Stability of falling liquid films on flexible substrates
Stability of falling liquid films on flexible substrates
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DOI：
10.1017/jfm.2020.538
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发表时间：
2020-08
影响因子：
3.7
通讯作者：
J. Alexander;Toby L. Kirk;D. Papageorgiou
中科院分区：
文献类型：
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作者：
J. Alexander;Toby L. Kirk;D. Papageorgiou
摘要采用分析和计算方法研究了重力作用下液体膜在倾斜柔性平面上的线性稳定性。采用柔性衬底的一般模型，利用切比雪夫分解对Orr-Sommerfeld问题进行了数值求解。对长波和小雷诺数的渐近极限进行了解析处理，并与计算相联系。对于长波，柔韧性具有破坏稳定的作用，其中临界雷诺数随着刚度的减小而减小，甚至在足够小的刚度下也会破坏斯托克斯流的稳定。为了进一步研究，考虑了斯托克斯流近似，它证实了长波的结果，但也揭示了长波膨胀没有捕捉到的短波不稳定性。增加表面张力对这些不稳定性几乎没有影响，因此它们被表征为壁模态。更广泛的探索揭示了色散关系中的模式切换，具有较高波数的壁面模式切换特性。零雷诺数结果表明，长波极限不足以确定不稳定性，因此寻求任意波数的数值解。Chebyshev tau谱法的实现和对解析解的验证。短波壁的不稳定性在较大的雷诺数下仍然存在，所有雷诺数的不稳定性都可以通过增加壁的柔韧性来实现，但是增加刚度会回到刚性壁的极限。提出了一种能量分解分析，并用于确定显著的不稳定机制，并将它们与物理起源联系起来。
Abstract The linear stability of a liquid film falling down an inclined flexible plane under the influence of gravity is investigated using analytical and computational techniques. A general model for the flexible substrate is used leading to a modified Orr–Sommerfeld problem addressed numerically using a Chebyshev tau decomposition. Asymptotic limits of long waves and small Reynolds numbers are addressed analytically and linked to the computations. For long waves, the flexibility has a destabilising effect, where the critical Reynolds number decreases with decreasing stiffness, even destabilising Stokes flow for sufficiently small stiffness. To pursue this further, a Stokes flow approximation was considered, which confirmed the long-wave results, but also revealed a short wave instability not captured by the long-wave expansions. Increasing the surface tension has little effect on these instabilities and so they were characterised as wall modes. Wider exploration revealed mode switching in the dispersion relation, with the wall and surface mode swapping characteristics for higher wavenumbers. The zero-Reynolds-number results demonstrate that the long-wave limit is not sufficient to determine instabilities so the numerical solution for arbitrary wavenumbers was sought. A Chebyshev tau spectral method was implemented and verified against analytical solutions. Short wave wall instabilities persist at larger Reynolds numbers and destabilisation of all Reynolds numbers is achievable by increasing the wall flexibility, however increasing the stiffness reverts back to the rigid wall limit. An energy decomposition analysis is presented and used to identify the salient instability mechanisms and link them to their physical origin.