A numerical procedure for scaling droplet deformation in a microfluidic expansion channel

Motivated by recent experiments, deformation and relaxation of a droplet flowing through a narrow channel opening to a planar sudden expansion are studied. Using the boundary element method (BEM), we numerically solve the Darcy equation in the two-dimensional microfluidic channel and investigate droplet motion as the droplet enters the sudden expansion channel. We find two regimes of deformation with a dependency on relative droplet size compared to narrow channel width. A first regime, for droplet smaller than the narrow channel width, is characterized by a deformation affected by the droplet size and capillary number. The second regime, in which droplet larger than the narrow channel width, deformation is characterized by dependence on the capillary number and the width ratio of two channels. Our numerical scalings are in good agreement with reported experimental scalings. Finally, by employing the Fourier analysis method, the relation between the Fourier coefficients and droplet shape is investigated.