Abstract
We study the time evolution of the free boundary of a viscous fluid for planar flows in Hele‐Shaw cells under injection. Applying methods from the theory of univalent functions, we prove the invariance in time of Φ‐likeness property (a geometric property which includes starlikeness and spiral‐likeness) for two basic cases: the inner problem and the outer problem. We study both zero and nonzero surface tension models. Certain particular cases are also presented.
Highlights
The time evolution of the free boundary of a viscous fluid for planar flows in Hele-Shaw cells under injection was studied by many authors
Starting with an initial bounded domain Ω 0 which is Φ-like, we prove that at each moment t ∈ 0, T the domain Ω t is Φ-like both for zero and nonzero surface tension models
The conclusion is immediate from the smoothness of the classical solution of 1.6
Summary
The time evolution of the free boundary of a viscous fluid for planar flows in Hele-Shaw cells under injection was studied by many authors. Let f ζ, t be the classical solution of the Polubarinova-Galin equation 1.1 with the initial condition f ζ, 0. The classical solution of the Polubarinova-Galin equation 1.1 with the initial condition f ζ, 0 f0 ζ is spiral-like of type α for t ∈ 0, T , where T is the blow-up time. The following result is a generalization of 6, Theorem 1 to the case of Φ-like functions. Φ-like on U and univalent on U, there exists t γ ≤ T such that the classical solution f ζ, t of 1.2 with the initial condition f ζ, 0 f0 ζ is Φ-like for t ∈ 0, t γ , where T is the blow-up time, Ω 0≤t
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