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  Scherk-type capillary graphs

Huff, R., & McCuan, J. (2006). Scherk-type capillary graphs. Journal of Mathematical Fluid Mechanics, 8(1), 99-119.

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Item Permalink: http://hdl.handle.net/11858/00-001M-0000-0013-4AA6-7 Version Permalink: http://hdl.handle.net/11858/00-001M-0000-0013-4AA7-5
Genre: Journal Article

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jmfm8_99.pdf (Publisher version), 326KB
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 Creators:
Huff, R.1, Author
McCuan, J., Author
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1Geometric Analysis and Gravitation, AEI-Golm, MPI for Gravitational Physics, Max Planck Society, ou_24012              

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Free keywords: capillarity; contact angle; free surface; mean curvature; minimal surface; wedge domain; Weierstrass representation
 Abstract: This paper concerns the regularity of a capillary graph (the meniscus profile of liquid in a cylindrical tube) over a corner domain of angle α. By giving an explicit construction of minimal surface solutions previously shown to exist (Indiana Univ. Math. J. 50 (2001), no. 1, 411–441) we clarify two outstanding questions. Solutions are constructed in the case α = π/2 for contact angle data (γ1, γ2) = (γ, π − γ) with 0 < γ < π. The solutions given with |γ − π/2| < π/4 are the first known solutions that are not C2 up to the corner. This shows that the best known regularity (C1, ∈) is the best possible in some cases. Specific dependence of the Hölder exponent on the contact angle for our examples is given. Solutions with γ = π/4 have continuous, but horizontal, normal vector at the corners in accordance with results of Tam (Pacific J. Math. 124 (1986), 469–482). It is shown that our examples are C0, β up to and including the corner for any β < 1. Solutions with |γ − π/2| > π/4 have a jump discontinuity at the corner. This kind of behavior was suggested by numerical work of Concus and Finn (Microgravity sci. technol. VII/2 (1994), 152–155) and Mittelmann and Zhu (Microgravity sci. technol. IX/1 (1996), 22–27). Our explicit construction, however, allows us to investigate the solutions quantitatively. For example, the trace of these solutions, excluding the jump discontinuity, is C2/3.

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Language(s): eng - English
 Dates: 2006-02
 Publication Status: Published in print
 Pages: -
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 Identifiers: eDoc: 298151
ISI: 000236200900005
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Title: Journal of Mathematical Fluid Mechanics
  Alternative Title : J. Math. Fluid Mech.
Source Genre: Journal
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Pages: - Volume / Issue: 8 (1) Sequence Number: - Start / End Page: 99 - 119 Identifier: ISSN: 1422-6928