## 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.

### MSC:

 76D05 Navier-Stokes equations for incompressible viscous fluids 76D08 Lubrication theory 35Q30 Navier-Stokes equations
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### References:

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