On the approximation of turbulent fluid flows by the Navier-Stokes-α\alpha equations on bounded domains

The Navier-Stokes-$\alpha$ equations belong to the family of LES (Large Eddy Simulation) models whose fundamental idea is to capture the influence of the small scales on the large ones without computing all the whole range present in the flow. The constant $\alpha$ is a regime flow parameter that has the dimension of the smallest scale being resolvable by the model. Hence, when $\alpha=0$, one recovers the classical Navier-Stokes equations for a flow of viscous, incompressible, Newtonian fluids. Furthermore, the Navier-Stokes-$\alpha$ equations can also be interpreted as a regularization of the Navier-Stokes equations, where $\alpha$ stands for the regularization parameter. In this paper we first present the Navier-Stokes-$\alpha$ equations on bounded domains with no-slip boundary conditions by means of the Leray regularization using the Helmholtz operator. Then we study the problem of relating the behavior of the Galerkin approximations for the Navier-Stokes-$\alpha$ equations to that of the solutions of the Navier-Stokes equations on bounded domains with no-slip boundary conditions. The Galerkin method is undertaken by using the eigenfunctions associated with the Stokes operator. We will derive local- and global-in-time error estimates measured in terms of the regime parameter $\alpha$ and the eigenvalues. In particular, in order to obtain global-in-time error estimates, we will work with the concept of stability for solutions of the Navier-Stokes equations in terms of the $L^2$ norm

An Error Estimate Uniform in Time for Spectral Galerkin Approximations for the Equations for the Motion of a Chemical Active Fluid

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The Initial Value Problem for the Equations of Magnetohydrodynamic Type in Non-Cylindrical Domain

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Magneto-micropolar fluid motion: existence of weak solutions

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Optimal error estimates of the penalty finite element method for micropolar fluids equations

An optimal error estimate of the numerical velocity, pressure and angular velocity, is proved for the fully discrete penalty finite element method of the micropolar equations, when the parameters ², ∆t and h are sufficiently small. In order to obtain above we present the time discretization of the penalty micropolar equation which is based on the backward Euler scheme; the spatial discretization of the time discretized penalty Micropolar equation is based on a finite elements space pair (Hh, Lh) which satisfies some approximate assumption

Global solution of nematic liquid crystals models

We prove existence of a global weak solution for a nematic liquid crystal problem by means of a penalization method using a simplified Ericksen-Leslie model and a new compactness property for the gradient of the director field

Optimal control problem for the generalized bioconvective flow

In this work, we consider an optimal control problem for the generalized bioconvective flow, which is a well known model to describe the convection caused by the concentration of upward swimming microorganisms in a fluid. Firstly, we study the existence and uniqueness of weak solutions for this model, moreover we prove the existence of the optimal control and we establish the minimum principle by using Dubovitskii-Milyutin’s formalism.DGI-MEC BFM2003- 06446CGCI MECD-DGU Brazil/Spain 117/06FONDECYT 103094