Advertisement

Journal of Mathematical Fluid Mechanics

, Volume 17, Issue 2, pp 261–277 | Cite as

Optimal Control by Multipoles in the Hele-Shaw Problem

  • Lev Lokutsievskiy
  • Vincent Runge
Article

Abstract

The two-dimensional Hele-Shaw problem for a fluid spot with free boundary can be solved using the Polubarinova–Galin equation. The main condition of its applicability is the smoothness of the spot boundary. In the sink-case, this problem is not well-posed and the boundary loses smoothness within finite time—the only exception being the disk centred on the sink. An extensive literature deals with the study of the Hele-Shaw problem with non-smooth boundary or with surface tension, but the problem remains open. In our work, we propose to study this flow from a control point of view, by introducing an analogue of multipoles (term taken from the theory of electromagnetic fields). This allows us to control the shape of the spot and to avoid non-smoothness phenomenon on its border. For any polynomial contours, we demonstrate how all the fluid can be extracted, while the border remains smooth until the very end. We find, in particular, sufficient conditions for controllability and a link between Richardson’s moments and Polubarinova–Galin equation.

Keywords

Hele-Shaw flow multipoles controllability optimal control 

Mathematics Subject Classification

Primary 49K15 Secondary 76D27 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Darcy, H.: Les fontaines publiques de la ville de Dijon: exposition et application des principes suivre et des formules employer dans les questions de distribution d’eau. Victor Dalmont, Paris (1856)Google Scholar
  2. 2.
    Entov V.M., Etingof P.I., Kleinbock D.Ya.: Hele-Shaw flows with a free boundary produced by multipoles. Eur. J. Appl. Math. 4, 97–120 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Galin L.A.: Unsteady filtration with a free surface. Dokl. Akad. Nauk SSSR 47, 250–253 (1945)MathSciNetGoogle Scholar
  4. 4.
    Hohlov Yu.E., Howison S.D.: On the classification of solutions to the zero-surface-tension model for Hele-Shaw free boundary flows. Q. Appl. Math. 51(4), 777–789 (1993)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Hörmander L.: Linear Partial Differential Operators. Springer, New York (1963)CrossRefzbMATHGoogle Scholar
  6. 6.
    Hörmander L.: Linear Partial Differential Operators, Fourth Printing. Springer, Berlin (1976)Google Scholar
  7. 7.
    Kuznetsova, O.S.: On polynomial solutions of the Hele-Shaw problem. Sib. Mat. Zh. 42(5), 1084–1093 (2001). [Engl. transl.: Sib. Math. J. 42(5), 907–915 (2001)]Google Scholar
  8. 8.
    Lamb, H.: Hydrodynamics, 6th edn.. Cambridge Univ. Press., Cambridge (1932)Google Scholar
  9. 9.
    Lavrentiev, M.A., Shabat, B.V.: Methods of Complex Function Theory. Nauka, Moscow (1987)Google Scholar
  10. 10.
    Leibenson, L.S.: Oil Producing Mechanics, Part II. Moscow, Neftizdat (1934)Google Scholar
  11. 11.
    Nie Q., Tian F.: Singularities in Hele-Shaw flows driven by a multipole. SIAM J. Appl. Math. 62(2), 385–406 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Polubarinova-Kochina, P.Ya.: Concerning unsteady motions in the theory of filtration. Prik. Mat. Mech. 9(1), 79-90 (1945). (In Russian)Google Scholar
  13. 13.
    Richardson S.: Hele-Shaw flows with a free boundary produced by the injection of fluid into a narrow channel. J. Fluid Mech. 56(4), 609–618 (1972)CrossRefADSzbMATHGoogle Scholar
  14. 14.
    Sakai M.: A moment problem on Jordan domains. Proc. Am. Math. Soc. 70(1), 35–38 (1978)CrossRefzbMATHGoogle Scholar
  15. 15.
    Stokes, G.G.: Mathematical proof of the identity of the stream lines obtained by means of viscous film with those of a perfect fluid moving in two dimensions. Br. Ass. Rep. 143 (Papers, V, 278) (1898)Google Scholar
  16. 16.
    Varchenko, A.N., Etingof, P.I.: Why the Boundary of a Round Drop Becomes a Curve of Order Four? University Lecture Series, vol. 3. AMS, New York (1992)Google Scholar
  17. 17.
    Zelikina, L.F.: K voprosu o reguliarnom sinteze (1982). (In Russian)Google Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Department General Problems of Control, Faculty of Mechanics and MathematicsMoscow State UniversityMoscowRussia
  2. 2.Université Paris-Dauphine, CEREMADEParisFrance

Personalised recommendations