Homogenization of a lubrication problem in oscillating domain by two-scale convergence method
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2021), pp. 31-68.

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In present paper we do a homogenization with respect to a small parameter of a boundary-value problem describing fluid flow between two moving in space and time rough surfaces. The two-scale convergence method was used to justify the behavior of the flow in the limit.
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Y. O. Koroleva; A. V. Korolev. Homogenization of a lubrication problem in oscillating domain by two-scale convergence method. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2021), pp. 31-68. https://geodesic-test.mathdoc.fr/item/BASM_2021_1_a1/

[1] Allaire G., “Homogenization of the Stokes flow in a connected porous medium”, Asympt. Analysis, 2 (1989), 203–222 | DOI | MR | Zbl

[2] Allaire G., “Homogenization of the unsteady Stokes equations in porous media”, Progress in partial differential equations: calculus of variations, applications (Pont-Mousson, 1991), Pitman Res. Notes Math. Ser., 267, Longman Sci. Tech., Harlow, 1992, 109123 | MR

[3] Allaire G., “Homogenization and two-scale convergence”, SIAM J. Math. Anal., 23 (1992), 1482–1518 | DOI | MR | Zbl

[4] Amirat Y., Chechkin G. A., Romanov M. S., “On Multiscale Homogenization Problems in Boundary Layer Theory”, Zeitschrift für angewandte Mathematik und Physik, 63:3 (2012), 475–502 | DOI | MR | Zbl

[5] Amirat Y., Bodart O., Chechkin G. A., Piatnitski A. L., “Boundary homogenization in domains with randomly oscillating boundary”, Stochastic Processes and their Applications, 121 (2011), 1–23 | DOI | MR | Zbl

[6] D'Apice Ciro, Chechkin G. A., De Maio Umberto, “On the Rate of Convergence of Solutions in Domain with Periodic Multilevel Oscillating Boundary”, Mathematical Methods in the Applied Sciences (M2AS), 33:3 (2010), 2019–2036 | MR

[7] Bayada G., Chambat M., “The transition between the Stokes equation and the Reynolds equation: A mathematical proof”, Appl. Math. Optim., 14 (1986), 73–93 | DOI | MR | Zbl

[8] Bayada G., Chambat M., “New models in the theory of the hydrodynamic lubrication of rough surfaces”, J. of Trib., 110:3 (1988), 402–407 | DOI

[9] Bayada G., Chambat M., “Homogenization of the Stokes system in a thin film flow with rapidly varying thickness”, RAIRO Model. Math. Anal. Numer., 23:2 (1989), 205–234 | DOI | MR | Zbl

[10] Bayada G., Chambat M., Ciuperca I., “Asymptotic Navier-Stokes equations in a thin moving boundary domain”, Asympt. Anal., 21 (1999), 117–132 | MR | Zbl

[11] Belyaev A. G., Mikheev A. G., Shamaev A. S., “Plane wave diffraction by a rapidly oscillating surface”, Comput. Math. Math. Phys., 32 (1992), 1121–1133 | MR | Zbl

[12] Benhaboucha N., Chambat M., Ciuperca I., “Asymptotic behaviour of pressure and stresses in a thin film flow with a rough boundary”, Quarterly of Applied Mathematics, 63:2 (2005), 369–400 | DOI | MR | Zbl

[13] Bouchitte G., Lidouh A., Sequet P., “Homogénéisation de frontière pour la modélisation du contact entre un corps déformable non linéaire et un corps rigide [Boundary homogenization and modeling of friction between a nonlinear deformable body and a rigid body]”, C. R. Acad. Sci. Paris Ser. I Math., 313 (1991), 967–972 (French) | MR | Zbl

[14] Bunoiu R., Saint Jean Paulin J., “Linear flow in porous media with double porosity”, Portugalie Matematica, 56:2 (1999), 308–340 | MR

[15] Cattabriga L., “Su un problema al contorno relativo al sistema di equazioni di Stokes”, Rend. Sem. Mat. Univ. Padova, 31 (1961), 308–340 (Italian) | MR | Zbl

[16] Chechkin G. A., Linkevich A. Yu., Ratiu T. S., Spiridonov S. V., “On a Thin Layer of Non-Newtonian Fluid on Rough Surface Percolating through Perforated Obstacle”, Journal of Mathematical Sciences, 189:3 (2013), 525–535 | DOI | MR | Zbl

[17] Chechkin G. A., Linkevich A. Yu., Spiridonov S. V., “On Boundary Layer of Newtonian Fluid Passing Rough Surface and Penetrating Through Perforated Wall”, Ufa Mathematical Journal, 3:3 (2011), 90–101 | MR

[18] Elrod H. G., “Thin-film lubrication theory for Newtonian fluids with surfaces possessing striated roughness or grooving”, J. of Lub. Tech., 95 (1973), 484–489 | DOI

[19] Fabricius J., Koroleva Y. O., Wall P., “A rigorous derivation of the time-dependent Reynolds equation”, Asympt. Anal., 84:1–2 (2013), 103–121 | MR | Zbl

[20] Fabricius J., Koroleva Y. O., Tsandzana A., Wall P., “Asymptotic behavior of Stokes flow in a thin domain with a moving rough boundary”, Proc. R. Soc. A, 470:2167 (2014), 20130735, 21 pp. | DOI | MR | Zbl

[21] Girault V. and Raviart P. A., Finite element approxomation of the Navier-Stokes equations, Rvd Reprinted, Springer Verlag, Berlin, 1981 | MR

[22] Jäger W. and Mikelić A., “On the flow conditions at the boundary between a porous medium and an impervious solid”, Progress in partial differential equations: the Metz surveys, v. 3, Pitman Res. Notes Math. Ser., 314, Longman Sci. Tech., Harlow, 1994, 145160 | MR

[23] D. Lukkassen, A. Meidell, P. Wall, “Homogenization of some variational problems connected to the theory of lubrication”, International Journal of Engineering Science, 47:1 (2009), 153–169 | DOI | MR

[24] Marušić S., Marušić–Paloka E., “Two-scale convergence for thin domains and its applications to some lower-dimensional models in fluid mechanics”, Asymptot. Anal., 23:1 (2000), 23–57 | MR | Zbl

[25] Siberian Math. J., 31:2 (1990), 296–307 | DOI | MR | Zbl

[26] Nazarov S. A., Videman J. H., “A modified nonlinear Reynolds equation for thin viscous flows in lubrication”, Asymptot. Anal., 52:1–2 (2007), 1–36 | MR | Zbl

[27] Lithuanian Math. J., 30:4 (1990), 366–375 | DOI | MR | Zbl

[28] Nguetseng G., “A general convergence result for a functional related to the theory of homogenization”, SIAM J. Math. Anal., 20:3 (1989), 608–623 | DOI | MR | Zbl

[29] Patir Thien N., Cheng H. S., “An average flow model for determining effects of three-dimensional roughness on partial hydrodynamic lubrication”, J. of Lub. Tech., Trans. ASME, Ser. F, 100 (1978), 12–17 | DOI

[30] Reynolds O., “On the theory of lubrication and its aplication to Mr. Beauchamp Tower's experiments, incluing an experimental dertemination of the viscosity of olive oil”, Phi. Trans. of the Roy. Soc. of London, 177 (1886), 157–234 | DOI

[31] Sanchez-Palencia E., Non–Homogeneous Media and Vibration Theory, Springer-Verlag, Berlin, 1980 | MR | Zbl

[32] Tartar L., “Incompressible fluid flow in a porous medium-convergence of the homogenization process”: Sanchez-Palencia E., Non–Homogeneous Media and Vibration Theory, Springer-Verlag, Berlin, 1980, 368–377 | MR

[33] Temam R., Navier-Stokes equations, North Holland, Amsterdam, 1979 | MR | Zbl