zbMATH — the first resource for mathematics

Stokes-Leibenson problem for Hele-Shaw flow: a critical set in the space of contours. (English) Zbl 1342.76039
Summary: 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.
76D07 Stokes and related (Oseen, etc.) flows
Full Text: DOI
[1] Almgren, R., Crystalline Saffman-Taylor fingers, SIAM J. Appl. Math., 55, 1511-1535, (1995) · Zbl 0838.76094
[2] 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).
[3] Bazalii, B. V., On a proof of the classical solvability of the Hele-Shaw problem with a free boundary, Ukrainian Math. J., 50, 1452-1462, (1998) · Zbl 0938.35200
[4] Caginalp, G., Stefan and Hele-Shaw type models as asymptotic limits of the phase-field equations, Phys. Rev. A, 39, 5887-5896, (1989) · Zbl 1027.80505
[5] Caginalp, G.; Chen, X., Convergence of the phase field model to its sharp interface limits, European J. Appl. Math., 9, 417-445, (1998) · Zbl 0930.35024
[6] Demidov, A. S., A polygonal model for the Hele-Shaw flow, Uspekhi Mat. Nauk., 4, 195-196, (1998)
[7] Demidov, A. S., On evolution of a small perturbation of a circle in a problem for Hele-Shaw flows, Russian Math. Surveys, 57, 1212-1214, (2002) · Zbl 1062.35081
[8] Demidov, A. S., Evolution of the perturbation of a circle in the Stokes-leibenson problem for a Hele-Shaw flow, Sovrem. Mat. Prilozh. No. 2, 123, 4381-4403, (2004) · Zbl 1072.76025
[9] Demidov, A. S., Evolution of the perturbation of a circle in the Stokes-leibenson problem for a Hele- Shaw flow. II, Sovrem. Mat. Prilozh, 139, 7064-7078, (2006) · Zbl 1127.76021
[10] Demidov, A. S., A functional-geometric method for solving problems with a free boundary for harmonic functions, Uspekhi Mat. Nauk, 65, 3-96, (2010)
[11] Demidov, A. S.; Lohéac, J.-P., The Stokes-leibenson problem for Hele-Shaw flows, 103-124, (2003)
[12] Demidov, A. S.; Lohéac, J.-P., Numerical scheme for Laplacian growth models based on the Helmholtz-Kirchhoff method, 107-114, (2009) · Zbl 1297.76130
[13] Demidov, A. S.; Lohéac, J.-P.; Runge, V., Problème de Cauchy pour l’approximation de Stokes-leibenson d’une cellule de Hele-Shaw en coin, Comptes Rendus Mécanique, 341, 755-759, (2013)
[14] Demidov, A. S.; Vasilieva, O. A., The finite point model of the Stokes-leibenson problem for the Hele-Shaw flow, Fundam. Prikl. Mat., 1, 67-84, (1999) · Zbl 0988.76029
[15] Escher, J.; Simonett, G., Classical solutions of multidimensional Hele-Shaw models, SIAM J. Math. Anal., 28, 1028-1047, (1997) · Zbl 0888.35142
[16] F. D. Gakhov, Boundary Value Problems (Oxford, NY, Pergamon Press, 1966). · Zbl 0141.08001
[17] Galin, L. A., Unsteady filtration with a free surface, Dokl. Akad. Nauk SSSR, 47, 250-253, (1945) · Zbl 0061.46202
[18] Gustafsson, B., Applications of variational inequalities to a moving boundary problem for Hele-Shaw flows, SIAM J. Math. Anal., 16, 279-300, (1985) · Zbl 0605.76043
[19] B. Gustafsson and A. Vasil’ev, Conformal and Potential Analysis in Hele-Shaw Cells (Birkhäuser Verlag, Basel, 2006). · Zbl 1122.76002
[20] Hohlov, Yu. E.; Howison, S. D., On the classification of solutions to the zero-surface-tension model for Hele-Shaw free boundary flows, Quart. Appl. Math., 51, 777-789, (1993) · Zbl 0793.76093
[21] Hele-Shaw Flows and Related Problems, S. D. Howison and J. R. Ockendon, eds., European J. Appl. Math.10 (6), 511-709 (1999).
[22] H. Helmholtz, Über discontinuirliche Flussigkeitsbewegungen (Monatsber. Konigl.Akad.Wissenschaften, Berlin, 1868). · JFM 01.0341.03
[23] Khokhlov, Yu. E.; Howison, S. D., On the classification of solutions to the zero-surface-tension model for Hele-Shaw free boundary flow, Quart. Appl. Math., 51, 777-789, (1993) · Zbl 0793.76093
[24] G. Kirchhoff, Zür Theorie freier Flussigkeitsstrahlen (Borchardt’s J., Bd. 70, 1869).
[25] P. Ya. Kochina, Selected Works. Hydrodynamics and Filtration Theory (Nauka, Moscow, 1991) [in Russian]. · Zbl 0732.76084
[26] Kufarev, P. P., A solution of the boundary problem for an oil well in a circle, Doklady Akad. Nauk SSSR (N. S.), 60, 1333-1334, (1948)
[27] H. Lamb, Hydrodynamics. (Cambridge Univ. Press., 6th ed., Cambridge, 1932). · JFM 58.1298.04
[28] L. S. Leibenson, Oil Producing Mechanics, Part II (Moscow, Neftizdat, 1934) [in Russian]. · JFM 60.1387.03
[29] Meirmanov, A. M.; Zaltzman, B., Global in time solution to the Hele- Shaw problem with a change of topology, European J. Appl. Math., 13, 431-447, (2002) · Zbl 1068.76022
[30] N. I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity [in Russian] (Leningrad, AN SSSR, 1949; Noordhoff Int. Publishing, Leiden, 1977).
[31] Ockendon, J. R.; Howison, S. D., Kochina and Hele-Shaw in modern mathematics, natural sciences and technology, J. Appl. Math. Mech., 66, 505-512, (2002) · Zbl 1066.01513
[32] Plotnikov, P. I.; Starovöikov, V. N., The Stefan problem with surface tension as a limit of the phase field model, Differ. Uravn., 29, 461-471, (1993)
[33] Polubarinova-Kochina, P. Ya., On the motion of the oil contour, Dokl. Akad. Nauk SSSR, 47, 254-257, (1945)
[34] Polubarinova-Kochina, P. Ya., Concerning unsteady motions in the theory of filtration, Prikl. Mat. Mech., 9, 79-90, (1945) · Zbl 0061.46109
[35] Reissig, M.; Wolfersdorf, L., A simplified proof for a moving boundary problem for Hele-Shaw flows in the plane, Ark. Mat., 31, 101-116, (1993) · Zbl 0802.35168
[36] Sakai, M., Regularity of a boundary having a Schwarz function, Acta Math., 166, 263-297, (1991) · Zbl 0728.30007
[37] Sakai, M., Regularity of free boundaries in two dimensions, Ann. Sc. Norm. Super. Pisa Cl. Sci., 20, 323-339, (1993) · Zbl 0851.35022
[38] 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). · Zbl 0784.30036
[39] 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). · JFM 29.0645.04
[40] Vinogradov, Yu. P.; Kufarev, P. P., On a problem of filtration, Prikl. Mat. Mech., 12, 181-198, (1948) · Zbl 0032.27901
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.