On Proof of the Generalized Lagrange Variational Principle
The proof of the generalized Lagrange variational principle for the case when the velocity field is a vortex of some auxiliary vector field is given in the present paper. The proof is obtained for Newtonian fluids. It is demonstrated that the generalized Lagrange functional takes a minimum value on a real field. The generalized Lagrange variational principle extends the class of solving problems to quasistationary ones and can be applied to solve problems in the hydrodynamic lubrication theory. To verify the theoretical results, a numerical solution of the variational problem of fluid flow in a thin layer between rigid parallel plates is performed. The numerical results match with the analytical results with a high accuracy.
KeywordsVariational principle Mechanics of continua Hydrodynamic lubrication theory Newtonian fluid
- 3.Koutcheryaev BV (2000) Mechanics of continua. MISIS, Moscow (in Russian)Google Scholar
- 6.Korovchinsky MV (1959) Theoretical basics of journal bearings. Mashgiz, Moscow (in Russian)Google Scholar
- 13.Roache P (1998) Fundamentals of computational fluid dynamics. Hermosa Pub, SocorroGoogle Scholar
- 15.GNU Octave (2017) Official site. http://www.gnu.org/software/octave. Accessed 30 Dec 2017
- 16.Salvadori M, Baron M (1961) Numerical methods in engineering, 2nd edn. Prentice-Hall, Englewood CliffsGoogle Scholar