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

Abstract

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 rigorous 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 flux constant $C$ is the eigenvalue and the corresponding self-similar pattern $\mathbf{x}$ 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. The influence of initial guesses on the self-similar patterns is investigated. We are able to obtain a desired self-similar shape once the initial guess is properly chosen. Our results go beyond the predictions of linear theory and establish a bridge between the linear theory and simulations.

Figures (8)

• Figure 1: A schematic diagram for an air-oil interface system.
• Figure 2: The difference of flux constants between the linear theory and nonlinear theory for (a) 3-fold dominant self-similar shapes and (b) 4-fold dominant self-similar shapes.
• Figure 3: Flux constants of self-similar shapes and selected morphologies. Flux constants from linear theory (solid line) are given by $C = \frac{k(k^2-1)}{k-2}$. Symbols denote the nonlinear results. The dashed line represents the best fit for the nonlinear self-similar shapes with a large shape factor $\delta/R$, $C = \frac{k(k^{1.939}-1)}{k-2}$ ($k\ge 4$).
• Figure 4: The effect of the initial guess of a single mode on (a) the flux constant and (b) the shape factor $\delta/R$. We set $C_0 = 30$. The different nonlinear simulations are obtained by varying $\Tilde{\delta}_4$ in the initial guess.
• Figure 5: The effect of the initial guess with mixed modes on the flux constant. We set $C_0 = 50$ and $\Tilde{\delta}_5 = 0.1$. The different nonlinear simulations are obtained by varying $\Tilde{\delta}_6$.