From Phenomenology to Variational Closure: Reinterpreting Electro–Co-Flow Emission Modes Through the Universal Ψ Lagrangian

Classical electro–co-flow emission modes—dripping, microdripping, cone-jet formation, axisymmetric whipping, and fully 3D whipping—are traditionally interpreted through phenomenological balances between viscous, interfacial, and electric stresses. These models, including the benchmark analysis of Fernández-Nieves et al. (Phys. Fluids 31, 082009 (2019)), lack a closed predictive dynamics and rely on empirical scaling laws rather than a variational foundation. This work derives all emission regimes directly from the Universal Ψ Lagrangian, a closed and self-referential variational principle whose second-order operator generates hydrodynamic, scalar, vector, and torsional modes without external parameters. Classical “regimes” correspond to eigenmodes of the universal operator: scalar modes → dripping & microdripping, vector modes → stable cone-jet formation, torsional modes → axisymmetric & 3D whipping instabilities. This establishes a complete variational reinterpretation of electro–co-flow phenomena, revealing classical phenomenology as an effective limit of a parameter-free coherent dynamical structure.