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Droplet migration in a Hele--Shaw cell is a fundamental multiphase flowproblem which is crucial for many microfluidics applications. We focus on theregime at low capillary number and three-dimensional direct numericalsimulations are performed to investigate the problem. In order to reduce thecomputational cost, an adaptive mesh is employed and high mesh resolution isonly used near the interface. Parametric studies are performed on the droplethorizontal radius and the capillary number. For droplets with an horizontalradius larger than half the channel height the droplet overfills the channeland exhibits a pancake shape. A lubrication film is formed between the dropletand the wall and particular attention is paid to the effect of the lubricationfilm on the droplet velocity. The computed velocity of the pancake droplet isshown to be lower than the average inflow velocity, which is in agreement withexperimental measurements. The numerical results show that both the strongshear induced by the lubrication film and the three-dimensional flow structurecontribute to the low mobility of the droplet. In this low-migration-velocityscenario the interfacial flow in the droplet reference frame moves toward therear on the top and reverses direction moving to the front from the two sideedges. The velocity of the pancake droplet and the thickness of the lubricationfilm are observed to decrease with capillary number. The droplet velocity andits dependence on capillary number cannot be captured by the classic Hele--Shawequations, since the depth-averaged approximation neglects the effect of thelubrication film.

Droplet migration in a Hele-Shaw cell: Effect of the lubrication film on the droplet dynamics