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A simple numerical method for Hele–Shaw type problems by the method of fundamental solutions

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Abstract

Hele-Shaw flows with time-dependent gaps create fingering patterns, and magnetic fluids in Hele–Shaw cells create intriguing patterns. We propose a simple numerical method for Hele–Shaw type problems by the method of fundamental solutions. The method of fundamental solutions is one of the mesh-free numerical solvers for potential problems, which provides a highly accurate approximate solution despite its simplicity. Moreover, the numerical method satisfies the volume-preserving property combining with the asymptotic uniform distribution method. We use Amano’s method to arrange the singular points in the method of fundamental solutions. We show several numerical results to exemplify the effectiveness of our numerical scheme.

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References

  1. Amano, K., Okano, D., Ogata, H., Sugihara, M.: Numerical conformal mappings onto the linear slit domain. Jpn. J. Ind. Appl. Math. 29(2), 165–186 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barnett, A., Betcke, T.: Stability and convergence of the method of fundamental solutions for Helmholtz problems on analytic domains. J. Comput. Phys. 227(14), 7003–7026 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cowley, M., Rosensweig, R.: The interfacial stability of a ferromagnetic fluid. J. Fluid Mech. 30, 671–688 (1967)

    Article  MATH  Google Scholar 

  4. Dockery, J., Klapper, I.: Finger formation in biofilm layers. SIAM J. Appl. Math. 62(3), 853–869 (2001/02)

  5. Elias, F., Flament, C., Bacri, J.C., Neveu, S.: Macro-organized patterns in ferrofluid layer: Experimental studies. Journal de Physique I 7(5), 711–728 (1997)

    Article  Google Scholar 

  6. Gustafsson, B., Vasil’ev, A.: Conformal and potential analysis in Hele-Shaw cells. Advances in Mathematical Fluid Mechanics. Birkhäuser Verlag, Basel (2006)

  7. Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration, Springer Series in Computational Mathematics, vol. 31. Springer, Heidelberg (2010). Structure-preserving algorithms for ordinary differential equations, Reprint of the second (2006) edition

  8. Hele-Shaw, H.S.: The flow of water. Nature 58, 34–36 (1898)

    Article  Google Scholar 

  9. Hou, T., Lowengrub, J., Shelley, M.: Removing the stiffness from interfacial flows with surface tension. J. Comput. Phys. 114(2), 312–338 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  10. Howison, S.: Fingering in Hele-Shaw cells. J. Fluid Mech. 167, 439–453 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  11. Katsurada, M.: Asymptotic error analysis of the charge simulation method in a Jordan region with an analytic boundary. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 37(3), 635–657 (1990)

    MathSciNet  MATH  Google Scholar 

  12. Katsurada, M., Okamoto, H.: A mathematical study of the charge simulation method. I. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 35(3), 507–518 (1988)

    MathSciNet  MATH  Google Scholar 

  13. Kemmochi, T.: Energy dissipative numerical schemes for gradient flows of planar curves. BIT 57(4), 991–1017 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lamb, H.: Hydrodynamics, sixth edn. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1993). With a foreword by R. A. Caflisch [Russel E. Caflisch]

  15. Mikula, K., Ševčovič, D.: A direct method for solving an anisotropic mean curvature flow of plane curves with an external force. Math. Methods Appl. Sci. 27(13), 1545–1565 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Mikula, K., Ševčovič, D.: Evolution of curves on a surface driven by the geodesic curvature and external force. Appl. Anal. 85(4), 345–362 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Otto, F.: Dynamics of labyrinthine pattern formation in magnetic fluids: a mean-field theory. Arch. Rational Mech. Anal. 141(1), 63–103 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. Rosensweig, R.: Magnetic fluids. Annu. Rev. Fluid Mech

  19. Rosensweig, R., Zahn, M., Shumovich, R.: Labyrinthine instability in magnetic and dielectric fluids. J. Magn. Magn. Mater. 39(1–2), 127–132 (1983)

    Article  Google Scholar 

  20. Saffman, P., Taylor, G.: 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, 312–329 (1958). ((2 plates))

    Article  MathSciNet  MATH  Google Scholar 

  21. Sakakibara, K.: Asymptotic analysis of the conventional and invariant schemes for the method of fundamental solutions applied to potential problems in doubly-connected regions. Jpn. J. Ind. Appl. Math. 34(1), 177–228 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sakakibara, K.: Bidirectional numerical conformal mapping based on the dipole simulation method. Eng. Anal. Bound. Elem. 114, 45–57 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  23. Sakakibara, K., Miyatake, Y.: A fully discrete curve-shortening polygonal evolution law for moving boundary problems. J. Comput. Phys. 424, 109857,22 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  24. Sakakibara, K., Yazaki, S.: Structure-preserving numerical scheme for the one-phase Hele-Shaw problems by the method of fundamental solutions. Comput. Math. Methods 1(6), e1063,25 (2019)

    Article  MathSciNet  Google Scholar 

  25. Ševčovič, D., Yazaki, S.: Evolution of plane curves with a curvature adjusted tangential velocity. Jpn. J. Ind. Appl. Math. 28(3), 413–442 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ševčovič, D., Yazaki, S.: On a gradient flow of plane curves minimizing the anisoperimetric ratio. IAENG Int. J. Appl. Math. 43(3), 160–171 (2013)

    MathSciNet  Google Scholar 

  27. Shelley, M., Tian, F.R., Wlodarski, K.: Hele-Shaw flow and pattern formation in a time-dependent gap. Nonlinearity 10(6), 1471–1495 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  28. Tanveer, S.: Surprises in viscous fingering. J. Fluid Mech. 409, 273–308 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  29. Tatulchenkov, A., Cebers, A.: Magnetic fluid labyrinthine instability in Hele-Shaw cell with time dependent gap. Phys. Fluids 20(5), 054101,10 (2008)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Science, Okayama University of Science, 1-1 Ridaicho, Kita-ku, Okayama-shi, Okayama, 700-0005, Japan

    Koya Sakakibara

  2. RIKEN iTHEMS, 2-1 Hirosawa, Wako-shi, Saitama, 351-0198, Japan

    Koya Sakakibara

  3. Graduate School of Science and Technology, Meiji University, 1-1-1 Higashi-Mita, Tama-ku, Kawasaki-shi, Kanagawa, 214-8571, Japan

    Yusaku Shimoji

  4. Department of Mathematics, School of Science and Technology, Meiji University, 1-1-1 Higashi-Mita, Tama-ku, Kawasaki-shi, Kanagawa, 214-8571, Japan

    Shigetoshi Yazaki

Authors
  1. Koya Sakakibara
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  2. Yusaku Shimoji
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  3. Shigetoshi Yazaki
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Correspondence to Yusaku Shimoji.

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Sakakibara, K., Shimoji, Y. & Yazaki, S. A simple numerical method for Hele–Shaw type problems by the method of fundamental solutions. Japan J. Indust. Appl. Math. 39, 869–887 (2022). https://doi.org/10.1007/s13160-022-00530-1

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