An eigenvalue problem for self-similar patterns in Hele-Shaw flows

Hele-Shaw problems are prototypes to study the interface dynamics. Linear theory suggests the existence of self-similar patterns in a Hele-Shaw flow. That is, with a specific injection flux the interface shape remains unchanged while its size increases. In this paper, we explore the existence of self-similar patterns in the nonlinear regime and develop a nonlinear theory characterizing their fundamental features. Using a boundary integral formulation, we pose the question of self-similarity as a generalized nonlinear eigenvalue problem, involving two nonlinear integral operators. The nonlinear flux constant $C_f$ is the eigenvalue and the corresponding self-similar pattern $\tilde{x}$ is the eigenvector. We develop a quasi-Newton method to solve the problem and show the existence of nonlinear shapes with $k$-fold dominated symmetries. Nonlinear results are compared with the established linear theory, demonstrating a divergence between the two due to non-linear effects absent in the linear stability analysis. Further, we investigate how sensitive the shape of the interface is to the viscosity. Additionally, we conduct numerous numerical experiments using a wide range of initial guesses and initial flux constants. Through these experiments, one is able to obtain a diagram of self-similar shapes and the corresponding flux. It could be used to verify possible self-similar shapes with a proper initial guess and initial flux constant. Our results go beyond the predictions of linear theory and establish a bridge between the linear theory and simulations.