Broadley H, Papageorgiou D T
School of Mathematics, University of Manchester, Manchester, M13 9PL UK.
Department of Mathematics, Imperial College London, 180 Queen's Gate, London, SW7 2BZ UK.
J Eng Math. 2022;133(1):6. doi: 10.1007/s10665-021-10182-8. Epub 2022 Mar 9.
Starting from the Euler equations governing the flow of two immiscible incompressible fluids in a horizontal channel, allowing gravity and surface tension, and imposing an electric field across the channel, a nonlinear long-wave analysis is used to derive a system of evolution equations describing the interface position and a modified tangential velocity jump across it. Travelling waves of permanent form are shown to exist and are constructed in the periodic case producing wave trains and the infinite case yielding novel gravity electro-capillary solitary waves. Various regimes are analysed including a hydrodynamically passive but electrically active upper layer, pairs of perfect dielectric fluids and a perfectly conducting lower fluid. In all cases, the presence of the field produces both depression and elevation waves travelling at the same speed, for given sets of parameters. The stability of the non-uniform travelling waves is investigated by numerically solving appropriate linearised eigenvalue problems. It is found that depression waves are neutrally stable whereas elevation ones are unstable unless the surface tension is large. Stability or instability is shown to be linked mathematically to the type of local eigenvalues of the nonlinear flux matrix used to obtain travelling and solitary waves; if these are real (hyperbolic flux matrix), the system is stable, and if they are complex (elliptic), the system is unstable. The latter is a manifestation of Kelvin-Helmholtz instability in electrified flows.
从描述两种不混溶不可压缩流体在水平通道中流动的欧拉方程出发，考虑重力和表面张力，并在通道上施加电场，采用非线性长波分析方法推导了一组演化方程，用于描述界面位置以及跨越界面的修正切向速度跃变。结果表明，在周期性情况下存在并构造出了具有永久形式的行波，产生了波列；在无限情况下则产生了新型的重力电毛细管孤立波。分析了各种情况，包括流体动力学上被动但电动力学上活跃的上层、成对的理想介电流体以及完全导电的下层流体。在所有情况下，对于给定的参数集，电场的存在都会产生以相同速度传播的凹陷波和凸起波。通过数值求解适当的线性化特征值问题，研究了非均匀行波的稳定性。结果发现，凹陷波是中性稳定的，而凸起波是不稳定的，除非表面张力很大。稳定性或不稳定性在数学上与用于获得行波和孤立波的非线性通量矩阵的局部特征值类型相关；如果这些特征值是实的（双曲通量矩阵），系统是稳定的，如果它们是复的（椭圆），系统是不稳定的。后者是带电流动中开尔文 - 亥姆霍兹不稳定的一种表现。
