Determination of the characteristic length of electrosprays operating in the cone-jet mode

Gamero-Castaño and colleagues have reported that a large number of calculated shapes for electrified cone jets collapse into a nearly universal geometry when scaled with a characteristic length $R_G$ previously introduced by Gañán-Calvo et al. ( J. Aerosol Sci. , vol. 25, 1994, pp. 1121–1142). The theoretical reasons for that unexpected success were, however, unclear. Recently, Pérez-Lorenzo & Fernández de la Mora ( J. Fluid Mech. , vol. 931, 2022, A4) have noted that a slightly different length scale $L_j$ is suggested by the asymptotic jet structure inferred by Gañán-Calvo ( Phys. Rev. Lett. , vol. 79, 1997, pp. 217–220) from energy conservation and the hypothesis that the asymptotic electric field is that given by Taylor’s static model. This article aims to identify which of these two scales best collapses calculated cone-jet structures, and whether there is an alternative superior one. The characteristic lengths are tested against a large set of numerical solutions of a cone-jet model. The effectiveness of each scaling is determined through analyses based on the standard deviation of the numerical solutions. Despite the slight difference between $R_G$ and $L_j$ , this analysis clearly identifies $L_j$ as the most accurate scaling for all cone-jet parameters tested. Differentiating between both scales would not have been possible with experimental measurements, but requires the use of high-fidelity numerical solutions. Surprisingly, the success of $L_j$ is not limited to the jet region, but extends to the cone and the neck. These findings provide a slightly superior scaling enjoying a considerably firmer theoretical basis.