Dynamique des fluides de grade deux

Abstract : This thesis is devoted to the study of the second grade fluid system. When the material coefficient $\alpha$ is small, these equations can be considered as a singular perturbation of the Navier-Stokes equations since they involve a third order derivative term. In the first part, we consider the equations of a rotating incompressible non-Newtonian fluid flow of grade two in a three dimensional torus. We obtain two different results of global existence of strong solutions. In the first case, we consider an arbitrary coefficient $\alpha$ and we suppose that the third components of the vertical average of the initial data and of the forcing term are small compared to the horizontal components. In the second case, we consider a forcing term of arbitrary size and large initial data but we need to restrict the size of $\alpha$. In both cases, we show that the system of a rotating second grade fluid converges to a limit system composed of a linear system and a second grade fluid system with two variables and three components. The global existence of solutions of this limit system with three components plays a big role in the proof. In the second part, we study the large time behavior of solutions of the second grade fluid system in the space $\mathbb{R}^2$. Using scaling variables and performing energy estimates in weighted Sobolev spaces, we prove that the solutions of the second grade fluid equations converge to the Oseen vortex, if the initial data are small enough. We also give an estimate of the rate of convergence. The last part of this thesis concerns the study of the comparaison of the dynamics of the second grade fluid system with the ones of the Navier-Stokes equations, in the two-dimensional case. We show that, if $z_0$ is an hyperbolic equilibrium point of the Navier-Stokes equations, the second grade fluid system has a unique equilibrium point $z_{\alpha}$ in a neighborhood of $z_0$, if $\alpha$ is small enough. Next, we construct the local unstable manifold of $z_{\alpha}$ and we compare it to the local unstable manifold of $z_0$.
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Basma Jaffal. Dynamique des fluides de grade deux. Equations aux dérivées partielles [math.AP]. Université Paris Sud - Paris XI, 2010. Français. ⟨tel-00640385⟩

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