In this paper we prove the logarithmically improved Beale-Kato-Majda's criterion to the three-dimensional incompressible Euler and Navier-Stokes equations as well as the logarithmically improved Serrin's criteria to the three-dimensional incompressible Navier-Stokes equations.

This chapter illustrates the use of algebraic flux correction in the context of finite element methods for the incompressible Navier-Stokes equations and related models. In the convection-dominated flow regime, nonlinear stability is enforced using algebraic flux correction. The numerical treatment of the incompressibility constraint is based on the ‘Multilevel Pressure Schur Complement’ (MPSC)...

Two-dimensional shallow water systems are frequently used in engineering practice to model environmental flows. The benefit of these systems are that, by integration over the water depth, a two-dimensional system is obtained which approximates the full three-dimensional problem. Nevertheless, for most applications the need to propagate waves over many wavelengths means that the numerical soluti...

Several inviscid models in hydrodynamics and geophysics such as the incompressible Euler vorticity equations, the surface quasi-geostrophic equation and the Boussinesq equations are not known to have even local well-posedness in the corresponding borderline Sobolev spaces. Here H is referred to as a borderline Sobolev space if the L∞-norm of the gradient of the velocity is not bounded by the H-...

The explosive growth in integration technology and the parallel nature of rasterization-based graphics APIs changed the panorama of consumer-level graphics: today, GPUs are cheap, fast and ubiquitous. We show how to harness the computational power of GPUs and solve the incompressible Navier-Stokes fluid equations significantly faster (more than one order of magnitude in average) than on CPU sol...

In this paper we study finite dimensional approximations to Boussinesq type equations. Our methods are based on infinite dimensional center manifold theory. The main advantage of our approach is that we can handle both well-posed and ill-posed versions of the Boussinesq equation. We show that for suitable initial conditions, our approximations describe the dynamics accurately for long enough ti...

1 Institute for Physical Science and Technology and Department of Mathematics, University of Maryland, College Park, Maryland 20742. 2 Institute for Scientific Computing and Applied Mathematics and Department of Mathematics, Indiana University, Bloomington, Indiana 47405. E-mail: [email protected] 3 Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109. Received February ...

This paper reports the result of an analytical investigation of a steady, incompressible and viscous flow between two eccentric, rotating cylinders at high Reynolds number. A one dimensional case is far from reality because the gap between the cylinders is very small. Further, when their axes are displaced by a small distance, usually caused by bearing loads, two dimensional effects become obvi...

A finite element approach for solving gas-liquid flows with mass transfer and chemical reactions is presented. The incompressible Navier-Stokes equations coupled with a modified k − ε turbulence model for computation of the eddy viscosity are solved by a discrete projection scheme from the family of Pressure Schur Complement methods. Additionally, the simplified two-fluid (Euler-Euler) model ba...