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 Section Mathematics Title Modeling the velocity field of two-layered creeping flow and some geophysical applications Author(-s) Pak V.V.a Affiliations Pacific Oceanological Institute, Far Eastern Branch of the Russian Academy of Sciencesa Abstract We study the long-time evolution of axisymmetric free-surface two-layered creeping flow subject to the initial topography of its boundaries and bottom velocities. Each layer has uniform density and viscosity. The upper layer is assumed to have a smaller density than the lower layer. Based on lubrication approximation (the Reynolds equations) the nonlinear system of diffusion-type equations with respect to the surface and interface between the layers is obtained to describe this flow. Taking the dimensionless density contrast between the layers as a small parameter, we apply the method of asymptotic expansions to extract leading-term approximation for the slowly varying large-time evolution of the governing equations. An asymptotic equation relating both surface and interface displacement to the bottom velocities is derived. Based on this equation, we develop the algorithm to calculate velocity fields within layers for large time. Streamlines are used to visualize the flow. Numerical results reveal stability of the streamlines in the upper layer under variation of the bottom velocity. As geophysical applications, the developed algorithm is used to evaluate the velocity field in the crust (the upper layer) beneath the large-scale lunar multi-ring basins influenced by deep movements in the underlying mantle (the lower layer). To validate the results of modeling, we compare the calculated velocity fields with basin ridge systems obtained by experimental observations. The model comparison has shown proximity of radii of basin rings and critical points of the surface velocity. Keywords multi-layered flow, long-wave approximation, Reynolds equations, nonlinear diffusion, ring structures UDC 532.5.032 MSC 76D50 DOI 10.20537/vm140106 Received 21 November 2013 Language Russian Citation Pak V.V. Modeling the velocity field of two-layered creeping flow and some geophysical applications, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 1, pp. 66-75. References Craster R.V., Matar O.K. Dynamics and stability of thin liquid films, Rev. of Modern Phys., 2009, vol. 81, no. 3, pp. 1131-1198. Merkt D., Pototsky A., Bestehorn M. Long-wave theory of bounded two-layer films with a free liquid–liquid interface: Short- and long-time evolution, Phys. Fluids, 2005, vol. 17, 064104. Pak V.V. The nonlinear model of axisymmetric free-surface two-layered creeping flow, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2010, no. 2, pp. 91-100 (in Russian). Pak V.V. An asymptotic study of three-layered creeping flow and some geophysical applications, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2011, no. 4, pp. 107-115 (in Russian). Nayfeh A.H. Metody vozmushchenii (Perturbation methods), Moscow: Mir, 1976, 456 p. Borisov V.G. On parabolic boundary value problems with a small parameter on the derivatives with respect to $t$, Math. USSR Sb., 1988, vol. 59, no. 2, pp. 287-302. Turcotte D.L., Schubert, G. Geodinamika. Geologicheskoe prilozhenie fiziki sploshnykh sred. Tom 2 (Geodynamics. Applications of Continuum Physics to Geological Problems. Vol. 2), Moscow: Mir, 1985, 360 p. Hikida H., Wieczorek M.A. Crustal thickness of the Moon: New constraints from gravity inversions using polyhedral shape models, Icarus, 2007, vol. 192, pp. 150-166. Pike R.J., Spudis P.D. Basin-ring spacing on the Moon, Mercury, and Mars, Earth, Moon, and Planets, 1987, vol. 39, pp. 129-194. Pak V.V. Modeling the velocity field of the axisymmetric two-layered flow, Fluxes and Structures in Fluids: Proceedings of Int. Conf., Russian Hydrometeorological State University, Saint-Petersburg, 2013, pp. 229-231. Full text