Nonlinear waves on the surface of a dielectric liquid in a strong tangential electric field

Abstract

The nonlinear dynamics of the free surface of an ideal dielectric liquid in a strong electric field is studied. The equation for the evolution of surface electrohydrodynamic waves is derived in the approximation of small surface-slope angles. It is established that the equation can be solved for liquids with sufficiently high values of the permittivity. This makes it possible to describe the interaction of the counter-propagating waves.

Model Equations for Three-Dimensional Nonlinear Water Waves under Tangential Electric Field

“…Interfacial electrohydrodynamic waves have many important applications in mechanical, chemical, and electrical industries, such as electrospray ionization, cooling systems, coating process, and electrowetting (see [1][2][3][4][5][6][7][8][9][10] and the references therein). The electric stresses have great impact on electrified free surfaces or interfaces, which can not only modify wave patterns but also change stability characteristics of the system.…”

“…The study of the stability of interfacial electrohydrodynamic waves was initiated by Melcher [3] and Taylor and McEwan [13], and the role of interfacial stresses resulting from electrodes was reviewed by Melcher and Taylor [4]. Recent theoretical research in this field has focused on the nonlinear phenomena and corresponding mechanisms, such as nonlinear coherent structures (e.g., [2,5,8,10,11,[14][15][16][17][18][19][20]) and touchdown singularities (e.g., [7,21]). Considerable effects have been put into the modeling and numerical studies of nonlinear interfacial electrohydrodynamic waves.…”

“…The forms of functions ( ) and ( ) depend on the specific problem; for example, ( ) = for normal electric field acting on a perfect conducting fluid ( [2,8]) and ( ) = ( − 1) 2 /( + 1) for a dielectric fluid under a tangential electric field ( [10]). Linearly speaking, a tangential electric field provides a dispersive contribution so as to stabilize the system, while a considerably strong 2 Advances in Mathematical Physics normal electric field can provide energy to a certain range of wavenumbers to induce instability.…”

“…On the contrary, it can be deduced from the work by Gleeson et al [16], Papageorgiou et al [21], Lin et al [2], and Wang [8] that the normal electric field has a destabilizing effect on the interface. N. M. Zubarev and O. V. Zubareva [10,20] and Tao and Guo [22] considered the electrified gas-fluid or vacuumfluid interface so that they can make the assumption that the permittivity of the fluid was much larger compared to that of gas (the permittivities for pure water and air are 80 and 1, resp.). As a consequence, the actual two-layer problem could be reduced to one layer, and the theoretical and numerical techniques developed for free-surface water wave problems can be generalized to include the electric field.…”

“…On the numerical side, most researches focus on periodic waves (see, e.g., [5,7,10,14,21]); however the studies of electrified solitary waves have begun recently. Easwaran [23] first derived the Korteweg-de Vries (KdV) equation in the context of electrohydrodynamic waves which admits sech 2 soliton solutions naturally.…”