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Analyticity and nonexistence of classical steady Hele‐Shaw fingers
Author(s) -
Tanveer Saleh,
Xie Xuming
Publication year - 2003
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3030
Subject(s) - mathematics , surface tension , algebraic number , space (punctuation) , surface (topology) , function (biology) , hele shaw flow , section (typography) , mathematical analysis , tension (geology) , geometry , physics , classical mechanics , computer science , thermodynamics , open channel flow , flow (mathematics) , evolutionary biology , biology , operating system , moment (physics)
This paper concerns analyticity of a classical steadily translating finger in a Hele‐Shaw cell and nonexistence of solutions when relative finger width λ is smaller than ½. It is proved that any classical solution to the finger problem, if it exists for sufficiently small but nonzero surface tension and is close to some Saffman‐Taylor zero‐surface‐tension solution and satisfies some algebraic decay conditions at ∞, must belong to the analytic function space A 0 , as defined in Section 1, and chosen in a previous study [34] of existence of finger solutions. Further, it is proved that for any fixed λ ∈ (0, ½), there can be no classical steady finger solution when surface tension is sufficiently small, in disagreement with a previous conclusion based on numerical simulation. © 2002 Wiley Periodicals, Inc.