Injection of bubbles in a quiescent inviscid liquid under a uniform electric field

Abstract

Numerical computations and order of magnitude estimates are presented for the periodic generation and coalescence of bubbles due to the injection of a constant flow rate of a gas through a circular orifice at the bottom wall of an inviscid dielectric or very polar liquid that is at rest and subject to a uniform vertical electric field far from the orifice. The problem depends on five dimensionless parameters: a Bond number based on the radius of the orifice; Weber and electric Bond numbers whose square roots are dimensionless measures of the flow rate of gas and the applied electric field; the dielectric constant of the liquid; and the contact angle of the liquid with the bottom wall. The bubbles that grow quasi-statically at the orifice for small values of the Weber number are always elongated vertically by the electric stress that acts on their surface when an electric field is applied. The volume of these bubbles at detachment may reach a maximum at a certain value of the electric Bond number, if the Bond number is sufficiently small, or decrease monotonically with the electric Bond number if the Bond number is larger. In both cases the bubbling ceases to be periodic beyond a certain value of the electric Bond number, apparently giving way to more complex bubbling regimes, which are not investigated here. Bubble interaction and eventually coalescence occur when the Weber number is increased keeping the electric Bond number in the range of periodic bubbling. Different periodic regimes are described. It is shown that a moderate electric field may increase the value of the Weber number above which coalescence occurs without changing the shape of the bubbles much. A large electric field may suppress coalescence but it also favours the development of upward and downward jets that cross the bubbles and may cause their breakdown.