# Sur quelques problèmes de lubrification par des fluides newtoniens non isothermes avec des conditions aux bords non linéaires. Etude mathématique et numérique

Abstract : In the first chapter of this thesis, we recall the basic principles of the continuous media its mechanics from which we deduce the equations modelling the nonisothermal flow of an incompressible Newtonian fluid. In the second chapter, we consider the stationary case in a thin domain and we add the boundary conditions of Tresca's type on one part of the boundary of the domain. We deduce the corresponding variational problem which is strongly coupled, composed by a variational inequality and an equality, whose unknowns are the velocity field of the fluid, his pressure and its temperature. The principal difficulty is the presence in the variational equality of a term comprising the square of the deformation rate tensor, which does not make possible to give sense to the variational problem, if we search the velocity in $H^1$-convex. To overcome this difficulty, we search the $H^2$-regularity of the velocity, which requires the $\mathcal(C)^(0,1)$-regularity of the temperature, which is in the coefficients of the variational inequality. By using the Banach fixed point theorem, we show the existence, uniqueness and regularity of weak solution. The third chapter is devoted to the asymptotic analysis of this coupled variational problem in $\Om^\eps$. We estabilsh the $\eps$-estimates of the $H^1$-norm for the partial derivative of velocity and temperature, and of the $L^2$-norm for the partial derivative of the pressure. This enables us to obtain the strong limits. We then obtain the limit problem, the generalized Reynold equation and we prove the uniqueness of solutions to this limit problem. In the fourth chapter, we present an approximation of the limit problem by the finite elements methods, we study the convergence of the approximate solutions and we give the error estimates of the approximation. In the final chapter, we replace in preceding study the Tresca's boundary condition by Coulomb's and we obtain the similar results.
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Submitted on : Thursday, March 10, 2005 - 3:40:05 PM
Last modification on : Wednesday, November 20, 2019 - 3:20:20 AM
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• HAL Id : tel-00008745, version 1

### Citation

Fouad Saidi. Sur quelques problèmes de lubrification par des fluides newtoniens non isothermes avec des conditions aux bords non linéaires. Etude mathématique et numérique. Mathématiques [math]. Université Jean Monnet - Saint-Etienne, 2004. Français. ⟨tel-00008745⟩

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