Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-19T11:52:16.525Z Has data issue: false hasContentIssue false
  • English
  • Français

On the existence of solutions in plane quasistationary Stokes flow driven by surface tension

Published online by Cambridge University Press:  26 September 2008

G. Prokert
G. Prokert
Affiliation:
Faculty of Mathematics and Computer Science, Technical University Eindhoven, P. 0. Box 513, 5600 MB Eindhoven, The Netherlands (e-mail prokert@win.tue.nl)
Get access Rights & Permissions [Opens in a new window]

Abstract

Recently, the free boundary problem of quasistationary Stokes flow of a mass of viscous liquid under the action of surface tension forces has been considered by R. W. Hopper, L. K. Antanovskii, and others. The solution of the Stokes equations is represented by analytic functions, and a time dependent conformal mapping onto the flow domain is applied for the transformation of the problem to the unit disk. Two coupled Hilbert problems have to be solved there, which leads to a Fredholm boundary integral equation. The solution of this equation determines the time evolution of the conformal mapping. The question of the existence of a solution to this evolution problem for arbitrary (smooth) initial data has not yet been answered completely. In this paper, local existence in time is proved using a theorem of Ovsiannikov on Cauchy problems in an appropriate scale of Banach spaces. The necessary estimates are obtained in a way that is oriented at the a priori estimates for the solution given by Antanovskii. In the case of small deviations from the stationary solution represented by a circle, these a priori estimates, together with the local results, are used to prove even global existence of the solution in time.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 6 , Issue 5 , October 1995 , pp. 539 - 558
Copyright
Copyright © Cambridge University Press 1995

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]Antanovskii, L. K. 1991 Bianalytic stress-stream function in plane quasi-steady problems of capillary fluid mechanics. Sibirsk. Mat. Zh. 33(1), 3–15.Google Scholar
[2]Antanovskii, L. K. 1991 Analyticity of a free boundary in plane quasi-steady flow of a liquid form subject to variable surface tension. In Proc. Conf.: The Navier-Stokes Equations II: Theory and Numerical Methods. Oberwolfach, Germany. (Lecture Notes in Mathematics Volume 1530, pp. 1–16, Springer-Verlag, 1993.)Google Scholar
[3]Antanovskii, L. K. 1992 Creeping thermocapillary motion of a two-dimensional deformable bubble: existence theorem and numerical simulation. Euro. J. Mech. B/Fluids 11(6), 741–758.Google Scholar
[4]Constantin, P. & Pugh, M. 1993 Global solution for small data to the Hele-Shaw problem. Nonlinearity 6, 393–415.CrossRefGoogle Scholar
[5]Duchon, J. & Robert, R. 1985 Quasi-differential perturbation of a smoothing semigroup in a scale of Banach spaces. C. R. Acad. Sci. Paris, t. 301 Serie I (11), 561–564 (in French).Google Scholar
[6]Duchon, J. & Robert, R. 1986 Estimates of integral operators of Cauchy type in Ovsiannikov scales and application. Ann. Inst. Fourier 36(1), 83–95 (in French).CrossRefGoogle Scholar
[7]de Graaf, J. 1992 Mathematical addenda to Hopper's model of plane Stokes flow driven by capillarity on a free surface. In Proc. Conf.: Geometrical and quantum aspects of integrable systems, Scheveningen, The Netherlands. (Lecture Notes in Physics Volume 424, Springer-Verlag, 1993, pp. 167–185.)Google Scholar
[8]Hohlov, Yu. E. & Reissig, M. 1995 On classical solvability for the Hele-Shaw moving boundary problems with kinetic undercooling regularization. Euro. J. Appl. Math. 6.CrossRefGoogle Scholar
[9]Hopper, R. W. 1990 Plane Stokes flow driven by capillarity on a free surface. J. Fluid Mech. 213, 349–375.CrossRefGoogle Scholar
[10]Nishida, T. 1977 A note on a theorem of Nirenberg. J. Diff. Geom. 12 629–633.Google Scholar
[11]Ovsiannikov, L. V. 1971 A nonlinear Cauchy problem in a scale of Banach spaces. Dokl. Akad. Nauk SSSR 200(4), 789–792.Google Scholar
[12]Ovsiannikov, L. V. 1974 To the shallow water theory foundation. Arch. Mech. Stos. 26(3), 407–422.Google Scholar
[13]Prokert, G. 1993 On a quasistatic model for the motion of a viscous capillary liquid drop. RANA 93-16, Eindhoven University of Technology.Google Scholar
[14]Reissig, M. & Wolfersdorf, L. v. 1992 A simplified proof for a moving boundary problem for Hele-Shaw flows in the plane. Arkiv för matematik 31, 101–116.CrossRefGoogle Scholar
[15]Reissig, M. 1994 About the existence and uniqueness of analytic solutions for a moving boundary problem for Hele-Shaw flows in the plane. Nonlinear Analysis (to appear).CrossRefGoogle Scholar
[16]Richardson, S. 1992 Two-dimensional slow viscous flow with time-dependent free boundaries driven by surface tension. Euro. J. Appl. Math. 3 193–207.CrossRefGoogle Scholar
[17]Shelukhin, V. V. 1991 The justification of the conjugate conditions for the Euler's and Darcy's equations. In Proc. Conf.: Free boundary problems in continuum mechanics, Novosibirsk, Russia. (Int. Ser. Numer. Math. 106, Birkhäuser, 1992, pp. 293–300.)Google Scholar
[18]Tutschke, W. 1989 Solution of initial value problems in classes of generalized analytic functions. Springer-Verlag.CrossRefGoogle Scholar
[19]van de Vorst, G. A. L. 1994 Modelling and Numerical Simulation of Viscous sintering. PhD thesis, Eindhoven University of Technology.CrossRefGoogle Scholar