On Boundary Conditions for Hele-Shaw Problem

  • Hisasi TaniEmail author
Conference paper
Part of the Mathematics for Industry book series (MFI, volume 26)


Weakly nonlinear stability analysis is carried out for a two-dimensional radially growing interface between two immiscible viscous fluids, known as the Hele-Shaw problem . In contrast to the previous studies, here we consider the boundary conditions with the wetting-layer effect and the VNS effect, respectively. The difference in fingering patterns is investigated by employing each boundary condition, and then it is discussed about which one reflects the interfacial phenomena more appropriately.


Capillary Number Couette Flow Linear Stability Analysis Perturbation Amplitude Linear Growth Rate 
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  1. 1.
    Bataille, J.: Rev. Inst. Pétrole 23, 1349 (1968)Google Scholar
  2. 2.
    Batchelor, G.: An Introduction to Fluid Dynamics. Cambridge University Press (2000)Google Scholar
  3. 3.
    Ben-Jacob, E., Godbey, R., Goldenfeld, N.D., Koplik, J., Levine, H., Mueller, T., Sander, L.M.: Phys. Rev. Lett. 55, 1315–1318 (1985)Google Scholar
  4. 4.
    Chen, J.-D.: Exp. Fluids 5, 363–371 (1987)Google Scholar
  5. 5.
    Dias, E.O., Miranda, J.A.: Phys. Rev. E 88, 013016 (2013)Google Scholar
  6. 6.
    Escher, J., Simonett, G.: Adv. Differ. Equ. 2, 619–642 (1997)Google Scholar
  7. 7.
    Fujimura, K., Mizushima, J.: Euro. J. Mech. B 10, 25–30 (1991)Google Scholar
  8. 8.
    Gustafsson, B., Vasil’ev, A.: Conformal and Potential Analysis in Hele-Shaw Cells, Birkhäuser (2006)Google Scholar
  9. 9.
    Kim, H., Funada, T., Joseph, D.D., Homsy, G.M.: Phys. Fluids 21, 074106 (2009)Google Scholar
  10. 10.
    Landau, L., Levich, B.: Acta Physicochim. U.R.S.S. 17, 42–54 (1942)Google Scholar
  11. 11.
    Martyushev, L.M., Birzina, A.I.: J. Phys. Condens. Matter 20, 045201 (2008)Google Scholar
  12. 12.
    McLean, J.W., Saffman, P.G.: J. Fluid Mech. 102, 455–469 (1981)Google Scholar
  13. 13.
    Miranda, J.A., Widom, M.: Phys. D 120, 315–328 (1998)Google Scholar
  14. 14.
    Mizushima, J., Fujimura, K.: J. Fluid Mech. 234, 651–667 (1992)Google Scholar
  15. 15.
    Park, C.W., Homsy, G.M.: J. Fluid Mech. 139, 291–308 (1984)Google Scholar
  16. 16.
    Paterson, L.: J. Fluid Mech. 113, 513–529 (1981)Google Scholar
  17. 17.
    Reinelt, D.A.: J. Fluid Mech. 183, 219–234 (1987)Google Scholar
  18. 18.
    Saffman, P.G., Taylor, G.I.: Proc. R. Soc. Lond. A 245, 312–329 (1958)Google Scholar
  19. 19.
    Tani, H., Tani, A.: J. Phys. Soc. Jpn. 83, 034401 (2014)Google Scholar
  20. 20.
    Ushijima, T., Yazaki, S.: Proc. Czech-Jpn. Semin. Appl. Math. 146–152 (2005)Google Scholar
  21. 21.
    Weinstein, S.J., Dussan, E.B., Ungar, L.H.: J. Fluid Mech. 221, 53–76 (1990)Google Scholar

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© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.Meiji UniversityKawasaki-shiJapan

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