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Stability of lubricated viscous gravity currents. Part 2. Global analysis and stabilisation by buoyancy forces

机译：润滑粘性重力电流的稳定性。 第2.浮力势力的全球分析和稳定

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摘要

The novel viscous fingering instability recently found in the experiments of Kowal & Worster (J. Fluid Mech., vol. 766, 2015, pp. 626-655), involving two superposed currents of viscous fluid, has been shown to originate at the lubrication front when the fluids are of equal density. However, when the densities are unequal, additional buoyancy forces associated with the underlying layer act to suppress this instability and are largest at the lubrication front, which is where the instability originates. In this paper, we investigate the interaction between the mechanism of the instability and the stabilising influence of these buoyancy forces by performing a global and fully time-dependent analysis, which does not use the frozen-time approximation. We determine a critical condition for instability in terms of the viscosity ratio and the density difference between the two layers. Consistently with the local analysis of the companion paper, instabilities occur when the jump in hydrostatic pressure gradient across the lubrication front is negative, or, equivalently, when the intruding fluid is less viscous than the overlying fluid, provided the two fluids are of equal densities. Once there is a non-zero density difference, these driving buoyancy forces suppress the instability for large wavelengths, giving rise to wavelength selection. As the density difference increases, the instability criterion requires higher viscosity ratios for any instability to occur, and the band of unstable wavenumbers becomes bounded. Large enough density differences suppress the instability completely.
机译：最近发现的新型粘性手指不稳定性在kowal和Lamster（J. Fluid Mech。，Vol.766,2015，第626-655页）中涉及两个粘性流体的叠加电流，已被证明源于润滑当流体具有相同的密度时，前面。然而，当密度不等时，与底层的额外浮力力与潜在的层行为相关，以抑制这种不稳定性，并且在润滑前是最大的，这是不稳定性起源的地方。在本文中，我们通过进行全局和完全依赖的分析来研究不稳定性的机制与这些浮力力稳定影响的相互作用，这不会使用冻结时间近似。在粘度比和两层之间的密度差方面，我们确定不稳定性的临界条件。始终如一地与伴随纸张的局部分析，当润滑前的液压压力梯度跳跃为负时，或者等于时，当侵入流体粘稠的粘性比上覆流体较小时，所以提供了两个流体，所以两个流体具有相等的密度。一旦存在非零的密度差，这些驱动浮力力抑制了大波长的不稳定性，从而产生波长选择。随着密度差的增加，不稳定标准需要较高的粘度比以发生任何不稳定性，并且不稳定的波数的频带变得界定。足够大的密度差异完全抑制不稳定。

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