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Invariant Geometric Properties in Hele-Shaw Flows

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Abstract

In this paper, we investigate some geometric properties of the moving frontier of a viscous fluid for planar flows in Hele-Shaw cells under injection. We study invariance in time properties of the free boundary for bounded and unbounded domains (with bounded complement) under the assumption of zero surface tension. By applying certain results in the theory of univalent functions we partially solve an open problem of Vasil’ev concerning the invariance in time of starlikeness of order \(\alpha , \alpha \in [0,1)\), for bounded domains. In the case of unbounded domains with bounded complement we analyze the invariance in time of convexity of order \(\alpha , \alpha \in [0,1)\).

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Authors and Affiliations

  1. Department of Statistics, Forecasts and Mathematics, Babeş-Bolyai University, 400591, Cluj-Napoca, Romania

    Paula Curt

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  1. Paula Curt
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Correspondence to Paula Curt.

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Communicated by Stephan Ruscheweyh.

This work has been supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0899.

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Curt, P. Invariant Geometric Properties in Hele-Shaw Flows. Comput. Methods Funct. Theory 16, 503–513 (2016). https://doi.org/10.1007/s40315-016-0158-y

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  • DOI: https://doi.org/10.1007/s40315-016-0158-y

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