Two-phase radial viscous fingering problem in a Hele-Shaw cell with surface tension. I. Classical solvability

We study the problem of radial fingering in immiscible liquid/liquid flow in a Hele-Shaw cell under injection or suction of liquid of less viscosity. The addition of new effects of viscosity and surface tension at the liquid/liquid interface brings our theory into a much better agreement with experiments than other theories. In this paper the classical solvability of the original Hele-Shaw problem (a nonlinear problem with a free boundary for elliptic equations) is established by means of its parabolic regularization with a small parameter ε (> 0) in the time-derivative term and the nonhomogeneous term (the parabolic regularized Hele-Shaw problem) and by vanishing along some subsequence of {ε > 0}. The similar result for a one-phase problem has been already studied in “H. Tani, Classical solvability of the radial viscous fingering problem in a Hele-Shaw cell with surface tension, Sib. J. Pure Appl. Math., 16, 79–92 (2016);” “A. Tani and H. Tani, On the uniqueness of the classical solution of the radial viscous fingering problem in a Hele-Shaw cell with surface tension, J. Appl. Mech. Tech. Phys., 65, No. 5, 178–191 (2024).”

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