An eigenvalue problem for self-similar patterns in Hele-Shaw flows (Q6629738)

A nonlinear model is given for the flow of two immiscible fluids in a radial Hele-Shaw cell. Thus the authors explain some self-similar patterns (the space-time variables of the interface are separate). The new element is to represent the pressure in the points \(\mathbf{x}\) of the interface \(\Gamma\) through a single-layer and double-layer potentials. Using this, two integral operators \(M(\mathbf{x}), G(\mathbf{x})\) are introduced in the formulas (25)--(26), containing the high-order derivatives of the interface and some singular integrals. The flow is described by the generalized eigenvalue problem \(M(\mathbf{x})\) \(+C_f G(\mathbf{x}) =0\), where the ``eigenvalue'' \(C_f\) is the nonlinear flux constant. A quasi-Newton method is introduced, by using a combination of \textit{cosine} Fourier modes for representing \(\Gamma\), and a left-hand side discretization of the above integral operators. Different numerical experiments are performed, which highlight the important effect of the viscosity ratios. An important result is the existence of nonlinear, noncircular self-similarly interfaces. The authors study the behaviour of the quasi-Newton algorithm in terms of the initial guess for \(C_f\). Some interesting comparisons with the results in the linear case are also given. The paper is an important step forward for understanding possible Hele-Shaw flows, with applications in physical, biological, and engineering systems.