Long-wave equation for a confined ferrofluid interface: periodic interfacial waves as dissipative solitons
Long-wave equation for a confined ferrofluid interface: periodic interfacial waves as dissipative solitons
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DOI：
10.1098/rspa.2021.0550
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发表时间：
2021-05
期刊：
影响因子：
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通讯作者：
Zongxin Yu;I. Christov
中科院分区：
文献类型：
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作者：
Zongxin Yu;I. Christov
本文研究了在倾斜非均匀磁场作用下铁磁流体薄膜的动力学特性。结果表明，铁磁流体与非粘性外流体（空气）之间的界面支持行波，由在长波近似下导出的新的修正kuramoto - sivashinsky型方程控制。在这个长波方程中，能量产生和耗散之间的平衡允许耗散孤子的存在。这些永久行波的传播速度和剖面形状可以通过外部磁场进行调节。通过多尺度分析得到了对传播速度线性预测的修正，揭示了非线性是如何抑制线性不稳定性的。所发现的行周期界面波被识别为能量相平面上的不动点。结果表明，状态（波剖面）之间会发生转换。这些跃迁可以通过行波的谱稳定性来解释。有趣的是，在模型长波方程下也发现了多周期波的传播，多周期波是双余弦波的不可积模拟。对这些多周期解进行了数值研究，发现它们是长寿命的瞬态，但最终会突然转变为确定的稳定周期态之一。
We study the dynamics of a ferrofluid thin film confined in a Hele-Shaw cell, and subjected to a tilted non-uniform magnetic field. It is shown that the interface between the ferrofluid and an inviscid outer fluid (air) supports travelling waves, governed by a novel modified Kuramoto–Sivashinsky-type equation derived under the long-wave approximation. The balance between energy production and dissipation in this long-wave equation allows for the existence of dissipative solitons. These permanent travelling waves’ propagation velocity and profile shape are shown to be tunable via the external magnetic field. A multiple-scale analysis is performed to obtain the correction to the linear prediction of the propagation velocity, and to reveal how the nonlinearity arrests the linear instability. The travelling periodic interfacial waves discovered are identified as fixed points in an energy phase plane. It is shown that transitions between states (wave profiles) occur. These transitions are explained via the spectral stability of the travelling waves. Interestingly, multi-periodic waves, which are a non-integrable analogue of the double cnoidal wave, are also found to propagate under the model long-wave equation. These multi-periodic solutions are investigated numerically, and they are found to be long-lived transients, but ultimately abruptly transition to one of the stable periodic states identified.