We study how a suspended liquid film is deformed by an external flow en route to forming a bubble through experiments and a model. We identify a family of nonminimal but stable equilibrium shapes for flow speeds up to a critical value beyond which the film inflates unstably, and the model accounts for the observed nonlinear deformations and forces. A saddle-node or fold bifurcation in the solution diagram suggests that bubble formation at high speeds results from the loss of equilibrium and at low speeds from the loss of stability for overly inflated shapes.

A variational model for the interaction between homogenization and phase separation is considered in the regime where the former happens at a smaller scale than the latter. The first order Γ−limit is proven to exhibit a separation of scales which has only been previously conjectured.

In this paper, a model for defects that was introduced in [ZANV21] is studied. In the literature, the setting of most models for defects is the function space SBV (special bounded variation functions) (see, e.g., [CGO15, GMPS21]). However, this model regularizes the director field to be in a Sobolev space by adding a second field to incorporate the defect. A relaxation result in the case of fixed parameters is proven along with some partial compactness results.

In this paper, a model for defects in nematic liquid crystals that was introduced in Zhang et al. (Physica D Nonlinear Phenom 417:132828, 2021) is studied. In the literature, the setting of many models for defects is the function space SBV (special functions of bounded variation). However, the model considered herein regularizes the director field to be in a Sobolev space by introducing a second vector field tracking the defect. A relaxation result in the case of fixed parameters is proved along with some partial compactness results as the defect width vanishes.