Droplet breakup regimes and critical capillary number in a cross-junction microchannels

Abstract A three-dimensional hybrid volume-of-fluid and level-set method is employed to investigate droplet breakup dynamics in a symmetric cross-junction microchannel under low-Reynolds-number conditions ( R e ≪ 1 ), where a single mother droplet is divided into three daughter droplets. Three distinct regimes—non-breakup (NB), tunnel-breakup (TB), and obstructed-breakup (OB)—are identified, and a regime map is constructed to determine the critical conditions for each mode. The transitions between these regimes are examined as functions of the capillary number ( C a ), droplet length ratio ( l / w ), and viscosity ratio ( λ ). The results show that the NB–TB transition depends on both C a and l / w , whereas the TB–OB transition is governed primarily by l / w . Increasing λ shifts the NB–TB transition toward lower C a , indicating that droplets with higher λ break more easily. A modified predictive model for the critical capillary number ( C a cr ) is proposed by extending an existing T-junction correlation to account for the broader viscosity effects in cross-junctions. The new model accurately captures the nonlinear variation of C a cr over a wide range of λ and agrees well with numerical observations. Additionally, the distribution of daughter droplet lengths ( l t / l h ) is quantified: droplets in the straight channel remain the longest, while the relative ratio l