  In this paper we present a comprehensive study of the step-emulsification
process for high-throughput production of (sub-)$\mu$m-size monodisperse
droplets. The microfluidic device combines a Hele-Shaw nanofluidic cell with a
step-like outlet to a deep and wide reservoir. The proposed theory based on
Hele-Shaw hydrodynamics provides the quasi-static shape of the free boundary
between the disperse liquid phase engulfed by the co-flowing continuous phase
prior to transition to oscillatory step-emulsification at low enough capillary
number. At the transition the proposed theory anticipates a simple condition
for critical capillary number as a function of the Hele-Shaw cell geometry. The
transition threshold is in excellent agreement with experimental data. A simple
closed-form expression for the size of the droplets generated in
step-emulsification regime derived using simple geometric arguments also shows
a very good agreement with the experimental results.
