Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-27T06:25:22.506Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalised Hele-Shaw flow: A Schwarz function approach

Published online by Cambridge University Press:  16 May 2011

N. R. McDONALD
N. R. McDONALD*
Affiliation:
Department of Mathematics, University College London, WC1E 6BT, UK email: robb@math.ucl.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

An equation governing the evolution of a Hele-Shaw free boundary flow in the presence of an arbitrary external potential – generalised Hele-Shaw flow – is derived in terms of the Schwarz function g(z, t) of the free boundary. This generalises the well-known equation ∂g/∂t = 2∂w/∂z, where w is the complex potential, which has been successfully employed in constructing many exact solutions in the absence of external potentials. The new equation is used to re-derive some known explicit solutions for equilibrium and time-dependent free boundary flows in the presence of external potentials, including those with singular potential fields, uniform gravity and centrifugal forces. Some new solutions are also constructed that variously describe equilibrium flows with higher order hydrodynamic singularities in the presence of electric point sources and an unsteady solution describing bubbles under the combined influence of strain and centrifugal potential.

Keywords

Hele-Shawfree boundarySchwarz function
Type
Papers
Information
European Journal of Applied Mathematics , Volume 22 , Issue 6 , December 2011 , pp. 517 - 532
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Abanov, A., Mineev-Weinstein, M. & Zabrodin, A. (2007) Self-similarity in Laplacian growth. Physica D 235, 62–71.CrossRefGoogle Scholar
[2]Crowdy, D. G. (1999) A class of exact multipolar vortices. Phys. Fluids 11, 2556–2564.CrossRefGoogle Scholar
[3]Crowdy, D. G. (2002) On a class of geometry-driven free boundary problems. SIAM J. Appl. Math. 62, 945–964.CrossRefGoogle Scholar
[4]Cummings, L. J., Hohlov, Y. E., Howison, S. D. & Kornev, K. (1999) Two-dimensional solidification and melting in potential flows. J. Fluid Mech. 378, 1–18.CrossRefGoogle Scholar
[5]Cummings, L. J., Howison, S. D. & King, J. R. (1999) Two-dimensional Stokes and Hele-Shaw flows with free surfaces. Eur. J. Appl. Math. 10, 635–680.CrossRefGoogle Scholar
[6]Davis, P. J. (1974) The Schwarz Function and Its Applications, Vol. 17 of The Carus Mathematical Monographs, The Mathematical Association of America, Washington, USA.CrossRefGoogle Scholar
[7]Entov, V. M. & Etingof, P. (2007) On a generalized two-fluid Hele-Shaw flow. Eur. J. Appl. Math. 18, 103–128.CrossRefGoogle Scholar
[8]Entov, V. M., Etingof, P. I. & Kleinbock, D. Y. (1995) On nonlinear interface dynamics in Hele-Shaw flows. Eur. J. Appl. Math. 6, 399–420.CrossRefGoogle Scholar
[9]Entov, V. M., Kleinbock, D. Y. & Etingof, P. I. (1993) Hele-Shaw flows with a free boundary produced by multipoles. Fluid Dyn. 28, 686–691.CrossRefGoogle Scholar
[10]Galin, L. A. (1945) Unsteady filtration with a free surface. Dokl. Akad. Nauk SSSR. 47, 246–249.Google Scholar
[11]Gustafsson, B. & Vasil'ev, A. (2006) Conformal and Potential Analysis in Hele-Shaw Cells, Birhäuser-Verlag, Basel, Switzerland. Boston, USA. Berlin, Germany.Google Scholar
[12]Magdaleno, F. X., Rocco, A. & Casademunt, J. (2000) Interface dynamics in Hele-Shaw flows with centrifugal forces: Preventing cusp singularities with rotation. Phys. Rev. E 62, R5887–R5890.CrossRefGoogle ScholarPubMed
[13]Millar, R. F. (1990) Inverse problems for a class of Schwarz functions. Complex Vars. Elliptic Eqs. 15, 1–10.Google Scholar
[14]Polubarinova-Kochina, P. Y. (1945) On a problem of the motion of the contour of a petroleum shell. Dokl. Akad. Nauk SSSR. 47, 254–257.Google Scholar
[15]Richardson, S. (1972) Hele-Shaw flows produced by injection of fluid into a narrow channel. J. Fluid Mech. 56, 609–618.CrossRefGoogle Scholar
[16]Richardson, S. (1981) Some Hele-Shaw flows with time-dependent free boundaries. J. Fluid Mech. 102, 263–278.CrossRefGoogle Scholar
[17]Richardson, S. (2001) Hele-Shaw flows with free boundaries in a corner or around a wedge. Part I: Liquid at the vertex. Eur. J. Appl. Math. 12, 665–676.CrossRefGoogle Scholar
[18]Richardson, S. (2001) Hele-Shaw flows with free boundaries in a corner or around a wedge. Part II: Air at the vertex. Eur. J. Appl. Math. 12, 667–688.Google Scholar
[19]Saffman, P. G. & Taylor, G. I. (1958) The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous fluid. Proc. R. Soc. A 245, 312–329.Google Scholar
[20]Taylor, G. I. & Saffman, P. G. (1959) A note on the motion of bubbles in a Hele-Shaw cell and porous medium. Quart. J. Mech. Appl. Math. 12, 265–279.CrossRefGoogle Scholar