Electrohydrostatically driven flow and instability in a vertical hele-shaw cell

The electrohydrostatic capillary-driven flow of a viscous poorly conducting Newtonian fluid rising between conducting parallel plates is studied both theoretically and experimentally. By scaling the problem with a pressure and time derived by considering Maxwell stresses along the interface, it is determined that the dimensionless parameters governing the flow are the hydrostatic bond (Bo(H)), electrostatic bond (Bo(E)) and electrostatic Reynolds (Re(E)) numbers. A lubrication theory analysis, in the limit Re(E) --> 0, of the momentum balance leads to an analytical solution for the elapsed time versus interface position that is analogous to one derived by Washburn (1921) for the capillary pressure-driven flow of a fluid in cylindrical capillaries (Washburn, E. W. Phys. Rev. 1921, 17 (3), 273-283). Experiments are performed using silicone and castor oil at gap spacing less than the capillary length for two ranges of electrostatic Reynolds numbers 0.001 < Re(E) < 0.01 and 10 < Re(E) < 1000. The experimental results for the interface displacement as a function of elapsed time are compared with the theoretical predictions. At large electrostatic Reynolds numbers (>1), a convective instability is observed in plots of the interface position as a function of time. The propagating front also reveals an interfacial instability for large electrostatic Reynolds numbers coupled with large fluid displacements. The theory and experiments for the static rise height show good agreement with theory while the flow dynamics show good qualitative agreement in the applicable limits at low electrostatic Reynolds numbers.