# Etude mathématique et analyse asymptotique de quelques problèmes de lubrification par des fluides incompressibles essentiellement non-Newtoniens avec des conditions de non adhérence aux bords.

Abstract : In this thesis, we study some problems of lubrication by isothermal and non-isothermal non-Newtonian fluids in a thin film $\Omega^{\varepsilon}$ with the condition of Tresca at the bottom of $\Omega^{\varepsilon}$.

In the first chapter, we consider an isothermal non-Newtonian fluids whose viscosity follows the power law, we show the existence and uniqueness of the solution by using theorical results of the pseudo-monotonous operators. Then, we study the asymptotic analysis when $\varepsilon \rightarrow 0$. Thus, we obtain a limit problem, and we show the uniqueness of its solutions.

In the second chapter, we study the problem in the non-isothermal case. The problem obtained is complex and strongly nonlinear. The difficulty here is the proof of the theorem giving the existence of the solutions, as well as the a priori estimates on the gradient of the temperature.

In the third chapter, we study the Navier-Stokes equation, where the parameter $\varepsilon$ is also present in the movement equation in the form of the Reynolds number $\varepsilon^{\gamma}$ , and in the condition of friction of Tresca. We show the existence and the uniqueness of the solutions under a condition on $\varepsilon$ and $\gamma$. By a similar technics to that used in the previous chapters, we obtain the convergence results of the solution towards the limit problem solution.

In the last chapter, we treat another model of non-Newtonian fluids, the viscoplastic fluids of Bingham. We suppose in more of the condition of Tresca, another condition of Fourier at the Top of $\Omega^{\varepsilon}$. The difficulties here are technical and relate to the a priori estimates, especially to estimate the terms at boundaries.
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https://tel.archives-ouvertes.fr/tel-00011638
Contributor : Rachid El Mir <>
Submitted on : Friday, February 17, 2006 - 2:46:06 PM
Last modification on : Wednesday, November 20, 2019 - 3:08:41 AM
Long-term archiving on: : Saturday, April 3, 2010 - 10:27:46 PM

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• HAL Id : tel-00011638, version 1

### Citation

Rachid El Mir. Etude mathématique et analyse asymptotique de quelques problèmes de lubrification par des fluides incompressibles essentiellement non-Newtoniens avec des conditions de non adhérence aux bords.. Mathématiques [math]. Université Jean Monnet - Saint-Etienne, 2005. Français. ⟨tel-00011638⟩

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