Abstract
The Stokes–Leibenson problem for Hele-Shaw flow is reformulated as a Cauchy problem of a nonlinear integro-differential equation with respect to functions a and b, linked by the Hilbert transform. The function a expresses the evolution of the coefficient longitudinal strain of the free boundary and b is the evolution of the tangent tilt of this contour. These functions directly reflect changes of geometric characteristics of the free boundary of higher order than the evolution of the contour point obtained by the classical Galin–Kochina equation. That is why we managed to uncover the reason of the absence of solutions in the sink-case if the initial contour is not analytic at at least one point, to prove existence and uniqueness theorems, and also to reveal a certain critical set in the space of contours. This set contains one attractive point in the source-case corresponding to a circular contour centered at the source-point. The main object of this work is the analysis of the discrete model of the problem. This model, called quasi-contour, is formulated in terms of functions corresponding to a and b of our integro-differential equation. This quasi-contour model provides numerical experiments which confirm the theoretical properties mentioned above, especially the existence of a critical subset of co-dimension 1 in space of quasi-contours. This subset contains one attractive point in the source-case corresponding to a regular quasi-contour centered at the source-point. The main contribution of our quasi-contour model concerns the sink-case: numerical experiments show that the above subset is attractive. Furthermore, this discrete model allows to extend previous results obtained by using complex analysis. We also provide numerical experiments linked to fingering effects.
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References
R. Almgren, “Crystalline Saffman–Taylor Fingers,” SIAM J. Appl. Math. 55, 1511–1535 (1995).
A. Antontsev, A. M. Meirmanov, and V. Yurinsky, Hele-Shaw Flow in Two Dimensions: Global-In-Time Classical Solutions (Universidade da Beira Interior, Portugal, preprint 6, 1999).
B. V. Bazalii, “On a Proof of the Classical Solvability of the Hele-Shaw Problem with a Free Boundary,” Ukrainian Math. J. 50 (11), 1452–1462 (1998).
G. Caginalp, “Stefan and Hele-Shaw Type Models as Asymptotic Limits of the Phase-Field Equations,” Phys. Rev. A 39 (11), 5887–5896 (1989).
G. Caginalp and X. Chen, “Convergence of the Phase Field Model to Its Sharp Interface Limits,” European J. Appl. Math. 9 (4), 417–445 (1998).
A. S. Demidov, “A Polygonal Model for the Hele-Shaw Flow,” Uspekhi Mat. Nauk. 4, 195–196 (1998).
A. S. Demidov, “On Evolution of a Small Perturbation of a Circle in a Problem for Hele-Shaw Flows,” Russian Math. Surveys 57 (6), 1212–1214 (2002).
A. S. Demidov, “Evolution of the Perturbation of a Circle in the Stokes–Leibenson Problem for a Hele-Shaw Flow,” Sovrem. Mat. Prilozh. No. 2, Differ. Uravn. Chast. Proizvod. (2003), 3–24 [J. Math. Sci. (N. Y.) 123 (5), 4381–4403 (2004)].
A. S. Demidov, “Evolution of the Perturbation of a Circle in the Stokes–Leibenson Problem for a Hele- Shaw Flow. II,” Sovrem. Mat. Prilozh. No. 24, Din. Sist. i Optim. (2005), 51–65 [J. Math. Sci. (N. Y.) 139 (6), 7064–7078 (2006)].
A. S. Demidov, “A Functional-Geometric Method for Solving Problems with a Free Boundary for Harmonic Functions,” Uspekhi Mat. Nauk 65 (1), 3–96 (2010) [Russian Math. Surveys 65 (1), 1–94 (2010)].
A. S. Demidov and J.-P. Lohéac, “The Stokes–Leibenson Problem for Hele-Shaw Flows,” Patterns and Waves (Saint Petersburg, 2002), AkademPrint, St. Petersburg, 103–124 (2003).
A. S. Demidov and J.-P. Lohéac, “Numerical Scheme for Laplacian Growth Models Based on the Helmholtz–Kirchhoff Method,” Analysis and Mathematical Physics, Trends Math., Birkhäuser, Basel, 107–114 (2009).
A. S. Demidov, J.-P. Lohéac, and V. Runge, “Problème de Cauchy pour l’approximation de Stokes–Leibenson d’une cellule de Hele–Shaw en coin,” Comptes Rendus Mécanique 341 (11–12), 755–759 (2013).
A. S. Demidov and O. A. Vasilieva, “The Finite Point Model of the Stokes–Leibenson Problem for the Hele-Shaw Flow,” Fundam. Prikl. Mat. 1, 67–84 (1999).
J. Escher and G. Simonett, “Classical Solutions of Multidimensional Hele-Shaw Models,” SIAM J. Math. Anal. 28, 1028–1047 (1997).
F. D. Gakhov, Boundary Value Problems (Oxford, NY, Pergamon Press, 1966).
L. A. Galin, “Unsteady Filtration with a Free Surface,” Dokl. Akad. Nauk SSSR 47, 250–253 (1945).
B. Gustafsson, “Applications of Variational Inequalities to a Moving Boundary Problem for Hele-Shaw Flows,” SIAM J. Math. Anal. 16 (2), 279–300 (1985).
B. Gustafsson and A. Vasil’ev, Conformal and Potential Analysis in Hele-Shaw Cells (Birkhäuser Verlag, Basel, 2006).
Yu. E. Hohlov and S. D. Howison, “On the Classification of Solutions to the Zero-Surface-Tension Model for Hele-Shaw Free Boundary Flows,” Quart. Appl. Math. 51 (4), 777–789 (1993).
Hele-Shaw Flows and Related Problems, S. D. Howison and J. R. Ockendon, eds., European J. Appl. Math. 10 (6), 511–709 (1999).
H. Helmholtz, Über discontinuirliche Flussigkeitsbewegungen (Monatsber. Konigl.Akad.Wissenschaften, Berlin, 1868).
Yu. E. Khokhlov and S. D. Howison, “On the Classification of Solutions to the Zero-Surface-Tension Model for Hele-Shaw Free Boundary Flow,” Quart. Appl. Math. 51 (4), 777–789 (1993).
G. Kirchhoff, Zür Theorie freier Flussigkeitsstrahlen (Borchardt’s J., Bd. 70, 1869).
P. Ya. Kochina, Selected Works. Hydrodynamics and Filtration Theory (Nauka, Moscow, 1991) [in Russian].
P. P. Kufarev, “A Solution of the Boundary Problem for an Oil Well in a Circle,” Doklady Akad. Nauk SSSR (N. S.) 60, 1333–1334 (1948) [in Russian].
H. Lamb, Hydrodynamics. (Cambridge Univ. Press., 6th ed., Cambridge, 1932).
L. S. Leibenson, Oil Producing Mechanics, Part II (Moscow, Neftizdat, 1934) [in Russian].
A. M. Meirmanov and B. Zaltzman, “Global in Time Solution to the Hele- Shaw Problem with a Change of Topology,” European J. Appl. Math. 13 (4), 431–447 (2002).
N. I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity [in Russian] (Leningrad, AN SSSR, 1949; Noordhoff Int. Publishing, Leiden, 1977).
J. R. Ockendon and S. D. Howison, “Kochina and Hele-Shaw in Modern Mathematics, Natural Sciences and Technology,” J. Appl. Math. Mech. 66 (3), 505–512 (2002).
P. I. Plotnikov and V. N. Starovöikov, “The Stefan Problem with Surface Tension as a Limit of the Phase Field Model,” Differ. Uravn. 29 (3), 461–471 (1993) [Differ. Equ. 29 (3), 395–404 (1993)].
P. Ya. Polubarinova-Kochina, “On the Motion of the Oil Contour,” Dokl. Akad. Nauk SSSR 47, 254–257 (1945) [in Russian].
P. Ya. Polubarinova-Kochina, “Concerning Unsteady Motions in the Theory of Filtration,” Prikl. Mat. Mech. 9, 79–90 (1945) [in Russian].
M. Reissig and L. Wolfersdorf, “A Simplified Proof for a Moving Boundary Problem for Hele-Shaw Flows in the Plane,” Ark. Mat. 31 (1), 101–116 (1993).
M. Sakai, “Regularity of a Boundary Having a Schwarz Function,” Acta Math. 166, 263–297 (1991).
M. Sakai, “Regularity of Free Boundaries in Two Dimensions,” Ann. Sc. Norm. Super. Pisa Cl. Sci. 20, 323–339 (1993).
H. S. Shapiro, The Schwarz Function and Its Generalization to Higher Dimensions (University of Arkansas lecture notes in the mathematical sciences, 9, New-York, John Wiley & Sons Inc, 1992).
G. G. Stokes, 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 (Brit. Ass. Rep., 143 (Papers, V, 278) 1898).
Yu. P. Vinogradov and P. P. Kufarev, “On a Problem of Filtration,” Prikl. Mat. Mech. 12, 181–198, (1948) [in Russian].
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Demidov, A.S., Lohéac, J.P. & Runge, V. Stokes–Leibenson problem for Hele-Shaw flow: a critical set in the space of contours. Russ. J. Math. Phys. 23, 35–55 (2016). https://doi.org/10.1134/S1061920816010039
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DOI: https://doi.org/10.1134/S1061920816010039