Oseen equations

The Oseen equations ( after Carl Wilhelm Oseen ) is a mathematical model of the flow of incompressible fluids and gases in stationary equilibrium. In general, such fluid streams are described by the time-dependent incompressible Navier -Stokes equations with which the Oseen equations are used.

Formulation

As the incompressible Navier -Stokes equations, the equations Oseen a system of partial differential equations in four unknowns (speed in three spatial dimensions and pressure), in terms of four equations. The momentum equation

Describes the flow velocity under the influence of convection of independent flow rate, as a result of diffusion of the kinematic viscosity, under the effects of an external body force. The pulse is coupled via the pressure equation to the equation of continuity, which guarantees freedom of divergence,

The main difference to the Navier -Stokes equations is the absence of the time derivative (since fixed) and the flow- independent convection as opposed to.

Relevance of the Oseen equations

The Oseen equations arise in the linearization of the stationary Navier -Stokes equations using a so-called Picard iteration. The non-linear momentum equation

Can be approximated numerically through an iterative process. You start with a suitable velocity field and then solves successively

Occurs taking into account the incompressibility for until convergence, ie the change between low and is.

In addition, the Oseen equations can be interpreted as an extension of the stationary Stokes equation for the convective.

In the numerical analysis the Oseen equations are mainly used for the analysis and development of Ortsdiskretisierungen the Navier -Stokes equations, without having to deal with time-integration and iterative resolution of the nonlinearity. In particular, in the field of numerical linear algebra in computational fluid dynamics, ie solving the linear system, the Oseen equations are a popular benchmark.

The Oseen equations involve two fundamental properties that occur in the discretization of the Navier -Stokes equations, namely the saddle point structure by velocity - pressure coupling, and a possibly dominant convection (compared to diffusion).