Exact Real Trinomial Solutions to the Inner and Outer Hele–Shaw Problems

详细信息    查看全文

• 作者：Vincent Runge
• 关键词：Primary 76B07 ; Secondary 30C50 ; Hele–Shaw cell ; exact real trinomial solution ; domain of univalence
• 刊名：Journal of Mathematical Fluid Mechanics
• 出版年：2015
• 出版时间：March 2015
• 年：2015
• 卷：17
• 期：1
• 页码：9-22
• 全文大小：1,053 KB
• 参考文献：1. Ben Amar M.: Exact self-similar shapes in viscows fingering. Phys. Rev. A 43(10), 5724-727 (1991) CrossRef
  2. Brannan D.A.: Coefficient regions for univalent polynomials of small degree. Mathematika 14(2), 165-69 (1967) CrossRef
  3. Cohn A.: Uber die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise. Math. Zeitsch. 14(1), 110-48 (1922) CrossRef
  4. Cowling V.F., Royster W.C.: Domains of variability for univalent polynomials. Proc. Am. Math. Soc. 19(4), 767-72 (1968) CrossRef
  5. Dieudonné J.: Recherche sur quelques problèmes relatifs aux polyn?mes et aux fonctions bornées d’une variable complexe. Ann. Sci. l’ENS 48, 247-58 (1931)
  6. Galin, L.A.: Unsteady filtration with a free surface. Dokl. Akad. Nauk USSR 47, 246-49 (1945) (in Russian)
  7. Gustafsson B.: On a differential equation arising in a Hele–Shaw flow moving boundary problem. Arkiv Mat. 22(1-), 251-68 (1984) CrossRef
  8. Hele-Shaw, H.S.: Flow of water. Nature 58(1489), 34-6 (1898) (520)
  9. Howison S.D.: Cusp development in Hele–Shaw flow with a free surface. SIAM J. Appl. Math. 46(1), 20-6 (1986) CrossRef
  10. Howison S.D., King J.: Explicit solutions to six free-boundary problems in fluid flow and diffusion. IMA J. Appl. Math. 42, 155-75 (1989) CrossRef
  11. Huntingford C.: An exact solution to the one-phase zero-surface-tension Hele–Shaw free-boundary problem. Comput. Math. Appl. 29(10), 45-0 (1995) CrossRef
  12. Kuznetsova, O.: On polynomial solutions of the Hele–Shaw problem. Sibirsk. Mat. Zh. 42(5), 1084-093 (2001) (English translation: Siberian Math J 42(5), 907-15, 2001)
  13. Lamb H.: Hydrodynamics, 3rd edn. Cambridge University Press, Cambridge (1906)
  14. Leibenson L.S.: Oil Producing Mechanics, Part II. Neftizdat, Moscow (1934)
  15. Nishida T.: A note on a theorem of Nirenberg. J. Differ. Geom. 12(4), 629-33 (1977)
  16. Polubarinova-Kochina, P. Ya.: Concerning unsteady motions in the theory of filtration. Prikl. Mat. Mech. 9(1), 79-0 (1945) (in Russian)
  17. Rahman Q.I., Szynal J.: On some classes of univalent polynomials. Can. J. Math. 30(2), 332-49 (1978) CrossRef
  18. Ruscheweyh S., Wirths K.J.: Uber die Koeffizienten spezieller schlichter polynome. Ann. Pol. Math. 28, 341-55 (1973)
  19. Saffman P.G., Taylor G.I.: The penetration of a fluid into a porous medium or Hele–Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. Ser. A 245 281, 312-29 (1958) CrossRef
  20. Stokes, G.G.: Mathematical proof of the identily of the stream-lines obtained by means of viscous film with those of a perfect fluid moving in two dimensions. Br. Assoc. Rep. 143 (papers, V, 278) (1898)
  21. Vinogradov, Yu.P., Kufarev, P.P.: On a problem of filtration. Akad. Nauk SSSR Prikl. Mat. Mech. 12, 181-98 (1948) (in Russian)
  22. Vinogradov, Yu.P., Kufarev, P.P.: On some particular solutions of the problem of filtration. Doklady Akad. Nauk SSSR (N.S.) 57, 335-38 (1947) (in Russian)
  
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Fluids
  Mathematical Methods in Physics
  Mechanics, Fluids and Thermodynamics
  
• 出版者：Birkh盲user Basel
• ISSN：1422-6952

文摘

We find an explicit representation of the evolution of \({ t \mapsto \gamma_t = \{z (\zeta, t), \zeta \in \mathbb{C}, |\zeta| = 1 \} }\) of the contour \({ \gamma_t = \partial \omega_t }\) of fluid spots \({\omega_t = \{z (\zeta, t), |\zeta| for \({t > 0}\) or \({t in the Hele–Shaw problem with a sink ( \({t > 0}\) ) or a source ( \({t ) localized at point \({z(0, t)}\) described by trinomials $$z(\zeta, t) = a_1(t)\zeta + a_N(t) \zeta^N + a_M(t) \zeta^M,{\rm where} \quad M = 2N - 1,\quad{\rm and\,integer} \quad N \ge 2,$$ for the classical formulation of the problem when \({\omega_t }\) is within \({ \gamma_t }\) (inner Hele–Shaw problem), or by $$z(\zeta, t) = a_{-1}(t)\zeta^{-1} + a_N(t) \zeta^N + a_M(t)\zeta^M, {\rm where} \quad M = 2N + 1,\quad{\rm and\,integer} \quad N \ge 1,$$ for the outer Hele–Shaw problem when \({ \omega_t }\) is outside of \({\gamma_t}\) . We obtained a sufficient condition for univalence of real trinomials, improving a result found by Ruscheweyh and Wirths (Ann Pol Math. 28:341-55, 1973). A sufficient condition is also found for functions used in the outer problem.