A justification for the thin film approximation of Stokes flow with surface tension. (English) Zbl 1387.35632

Summary: In the free boundary problem of Stokes flow driven by surface tension, we pass to the limit of small layer thickness. It is rigorously shown that in this limit the evolution is given by the well-known thin film equation. The main techniques are appropriate scaling and uniform energy estimates in Sobolev spaces of sufficiently high order, based on parabolicity.


35R35 Free boundary problems for PDEs
76D07 Stokes and related (Oseen, etc.) flows
35K35 Initial-boundary value problems for higher-order parabolic equations
35Q30 Navier-Stokes equations
76D08 Lubrication theory
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[1] Antanovskii, L. K., The bi-analytic stress-stream function in plane quasistationary problems of capillary hydrodynamics, Sibirsk. Mat. Zh.. Sibirsk. Mat. Zh., Siberian Math. J., 33, 1, 1-11 (1992), (in Russian); English transl.: · Zbl 0787.76019
[2] Antanovskii, L. K., Creeping thermocapillary motion of a two-dimensional deformable bubble: Existence theorem and numerical simulation, Eur. J. Mech. B Fluids, 11, 6, 741-758 (1992) · Zbl 0762.76021
[3] Beale, J. T., Large-time regularity of viscous surface waves, Arch. Ration. Mech. Anal., 84, 307-352 (1983/84) · Zbl 0545.76029
[4] Cummings, L.; Howison, S. D.; King, J. R., Conserved quantities in Stokes flow with free surfaces, Phys. Fluids, 9, 3, 477-480 (1997) · Zbl 1185.76510
[5] Duvaut, G.; Lions, J.-L., Les inéquations en mécanique et en physique (1972), Dunod: Dunod Paris · Zbl 0298.73001
[6] Escher, J.; Prokert, G., Analyticity of solutions to nonlinear parabolic equations on manifolds and an application to Stokes flow, J. Math. Fluid Mech., 8, 1-35 (2006) · Zbl 1102.35048
[7] Friedman, A.; Reitich, F., Quasistatic motion of a capillary drop. I. The two-dimensional case, J. Differential Equations, 178, 212-263 (2002) · Zbl 0993.76017
[8] Friedman, A.; Reitich, F., Quasistatic motion of a capillary drop. II. The three-dimensional case, J. Differential Equations, 186, 509-557 (2002) · Zbl 1146.76593
[9] Friedman, A.; Reitich, F., Nonlinear stability of a quasi-static Stefan problem with surface tension: A continuation approach, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 30, 341-403 (2001) · Zbl 1072.35208
[10] Günther, M.; Prokert, G., Existence results for the quasistationary motion of a capillary liquid drop, Z. Anal. Anwend., 16, 2, 311-348 (1997) · Zbl 0888.35140
[11] Hopper, R. W., Plane Stokes flow driven by capillarity on a free surface, J. Fluid Mech., 213, 349-375 (1990) · Zbl 0698.76040
[12] Howison, S. D.; Richardson, S., Cusp development in free boundaries, and two-dimensional viscous flows, European J. Appl. Math., 6, 441-454 (1995) · Zbl 0840.76012
[13] Ockendon, H.; Ockendon, J. R., Viscous Flows (1995), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0837.76001
[14] Pukhnachev, V. V., The quasistationary approximation in the problem of the motion of an isolated volume of a viscous incompressible capillary liquid, J. Appl. Math. Mech., 62, 927-932 (1998)
[15] Solonnikov, V. A., On a non-steady motion of an isolated volume of a finite liquid mass bounded by a free surface, Zap. Nauchn. Sem. LOMI, 152, 137-157 (1986), (in Russian) · Zbl 0614.76026
[16] Solonnikov, V. A., On quasistationary approximation in the problem of motion of a capillary drop, (Escher, J.; Simonett, G., Topics in Nonlinear Analysis (1999), Birkhäuser: Birkhäuser Basel, Boston, Berlin), 643-671 · Zbl 0919.35103
[17] Solonnikov, V. A., On the justification of the quasistationary approximation in the problem of motion of a viscous capillary drop, Interfaces Free Bound., 1, 125-173 (1999) · Zbl 0974.35097
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