New solutions for interfacial capillary waves of permanent form
New solutions for interfacial capillary waves of permanent form
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DOI：
10.1017/jfm.2022.882
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发表时间：
2022-11
影响因子：
3.7
通讯作者：
X. Guan;J. Vanden-Broeck;Z. Wang
中科院分区：
文献类型：
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作者：
X. Guan;J. Vanden-Broeck;Z. Wang
摘要研究了平行于未扰动界面的刚性壁通道中两种均匀流体界面上的前进毛细波。这个问题被制定为一个系统的积分微分方程，可以通过边界积分方程方法加上未知函数的级数展开数值求解。利用这种高精度格式和数值延拓，我们研究了周期行波的全局分支。结果表明，存在两种类型的极限剖面，即自相交型和边界接触型，它们要么沿着由无穷小周期波分叉的主分支出现，要么出现在存在于某一有限振幅以上的孤立分支上。对于特定的参数集，这两种类型的分叉曲线可以相交，这可以被视为发生在主分支上的二次分叉现象。通过对几乎极限波的渐近分析和数值分析，发现边界接触解具有圆形几何特征，即界面由相同半径的圆弧拼接而成。理论研究产生了这些极端波存在的必要条件，从而我们可以预测大多数参数集的极限配置。理论预测和数值计算结果之间的比较显示出良好的一致性。
Abstract Progressive capillary waves on the interface between two homogeneous fluids confined in a channel with rigid walls parallel to the undisturbed interface are investigated. This problem is formulated as a system of integrodifferential equations that can be solved numerically via a boundary integral equation method coupled with series expansions of the unknown functions. With this highly accurate scheme and numerical continuation, we explore the global bifurcation of periodic travelling waves. It is found that there are two types of limiting profile, self-intersecting and boundary-touching, which appear either along a primary branch bifurcating from infinitesimal periodic waves or on an isolated branch existing above a certain finite amplitude. For particular sets of parameters, these two types of bifurcation curves can intersect, which can be viewed as a secondary bifurcation phenomenon occurring on the primary branch. Based on asymptotic and numerical analyses of the almost limiting waves, it is found that the boundary-touching solutions feature a circular geometry, i.e. the interface is pieced together by circular arcs of the same radius. A theoretical investigation yields the necessary conditions for the existence of these extreme waves, whereby we can predict the limiting configurations for most parameter sets. The comparisons between theoretical predictions and numerical results show good agreement.