Asymptotic solution of the Navier-Stokes problem on the flow of a thin layer of fluid. (English. Russian original) Zbl 0712.76037

Sib. Math. J. 31, No. 2, 296-307 (1990); translation from Sib. Mat. Zh. 31, No. 2(180), 131-144 (1990).
In Sect. 1 of the present paper we find the complete asymptotic expansion of a solution of the Navier-Stokes problem for an arbitrary motion of the walls of a vessel (in particular, the influence of the lateral surface is taken into account.) For this one uses a version of the algorithm for constructing an asymptotic solution of an elliptic boundary problem in a thin domain. In Sect. 2 the asymptotic expansion obtained is justified: the existence of a nearly asymptotic solution of the nonlinear problem is proved, the rate of convergence is estimated, the limits of applicability of the two-dimensional approximation are discussed. In Sect. 3, formally (without justification of the asymptotics obtained) another two- dimensional equation describing the flow of a thin layer of fluid with free surface is derived. Thus, the question of applicability of the lubrication equation under separation of the flow from one of the walls is solved negatively. Finally, in point 3.3 we study the problem of deformation of a layer of a slightly shrinking material included between two absolutely rigid bodies which is similar in formulation.


76D05 Navier-Stokes equations for incompressible viscous fluids
76D08 Lubrication theory
35Q30 Navier-Stokes equations
Full Text: DOI


[1] O. Reynolds, Hydrodynamic Theory of Lubrication [Russian translation], Gostekhizdat, Moscow-Leningrad (1934).
[2] N. P. Petrov, Hydrodynamic Theory of Lubrication: Selected Works [in Russian], Izd. Akad. Nauk SSSR, Moscow (1948).
[3] G. Bayada and M. Chambat, ?The transition between the Stokes equations and the Reynolds equation: a mathematical proof,? Appl. Math. Optim.,14, No. 1, 73-93 (1986). · Zbl 0701.76039
[4] A. L. Gol’denveiser, ?Construction of an approximate theory of bending of a plate by the method of asymptotic integration of the equations of the theory of elasticity,? Prikl. Mat. Mekh.,26, No. 4, 668-686 (1962).
[5] M. G. Dzhavadov, ?Asymptotics of a solution of a boundary problem for second order elliptic equations in thin domains,? Differents. Uravn.,4, No. 10, 1901-1909 (1968). · Zbl 0165.12302
[6] E. I. Zino and E. A. Tropp, Asymptotic Methods in Problems of Thermal Conductivity and Thermelasticity [in Russian], LGU, Leningrad (1978).
[7] S. A. Nazarov, ?Structure of solutions of elliptic boundary problems in thin domains,? Vestn. LGU, No. 7, 65-68 (1982). · Zbl 0509.35008
[8] V. V. Kucherenko and V. A. Popov, ?Asymptotics of solutions of problems of the theory of elasticity in thin domains,? Dokl. Akad. Nauk SSSR,274, No. 1 58-61 (1984).
[9] V. L. Berdichevskii, Variational Principles of Mechanics of Continuous Media [in Russian], Nauk, Moscow (1983).
[10] S. N. Leora, S. A. Nazarov, and A. V. Proskura, ?Derivation of limit equations for elliptic problems in thin domains with the help of computers,? Zh. Vychisl. Mat. Mat. Fiz.,26, No. 7, 1032-1048 (1986). · Zbl 0626.65129
[11] S. Agmon and L. Nirenberg, ?Properties of solutions of ordinary differential equations in Banach spaces,? Comm. Pure Appl. Math.,16, No. 1, 121-239 (1963). · Zbl 0117.10001
[12] V. A. Kondrat’ev, ?Boundary problems for elliptic equations in domains with conical or corner points,? Tr. Mosk. Mat. Obshch.,16, 209-292 (1967).
[13] V. G. Maz’ya and B. A. Plamenevskii, ?Coefficients in asymptotic solutions of elliptic boundary problems in a domain with conical points,? Math. Nachr.,81, 25-82 (1978). · Zbl 0371.35018
[14] O. A. Ladyzhenskaya, Mathematical Questions of the Dynamics of a Viscous Fluid [in Russian], Nauka, Moscow (1970). · Zbl 0215.29004
[15] V. A. Solonnikov, ?Solvability of a problem on the plane motion of a turbid viscous incompressible fluid partially filling a vessel,? Izd. Akad. Nauk SSR, Ser. Mat.,43, No. 1, 203-236 (1979).
[16] R. Temam, Navier-Stokes Equation. Theory and Numerical Analysis [Russian translation], Mir, Moscow (1981). · Zbl 0529.35002
[17] V. G. Maz’ya and B. A. Plamenevskii, ?Estimates in Lp and in Hölder classes and the Miranda-Agmon principle for solutions of elliptic boundary problems in domains with singular points on the boundary,? Math. Nachr.,76, 29-60 (1977). · Zbl 0359.35024
[18] S. Agmon, A. Douglis, and L. Nirenberg, ?Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions,? Comm. Pure Appl. Math.,17, No. 1, 35-92 (1964). · Zbl 0123.28706
[19] V. A. Solonnikov, ?Solvability of a three-dimensional problem with free boundary for a stationary system of Navier-Stokes equations,? Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst.,84, 252-285 (1979). · Zbl 0414.35062
[20] V. G. Maz’ya, B. A. Planmenevskii, and L. I. Stupyalis, ?Three-dimensional problem of steady state motion of a fluid with free surface,? in: Differential Equations and their Application [in Russian], Vol. 23, Izdat. LitSSR, Vilnius (1979), pp. 29-153.
[21] V. I. Malyi, ?Asymptotic solution of the problem of compression of a layer of slightly compressible material,? in: Mechanics of Elastomers [in Russian], KPI, Krasnodar (1983), pp. 38-44.
[22] L. V. Milyakova and K. F. Chernykh, ?General linear theory of thinly-layered rubbermetallic elements,? Mekh. Tverd. Tela, No. 3, 110-120 (1986).
[23] V. M. Mal’kov, ?Linear theory of a thin layer of slightly compressible material,? Dokl. Akad. Nauk SSSR,293, No. 1, 42-44 (1987).
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