We address the problem of the uniqueness of the particle trajectories corresponding to a weak solution of the two-dimensional Navier-Stokes equations with square integrable initial condition and regular enough forcing function. In order to do this we obtain an estimate for H-norm of the weak solutions of the Navier-Stokes equations and show that t‖D3/2u‖ is bounded which provides a bound on the...

In this project report I present my research and accompanying implementation of the unconditionally stable Navier-Stokes partial differential equation (PDE) solver presented to the computer graphics community by Jos Stam at the 1999 SIGGRAPH conference [Stam 1999]. I give a brief introduction to computational fluid dynamics (CFD), the governing Navier-Stokes equations and an overview of previou...

We consider a multidimensional model for the dynamic combustion of compressible reacting fluids formulated by the Navier Stokes equations in Euler coordinates. For the chemical model we consider a one way irreversible chemical reaction governed by the Arrhenius kinetics. The existence of globally defined weak solutions of the Navier-Stokes equations for compressible reacting fluids is establish...

Formulated in terms of velocity, pressure and the extra stress tensor, the incompressible Navier-Stokes equations are discretized by stabilized finite element methods. The stabilized methods proposed are analyzed for a linear model and extended to the Navier-Stokes equations. The numerical tests performed confirm the good stability characteristics of the methods. These methods are applicable to...

We consider a finite volume scheme for the two-dimensional incompressible Navier-Stokes equations. We use a triangular mesh. The unknowns for the velocity and pressure are respectively piecewise constant and affine. We use a projection method to deal with the incompressibility constraint. The stability of the scheme has been proven in [15]. We infer from it its convergence. Mathematics Subject ...

We investigate the singularity formation of a 3D model that was recently proposed by Hou and Lei (2009) in [15] for axisymmetric 3D incompressible Navier–Stokes equations with swirl. The main difference between the 3D model of Hou and Lei and the reformulated 3D Navier–Stokes equations is that the convection term is neglected in the 3D model. This model shares many properties of the 3D incompre...

Continuum equations are derived for the large-scale behavior of a class of cellular automaton models for fluids. The cellular automata are discrete analogues of molecular dynamics, in which particles with discrete velocities populate the links of a fixed array of sites. Kinetic equations for microscopic particle distributions are constructed. Hydrodynamic equations are then derived using the Ch...

In this work we present a technique of fast numerical computation for solutions of Navier-Stokes equations in the case of flows of industrial interest. At first the partial differential equations are translated into a set of nonlinear ordinary differential equations using the geometrical shape of the domain where the flow is developing, then these ODEs are numerically resolved using a set of co...

We study the local exponential stabilization, near a given steady-state flow, of solutions of the Navier-Stokes equations in a bounded domain. The control is performed through a Dirichlet boundary condition. We apply a linear feedback controller, provided by a well-posed infinite dimensional Riccati equation. We give a characterization of the domain of the closed-loop operator which is obtained...

In this paper, we propose a new paradigm for solving Navier–Stokes equations. The proposed methodology is based on a streamfunction–velocity formulation of the two-dimensional steady-state Navier–Stokes equations representing incompressible fluid flows in two-dimensional domains. Similar formulations are also possible for three-dimensional fluid flows. The main advantage of our formulation is t...