Barannyk, LL; Papageorgiou, DT; Petropoulos, PG; Vanden-Broeck, J-M; (2015) Nonlinear Dynamics and Wall Touch-Up in Unstably Stratified Multilayer Flows in Horizontal Channels under the Action of Electric Fields. SIAM Journal on Applied Mathematics , 75 (1) pp. 92-113. 10.1137/140968070.

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Abstract

This study considers the nonlinear dynamics of stratified immiscible fluids when an electric field acts perpendicular to the direction of gravity. A particular setup is investigated in detail, namely, two stratified fluids inside a horizontal channel of infinite extent. The fluids are taken to be perfect dielectrics, and a constant horizontal field is imposed along the channel. The sharp interface separating the two fluids may or may not support surface tension, and the Rayleigh--Taylor instability is typically present when the heavier fluid is on top. A novel system of partial differential equations that describe the interfacial position and the leading order horizontal velocity in the fluid layers is studied analytically and computationally. The system is valid in the asymptotic limit of one layer being asymptotically thin compared to the second fluid layer, and as a result nonlocal electrostatic terms arise due to the multiscale nature of the physical setup. The initial value problem on spatially periodic domains is solved numerically, and it is shown that a sufficiently strong electric field can linearly stabilize the Rayleigh--Taylor instability to produce nonlinear quasi-periodic oscillations in time that are quite close to standing waves. In situations when the instability is present, the system is shown to generically evolve to touch-up singularities with the interface touching the upper wall in finite time while the leading order horizontal velocity blows up. Accurate numerical solutions allied with asymptotic analysis show that the terminal states follow self-similar structures that are different if surface tension is present or absent, but with the electric field present. In the presence of surface tension, the touch-up is found to take place with bounded interfacial gradients but unbounded curvature, with electrostatic effects relegated to higher order. If surface tension is absent, however, the electric field supports touch-up with a local cusp structure so that the interfacial gradients themselves are unbounded. The self-similar solutions are of the second kind and extensive simulations are used to extract the scaling exponents. Distinct and independent methods are described and implemented, and agreement between them is excellent.

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