive the fingering instability in a rigid-walled cell. In contrast, capillary effects only play a very minor role in the suppression of the instability.