Transient two-phase liquid-liquid flows of Newtonian fluid droplets in a primary Newtonian fluid with different viscosities, through constricted channels and double-bend channels, are numerically modelled and studied. The deformation and migration of various arrangements (arrays) of droplets, as they are carried by the primary phase into the low Reynolds number (Re), developing Hagen-Poiseuille laminar flow, is computed and discussed on the basis of the non-dimensional numbers: Capillary number (Ca), Reynolds number (Re), Weber number (We) and viscosity ratio (λ). The observations are well compared with the results of the literature. To predict the contours of droplets, we implement a level-set method (LSM) for capturing interfaces with high order WENO and TVD Runge-Kutta schemes. The LSM implicitly provides the conditions for the revised, nearly second-order, two-phase, time-dependent, 2-D Navier-Stokes momentum equations' solver, incorporating viscous and surface tension terms depending on the level-set function.

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