In a recent paper [D. T. Papageorgiou and P. G. Petropoulos, J. Eng. Math. 50, 223 (2004)] we considered the linear stability of a two-dimensional incompressible leaky dielectric viscous liquid sheet surrounded by a hydrodynamically passive conducting medium, when an electric field is applied parallel to the initially flat bounding fluid interfaces. It was established that for order-one Reynolds numbers and when the dielectric permittivity ratio, εp =εin/εout, and the electric conductivity ratio, σR=σout/ /σin, satisfy εpσR >1 the flow is linearly stable in the absence of an electric field. When a field is present a band of unstable long waves emerges whose size increases as the field increases (the band remains finite, that is short waves are stable, for large fields). In the present study we consider the nonlinear dynamics in the vicinity of the zero electric field bifurcation. The scalings determined from the linear stability calculations are used to derive canonical strongly nonlinear evolution equations for the leading order shape of the sheet and the corresponding horizontal velocity. Numerical simulations indicate that for a wide class of initial conditions, a quasisteady state is reached in the long time when the layer organizes into a number of lobes connected by slowly draining threads whose height vanishes asymptotically in time. The number of lobes and their volumes depend on initial conditions. Using this insight, we construct an ordinary differential equation which describes the shape of the sheet in the limit t → ∞.
All Science Journal Classification (ASJC) codes
- Computational Mechanics
- Condensed Matter Physics
- Mechanics of Materials
- Mechanical Engineering
- Fluid Flow and Transfer Processes