Asymptotic analysis of a contact Hele-Shaw problem in a thin domain
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Nonlinear Differential Equations and Applications NoDEA
Asymptotic analysis of a contact Hele-Shaw problem in a thin domain Taras Mel’nyk
and Nataliya Vasylyeva
Abstract. We analyze the contact Hele-Shaw problem with zero surface tension of a free boundary in a thin domain Ωε (t). Under suitable conditions on the given data, the one-valued local classical solvability of the problem for each fixed value of the parameter ε is proved. Using the multiscale analysis, we study the asymptotic behavior of this problem as ε → 0, i.e., when the thin domain Ωε (t) is shrunk into the interval (0, l). Namely, we find exact representation of the free boundary for t ∈ [0, T ]; derive the corresponding limit problem (ε = 0); define other terms of the asymptotic approximation and prove appropriate asymptotic estimates that justify this approach. We also establish the preserving geometry of the free boundary near corner points for t ∈ [0, T ] under assumption that free and fixed boundaries form right angles at the initial time t = 0. Mathematics Subject Classification. Primary 35R35, 35B40; Secondary 76D27, 76A20. Keywords. Hele-Shaw problem, Asymptotic approximation, Thin domain.
1. Introduction The Hele-Shaw problem was first introduced in 1897 by H.S. Hele-Shaw, a British engineer, scientist and inventor [24,25]. This problem models the pressure of fluid squeezed between two parallel plate, a small distance apart. For the last 70 years, this problem has merited a great research interest among the mathematical, physical, engineering and biological community due to its wide application in hydrodynamics, mathematical biology, chemistry and finance. In addition, many other problems of fluid mechanics are associated with HeleShaw flows and therefore the study of these flows is very important, especially for microflows. This is due to manufacturing technology that creates shallow flat configurations, and the typically low Reynolds numbers of microflows. As it is noted in [29], there is a vast literature on the Hele-Shaw and related problems. This literature includes, in particular, well-posedness proofs of the Hele-Shaw problem in various physical settings. 0123456789().: V,-vol
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Figure 1. Typical domain configuration Here we focus on the contact one-phase Hele-Shaw problem with zero surface tension (ZST) of a free (unknown) boundary in a thin domain. Let T > 0 be arbitrarily fixed, and let Q ⊂ R2 be a rectangle Q = (0, l) × (0, 2ε) for some given positive values l and ε. We denote QT = Q × (0, T )
∂QT = ∂Q × [0, T ]. ¯ which splits the rectangle Q Let Γ (t), t ∈ [0, T ], be a simple curve Γ (t) ⊂ Q ε ε into two subdomains Ω (t) and Q\Ω (t), such that for some unknown function ρ = ρ(y1 , t) : [0, l] × [0, T ] → R, the domain Ωε (t) is given by and
ε
ε
Ωε (t) = {y = (y1 , y2 ) ∈ Q : y1 ∈ (0, l), 0 < y2 < ε + ρ(y1 , t)}, t ∈ (0, T ) (1.1) (see Fig. 1). The mathematical setting of the contact one-phase Hele-Shaw problem consists in determining the
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