Stable electrospraying in the cone-jet mode is restricted to flow rates above a minimum, and understanding the physics of this constraint is important to improve this atomization technique. We study this problem by measuring the minimum flow rate of electrosprays of tributyl phosphate and propylene carbonate at varying electrical conductivity $K$ (all other physical properties such as the density $\unicode[STIX]{x1D70C}$, surface tension $\unicode[STIX]{x1D6FE}$ and viscosity $\unicode[STIX]{x1D707}$ are kept constant and equal to those of the pure liquids), and through the analysis of numerical solutions. The experiments show that the dimensionless minimum flow rate is a function of both the dielectric constant $\unicode[STIX]{x1D700}$ of the liquid and its Reynolds number, $Re=(\unicode[STIX]{x1D70C}\unicode[STIX]{x1D700}_{o}\unicode[STIX]{x1D6FE}^{2}/\unicode[STIX]{x1D707}^{3}K)^{1/3}$. This result is unexpected in the light of existing theories which, for the conditions investigated, predict a minimum flow rate that depends only on $\unicode[STIX]{x1D700}$ and/or is marginally affected by $Re$. The experimental dependency on the Reynolds number requires the viscous stress to be a factor in the determination of the minimum flow rate. However, the numerical solutions suggest that a balance of opposing forces including the fixing viscous stress, which at decreasing flow rates may lower the acceleration of the flow to the point of making it unstable, is unlikely to be the cause. An alternative mechanism is the significant viscous dissipation taking place in the transition from cone to jet, and which at low flow rates cannot be supplied by the work done by the tangential electric stress in the same area. Instead, mechanical energy injected into the system farther downstream must be transferred upstream where dissipation predominantly takes place. This mechanism is supported by the balance between the energy dissipated and the work done by the electric stress in the transition from cone to jet, which yields a relationship between the minimum flow rate, the Reynolds number and the dielectric constant that compares well with experiments.
Surface viscous stress over wind-driven waves with intermittent airflow separation
