Lift-off of a conducting sessile drop in an electric field
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RESEARCH PAPER
Lift-off of a conducting sessile drop in an electric field A. Glie`re • J.-M. Roux • J.-L. Achard
Received: 7 August 2012 / Accepted: 16 January 2013 / Published online: 13 February 2013 Springer-Verlag Berlin Heidelberg 2013
Abstract A conducting drop in partial wetting regime, placed on the lower electrode of a parallel-plate capacitor and surrounded by a dielectric fluid, is considered. The drop, initially flattened by gravity, is elongated by the electrostatic force and possibly lifts-off when a uniform DC electric field is applied. The electrostatic force and the lift-off condition were calculated in two previous articles, respectively, for undeformable and for slightly deformable drops in the absence of gravity (zero Eo¨tvos number). In this paper, numerical models are put to work to study accurately the complex lift-off process resulting from the competition between gravitational, electrical and capillary forces. Large deformations of the drop surface at any value of the Eo¨tvos number may be addressed by such a numerical procedure. Computational results allow assessing the accuracy and limits of previous analytical and asymptotic relations. Keywords Interface deformation Electric force Wetting hysteresis Drop lift-off Numerical model List of symbols a0 Radius of the spherical drop (m) au Radius of the undeformed spherical cap (m) B, B Buoyancy force (N) Be Electrical Bond number Bce Electrical Bond number threshold for lift-off A. Glie`re (&) J.-M. Roux CEA, LETI, MINATEC Campus, 17 av. des Martyrs, 38054 Grenoble Cedex 9, France e-mail: [email protected] J.-L. Achard Laboratoire des Ecoulements Ge´ophysiques et Industriels, 38041 Grenoble Cedex 9, France
Ble BSc e Bte cf cu g D Eo¨ E? c E1 l E1 Sc E1
F, F F* F*c FS* h p qs R, R Sd Si T Te u V
Electrical Bond number limited by receding angle or p/2 Electrical Bond number threshold for lift-off of spherical drop Turning point electrical Bond number Contact area radius of the drop at contact angle hf (m) Contact area radius of the undeformed spherical cap (m) Acceleration of gravity (m/s2) Deformation rate tensor (s-1) Eo¨tvos number E€o ðpi po Þga20 =c Uniform external electric field (V/m) Electric field threshold for lift-off (V/m) Electric field limited by receding angle or p/2 (V/ m) Electric field threshold for lift-off of spherical drop (V/m) Electrostatic force (N) Nondimensional electrostatic force F : eoa20E2?F* Nondimensional capillary force Nondimensional electrostatic force for spherical drop Dimensionless difference between radii of the interface and a sphere Pressure (subscripts i inside and o outside the drop) (Pa) Surface charge (C) Interaction force between drop and substrate (N) Drop surface (m2) Contact area surface (m2) Stress tensor (Pa) Electric stress tensor (Pa) Velocity field (m/s) Drop volume (m3)
123
208
c d D eo f h ha hf hr hs k l q u
Microfluid Nanofluid (2013) 15:207–218
Surface tension (N/m) Nondimensional contact area radius d : cf/a0 Excess pressure at the drop foot (Pa) Dielectric pe
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