About the obtaining self-similar solutions of the Navier-Stokes equations by methods of power geometry
by
Mikhail M. Vasiliev
Keldysh Institute of Applied Mathematics, Moscow 125047 Russia

The methods of Power Geometry [1] were applied to the hydrodynamical problem of the viscous incompressible fluid steady flow around the semi-infinite flat plate [2]. This problem is described by one partial differential equation for the stream function with the corresponding boundary conditions.
After that the Power Geometry was applied for the first time to the boundary problem for the system of the partial differential equations [3]. The system describes the plane steady flows of the viscous compressible heat conducting fluid. It was established that Newton polyhedra belong to the 3D subspace of the original 5D space. That essentially simplifies the study. Various truncated systems of the Navier-Stokes equations are determinated with the help of the Newton polyhedra. In order to demonstrate the method, one of these systems is used for the obtaining the well-known solution of the boundary layer problem. It is the problem of boundary layer on the semi-infinite flat plate.
Another system is used for the obtaining the principal part of the asymptotics in infinity for the flow in the plane diffuser. In this case the energy balance equation contains only one function (enthalpy) and could be integrated analytically.
The truncated systems of the Navier-Stokes equations were obtained also for the case of axial-symmetric flows. One of these systems was applied to the study of the asymptotic behaviour of the flow in infinity in a conical diffuser.
The approximate analytical solutions for small diffuser angles were obtained both for the plane and for the conical diffuser.

References
1. Bruno A.D. Power Geometry in Algebraic and Differential Equations. M.: Nauka, Fizmatlit, 1998, 288pp. (Russian) = Elsevier Science, Amsterdam, 2000
2. Bruno A.D., Vasiliev M.M. Asymptotic analysis of the viscous fluid flow around a flat plate by the Newton polyhedron. Nonlinear Analysis, Theory, Methods and Applications, 30:8, 1997, 4765-4770.
3. Vasiliev M.M. About the obtaining self-similar solutions of the viscous heat conducting gas equations. Preprint N 95, Keldysh Inst. of Appl. Math. of RAS, M.: 1997. (Russian) = Vasiliev M.M. About the asymptotic analysis of the viscous heat conducting gas flow equations. Preprint N 65, Keldysh Inst. of Appl. Math. of RAS, M.: 1998.

Date received: May 21, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahk-77.