Local bifurcation of electrohydrodynamic waves on a conducting fluid

Abstract

We are concerned with progressive waves propagating on a two-dimensional conducting fluid when a uniform electric field is applied in the direction perpendicular to the undisturbed free surface. The competing effects of gravity, surface tension, and electrically induced forces are investigated using both analytical and numerical techniques for an inviscid and incompressible fluid flowing irrotationally. We simplify the full Euler equations by expanding and truncating the Dirichlet-Neumann operators in the Hamiltonian formulation of the problem. The numerical results show that when the electric parameter is in a certain range, the bifurcation structure near the minimum of the phase speed is rich with Stokes, solitary, generalized solitary, and dark solitary waves. In addition to symmetric solutions, asymmetric solitary waves featuring a multi-packet structure are found to occur along a branch of asymmetric generalized solitary waves that itself bifurcates from Stokes waves of finite amplitude. The detailed bifurcation diagrams, together with typical wave profiles, are presented.

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

“…More than that, Barannyk et al showed in [15] that the tangential electric field can even suppress the RayleighTaylor instability in some situations. 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.).…”

“…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.…”

Phase Dynamics of the Dysthe Equation and the Bifurcation of Plane Waves

“…Such instabilities have been speculated to lead to the formation of rogue waves [34,41], or the decrease in the frequency of the monochromatic wave [28,7,39]. However, even when such waves are stable to an instability like this, they can undergo different morphologies and generate structures such as dark solitary waves [32,24,30], whose mechanism for formation remains unclear. Therefore, even such a heavily studied problem has a wealth of dynamics we have yet to understand.…”