The solvability of a free boundary problem for the stationary Navier-Stokes equations with a dynamic contact line. (English) Zbl 0853.35134

We consider the plane steady-state problem with two free boundaries for the Navier-Stokes equations describing a coating process with a heavy viscous incompressible fluid onto a horizontally moving rigid wall. One of the a priori unknown free surfaces is noncompact and the other free surface ends at an a priori unknown contact line on the right wall. The problem is considered in the plane \(\mathbb{R}^2\).


35R35 Free boundary problems for PDEs
35Q30 Navier-Stokes equations
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[1] Pukhnachov, V. V.; Solonnikov, V. A., On the problem of dynamic contact angle, PMM USSR, 46, 771-779 (1983) · Zbl 0532.76026
[2] Socolowsky, J.; Bergmann, J., Eine gemischte Randwertaufgabe für stationäre Mehrschichtenströmungen, Z. Anal. Anwend. (ZAA), 6, 107-119 (1987) · Zbl 0634.76109
[3] Dussan, V. E.B.; Davis, S. H., On the motion of a fluid-fluid interface along a solid surface, J. Fluid Mech., 65, 71-95 (1974) · Zbl 0282.76004
[4] Huh, C.; Scriven, L. E., Hydrodynamic model of steady movement of a solid/liquid/fluid contact line, J. Colloid Interface Sci., 35, 85-101 (1971)
[5] Lowndes, J., The numerical simulation of the steady movement of a fluid meniscus in a capillary tube, J. Fluid Mech., 101, 631-646 (1980) · Zbl 0456.76022
[6] Silliman, W. J.; Scriven, L. E., Separating flow near a static contact line: slip at a wall and shape of a free surface, J. Comput. Phys., 34, 287-313 (1980) · Zbl 0486.76064
[7] Socolowsky, J., Mathematische Untersuchungen freier Randwertaufgaben der Hydrodynamik viskoser Flüssigkeiten, Habilitation thesis (1989), Merseburg
[8] Verfürth, R., Finite element approximation of incompressible Navier-Stokes equations with slip boundary condition, Numer. Math., 50, 697-721 (1987) · Zbl 0596.76031
[9] Kröner, D., Asymptotische Entwicklungen für Strömungen von Flüssigkeiten mit freiem Rand und dynamischem Kontaktwinkel, SFB-72 Preprint 809, 1-105 (1986), Bonn · Zbl 0687.76025
[10] Kröner, D., The flow of a fluid with a free boundary and dynamic contact angle, Z. angew. Math. Mech. (ZAMM), 67, T304-T306 (1987) · Zbl 0632.76038
[11] Kistler, S. F.; Scriven, L. E., Coating flow computations, (Pearson, J. R.A.; Richardson, S. G., Computational Analysis of Polymer Processing (1983), Applied Science Publishers), 243-299 · Zbl 0508.76053
[12] Kistler, S. F.; Scriven, L. E., Coating flow theory by finite element and asymptotic analysis of the Navier-Stokes system, Int. J. numer. Meth. Fluids, 4, 207-229 (1984) · Zbl 0555.76026
[13] Socolowsky, J.; Bergmann, J., Leitlinientechnik zur Berechnung von Strömungen nichtlinear-viskoser Medien bei Beschichtungsvorgängen, Technische Mechanik, 7, 21-29 (1986)
[14] Socolowsky, J., Eine verallgemeinerte Leitlinienmethode zur Berechnung mehrschichtiger Strömungen nichtlinear-viskoser Fluide, J. Appl. Math. Phys. (ZAMP), 39, 221-232 (1988) · Zbl 0642.76011
[15] Bemelmans, J., On a free boundary problem for the stationary Navier-Stokes equations, Ann. Inst. H. Poincare, 4, 6, 517-547 (1987) · Zbl 0637.76027
[16] Erunova, I. B., Investigation of the problem of the stationary motion of two liquids in a container, Dokl. Akad. Nauk. SSSR, 279, 53-58 (1984), (In Russian.) · Zbl 0586.76171
[17] Osmolovskii, V. G., The free surface of a drop in a symmetric force field, Zap. Nauchn. Sem. LOMI, 52, 160-174 (1975), (In Russian.) · Zbl 0358.76026
[18] Pileckas, K., Solvability of a problem of plane motion of a viscous incompressible fluid with noncompact free boundary, Diff. Eqns Appl. Inst. Math. Cybern. Acad. Sci. Lit. SSR, 30, 57-96 (1981), (In Russian.) · Zbl 0504.76032
[19] Pileckas, K., On the problem of motion of heavy viscous incompressible fluid with noncompact free boundary, Lit. Math. J.. Lit. Math. J., Zap. Nauchn. Sem. LOMI, 110, 2, 174-179 (1981), (In Russian.)
[20] Pileckas, K.; Zajaczkowski, W. M., On the free boundary value problem for stationary compressible Navier-Stokes equations, Communs math. Phys., 129, 169-204 (1990) · Zbl 0715.35088
[21] Pukhnachov, V. V., Plane stationary free boundary problem for Navier-Stokes equation, Zh. Prikl. Mekh. I Tekhn. Fiz., 3, 91-102 (1972), (In Russian.)
[22] V. Ya, Rivkind, A priori estimates and the method of successive approximations for the motion of a drop, Trudy Mat. Inst. Steklov, 159, 150-166 (1983), (In Russian.)
[23] Socolescu, D., Existenz- und Eindeutigkeitabeweis für ein freies Randwertproblem für die stationären Navier-Stokesschen Bewegungsgleichungen, ARMA, 73, 3, 191-243 (1980) · Zbl 0442.76028
[24] Socolowsky, J., Solvability of a stationary problem on the plane motion of two viscous incompressible liquids with non-compact free boundaries, Z. angew. math. mech. (ZAMM). Z. angew. math. mech. (ZAMM), Zap. Nauchn. Sem. LOMI, 163, 146-153 (1987), (In Russian.)
[25] Solonnikov, V. A., Solvability of a three-dimensional boundary value problem with a free surface for the stationary Navier-Stokes system, Zap. Nauchn. Sem. LOMI, 84, 252-285 (1979), (In Russian.) · Zbl 0414.35062
[26] Solonnikov, V. A., Solvability of the problem of plane motion of a heavy viscous incompressible capillary liquid partially filling a container, Izv. Akad. Nauk SSSR. Izv. Akad. Nauk SSSR, Math. USSR-Izv., 14, 193-221 (1980), English translation · Zbl 0429.76020
[27] Solonnikov, V. A., On the Stokes equation in domains with nonsmooth boundaries and on a viscous incompressible flow with a free surface, Nonlinear Partial Differential Equations and their Applications, Vol. 3, 340-423 (1980/1981), College de France Seminar
[28] Solonnikov, V. A., On a unsteady motion of an isolated volume of a viscous incompressible fluid, Izv. Akad. Nauk SSSR Ser. Mat., 51, 1065-1087 (1987), (In Russian.)
[29] Teramoto, Y., The initial value problem for viscous incompressible flow down an inclined plane, Hiroshima math. J., 15, 619-643 (1985) · Zbl 0625.76030
[30] Kondrat’ev, V. A., Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Moskov. Mat. Obshch., 16, 209-292 (1967) · Zbl 0162.16301
[31] Solonnikov, V. A., On the solvability of boundary and initial-boundary problems for the Navier-Stokes system in domains with noncompact boundaries, Pacific J. math., 93, 443-458 (1981) · Zbl 0413.35062
[32] Solonnikov, V. A.; Shchadilov, V. E., On a boundary value problem for a stationary system of Navier-Stokes equations, Trudy Mat. Inst. Steklov. Trudy Mat. Inst. Steklov, Proc. Steklov Inst. Math., 125, 186-199 (1973), English translation · Zbl 0313.35063
[33] Socolowsky, J., Elementary investigations of the equation for a free surface, Technische Mechanik, 12, 257-259 (1991)
[34] Finn, R.; Shinbrodt, M., The capillary contact angle. I. The horizontal plane and stick-alip motion, J. math. Analysis Applic., 123, 1-17 (1987)
[35] Finn, R.; Shinbrodt, M., The capillary contact angle. II. The inclined plane, 1-41 (1987), (preprint).
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