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

Abstract

Droplet migration in a Hele-Shaw cell is a fundamental multiphase flow problem which is crucial for many microfluidics applications. We focus on the regime at low capillary number and three-dimensional direct numerical simulations are performed to investigate the problem. In order to reduce the computational cost, an adaptive mesh is employed and high mesh resolution is only used near the interface. Parametric studies are performed on the droplet horizontal radius and the capillary number.For droplets with an horizontal radius larger than half the channel height the droplet overfills the channel and exhibits a pancake shape. A lubrication film is formed between the droplet and the wall and particular attention is paid to the effect of the lubrication film on the droplet velocity. The computed velocity of the pancake droplet is shown to be lower than the average inflow velocity, which is in agreement with experimental measurements. The numerical results show that both the strong shear induced by the lubrication film and the three-dimensional flow structure contribute to the low mobility of the droplet. In this low-migration-velocity scenario the interfacial flow in the droplet reference frame moves toward the rear on the top and reverses direction moving to the front from the two side edges. The velocity of the pancake droplet and the thickness of the lubrication film are observed to decrease with capillary number. The droplet velocity and its dependence on capillary number cannot be captured by the classic Hele-Shaw equations, since the depth-averaged approximation neglects the effect of the lubrication film.

Numerical Models of Surface Tension

“…It is thus not obvious that the relaxed Stokes criterion is beneficial for these applications, unless one considers significantly more viscous fluids and/or smaller channels. In practice, explicit surface tension stability is thus a stringent constraint for microfluidics applications, see Ling et al (2016) for a detailed discussion.…”

Selection of a Hele-Shaw Bubble via Exponential Asymptotics

“…We expand g(η) as the local series g(η) = ∞ n=0 g n η 3/2+7r/2 (36) and substitute this expression into (35) to recursively obtain g n for arbitrarily large values of n.…”

Predicting droplet velocity in a Hele-Shaw cell

“…The important development of droplet-based microfluidics these last ten years has rekindled the interest in the study of droplets and bubbles dynamics in micro-channels [1][2][3][4][5].…”

“…This last case and its extension to viscous droplets have also been investigated using direct 3D numerical simulations [12], but for droplet velocities compelled to higher values than our experimental ones. Indeed, the very high aspect ratio between the wetting film thickness and the channel height at small droplet velocity observed experimentally [13] prevents from solving the flow using direct numerical simulation in the entire flow domain [2,12].…”