Hele-Shaw Flows: Historical Overview

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Selection Problem The parameter λ, the relative width of the finger, is freely defined in 0 < λ ≤ 1. But in experiments λ was found to be close to 1/2 except some very special cases (very slow flow, Saffman’s unsteady solution). Why λ = 1/2 selected? (Saffman-Taylor, 1958) They also proposed to use small surface tension β as a selection mechanism as β → 0. BIRS, Canada, July 2007 – p. 71
Selection Problem This proposal was realized in: • D. A. Kessler, J. Koplik, H. Levine: Pattern selection in fingered growth phenomena, Adv. Phys. 37 (1988), no. 3, 255–339. • X. Xie, S. Tanveer: Rigorous results in steady finger selection in viscous fingering, Arch. Ration. Mech. Anal. 166 (2003), no. 3, 219–286. Without use of surface tension: • M. Mineev-Weinstein: Selection of the Saffman-Taylor finger width in the absence of surface tension: an exact result, Phys. Rev. Lett. 80 (1998), no. 10, 2113–2116. BIRS, Canada, July 2007 – p. 72
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Hele-Shaw Flows: Historical Overvie
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Henri Selby Hele-Shaw ... an exampl
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1871-1876 At the age of 17 he finis
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1885-1904 In 1885 Hele-Shaw was inv
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1906-1941 Upon returning from South
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1906-1941 Upon returning from South
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1906-1941 Hele-Shaw took a very act
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1906-1941 One of his greatest contr
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1906-1941 Hele-Shaw was a man of gr
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1906-1941 Hele-Shaw was a man of gr
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Hele-Shaw’s inventions Two greate
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Hele-Shaw’s inventions • Specia
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Hele-Shaw’s inventions H. S. Hele
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Hele-Shaw’s inventions Further de
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Hele-Shaw Prizes • Hele-Shaw Priz
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Stream-line Flow Methods The most n
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Stream-line Flow Methods In 1897 He
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Experiment by Reynolds O. Reynolds
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Experiment by Reynolds O. Reynolds
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Hele-Shaw Cell Hele-Shaw got the Go
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Hele-Shaw Papers • Experiments on
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Page 49 and 50: Stokes-Leibenzon Model Leonid Samui
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Page 53 and 54: Stokes-Leibenzon Model Ω(t) 0 y
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Page 57 and 58: Free Boundary Problem • ∆p = Q
Page 59 and 60: P. Ya. Polubarinova-Kochina Pelagey
Page 61 and 62: P. Ya. Polubarinova-Kochina In 1921
Page 63 and 64: P. Ya. Polubarinova-Kochina In 1958
Page 65 and 66: L. A. Galin Lev Alexandrovich Galin
Page 67 and 68: Hele-Shaw problem P. Ya. Polubarino
Page 69 and 70: Papers • P. Ya. Polubarinova-Koch
Page 71 and 72: P. P. Kufarev Pavel Parfenievich Ku
Page 73 and 74: Kufarev’s exact solutions 1947-19
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Page 77 and 78: The proof of local existence BIRS,
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Page 89 and 90: Fingering Phenomenon Modelling by t
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Page 93 and 94: Saffman-Taylor Finger Mathematical
Page 95 and 96: Philip G. Saffman Philip Geoffrey S
Page 97: Conformal formulation Saffman-Taylo
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Page 103 and 104: MODERN PERIOD 1981-PRESENT • Nowa
Page 105 and 106: Evolution Geometry Björn Gustafsso
Page 107 and 108: Other Models Witten, Sander, Hastin
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Page 111 and 112: Integrable Systems (2000-2007) P.Wi
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Page 115 and 116: Recommended Reading Two monographs:
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