Governing Equations

This chapter presents the equations framing the tip streaming phenomenon. These equations provide the theoretical framework to analyze the microfluidic applications studied in other chapters. We consider simple models for describing the effects of viscoelasticity, surfactants, and externally applied electric fields. In this latter case, we introduce the frequently used charge-conservative approximation and leaky-dielectric model, which allow for examining electrohydrodynamic processes in a relatively simple way. The relationship between these approaches and the full electrokinetic model is discussed. We introduce some rheological models to obtain the polymer stress in dilute viscoelastic liquids. We present the equations describing the transport and conservation of surfactants in the hydrodynamic bulk and interface, simple kinetic models to account for the adsorption-desorption processes, and commonly used equations of state to calculate the interfacial tension as a function of the surfactant surface density. The surface viscous stresses caused by a Newtonian surfactant monolayer are considered as well. The chapter closes with the one-dimensional (1D) approximation for slender configurations, such as jets with relatively small spatial accelerations.