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On Hele–Shaw flow of power-law fluids

Published online by Cambridge University Press:  16 July 2009

Gunnar Aronsson
Affiliation:
Department of Mathematics, Linköping University, Sweden
Ulf Janfalk
Affiliation:
Department of Mathematics, Linköping University, Sweden

Abstract

This paper reviews the governing equations for a plane Hele–Shaw flow of a power-law fluid. We find two closely related partial differential equations, one for the pressure and one for the stream function. Some mathematical results for these equations are presented, in particular some exact solutions and a representation theorem. The results are applied to Hele–Shaw flow. It is then possible to determine the flow near an arbitrary corner for any power-law fluid. Other examples are also given.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1992

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References

[Ar1]Aronsson, G. 1986 Construction of singular solutions to the p–harmonic equation and its limit equation for p =∞. Manuscripta Math. 56, 135158.CrossRefGoogle Scholar
[Ar2]Aronsson, G. 1988 On certain p-harmonic functions in the plane. Manuscripta Math. 61, 79101.CrossRefGoogle Scholar
[Ar3]Aronsson, G. 1989 Representation of a p–harmonic function near a critical point in the plane. Manuscripta Math. 66, 7395.CrossRefGoogle Scholar
[Ar4]Aronsson, G. 1990 Aspects of p–harmonic Functions in the Plane. LiTH-MAT-R-91–38. Linköping. (Also in Lecture Notes from the Finnish Summer School in Potential Theory 1990, Lame, J., Editor. Univ. of Joensuu, Finland.)Google Scholar
[AL]Aronsson, G. & Lindqvist, P. 1988 On p–harmonic functions in the plane and their stream functions. J. Derential Equations 74, 157178.CrossRefGoogle Scholar
[AM]Astarita, G. & Marrucci, G. 1974 Principles of Non-Newtonian Fluid Mechanics. McGraw- Hill.Google Scholar
[BHW]Barnes, H. A., Hutton, J. F. & Walters, K. 1989 An Introduction to Rheology. Elsevier.Google Scholar
[B]Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
[BAH]Bird, R. B., Armstrong, R. C. & Hassager, O. 1987 Dynamics of Polymeric Liquids; Vol. 1, Fluid Mechanics. Wiley.Google Scholar
[C]Currie, J. G. 1974 Fundamental Mechanics of Fluids. McGraw-Hill.Google Scholar
[D]Dobrowolski, M. 1985 On finite element methods for nonlinear elliptic problems in domains with corners. Lecture Notes in Mathematics 1121, 85103, Springer-Verlag.Google Scholar
[G]Güçeri, S. L. 1989 Finite Difference Solution of Field Problems. In Tucker, C. L., Editor, Fundamentals of Computer Modelling for Polymer Processing. C. Hanser Verlag, 141236.Google Scholar
[HL]Hassager, O. & Lauridsen, T. L. 1988 Singular behaviour of power-law fluids in Hele-Shaw flow. J. Non-Newt. Fluid Mech. 29, 337346.CrossRefGoogle Scholar
[IM]Iwaniec, T. & Manfredi, J. J. 1989 Regularity of p–harmonic functions on the plane. Rev. Mat. Iberoamericana 5 (1&2), 119.CrossRefGoogle Scholar
[J]Janfalk, U. 1992 Representation of a p–harmonic function near an isolated singularity in the plane. Annal. Acad. Sci. Fenn. 17.Google Scholar
[K]Kirchhoff, R. H. 1985 Potential flows. Marcel Dekker.Google Scholar
[L1]Lewis, J. 1977 Capacitary functions in convex rings. Arch. Rational Mech. Anal. 66, 201224.CrossRefGoogle Scholar
[L2]Lewis, J. 1983 Regularity of the derivatives of certain degenerate elliptic equations. Indiana Univ. Math. J. 32, 849858.CrossRefGoogle Scholar
[M]Manfredi, J. J. 1988 p–harmonic functions in the plane. Proc. Amer. Math. Soc. 103, 473479.Google Scholar
[P]Panton, R. L. 1984 Incompressible Flow. Wiley.Google Scholar
[Pe]Persson, L. 1989 Quasi-radial solutions of the p-harmonic equation in the plane and their stream functions. A survey. Licentiate thesis. Luleå.Google Scholar
[S]Saffman, P. G. 1986 Viscous fingering in Hele-Shaw cells. J. Fluid Mech. 173, 7394.CrossRefGoogle Scholar
[Th]Thompson, B. W. 1968 Secondary flow in a Hele-Shaw cell. J. Fluid Mech. 31 (2), 379395.CrossRefGoogle Scholar
[T]Tolksdorf, P. 1985 Invariance properties and special structures near conical boundary points. Lecture Notes in Mathematics, Vol. 1121, Springer-Verlag, 308318.Google Scholar
[TG]Trafford, D. L. & Güçeri, S. L. 1987 Computational Analysis and Simulation of the Mold Filling Process for Hele-Shaw Flows. Center for Composite Materials, Report 87–04, University of Delaware.Google Scholar
[Tu]Tucker, C. L. III (Editor). 1989 Fundamentals of Computer Modeling for Polymer Processing. C. Hanser Verlag.Google Scholar