23. Navier Stokes Equations:

We consider the time-dependent incompressible Navier Stokes equations:

Find velocity \(u : \Omega \times [0,T] \rightarrow {\mathbb R}^d\) and pressure \(p : \Omega \times [0,T] \rightarrow {\mathbb R}\) such that

\[\begin{split} \begin{array}{ccccl} \frac{\partial u}{\partial t} - \nu \Delta u + u \nabla u & + & \nabla p & = & f \\ \operatorname{div} u & & & = & 0 \end{array} \end{split}\]

The equation contains several challenges:

• It is a time dependent equation. We will use a combinatiaon of the implicit and explicit Euler method.

• A simplified model for incompressible stationary flow are Stokes’ equations:

  \[\begin{split} \begin{array}{ccccl} - \nu \Delta u + & + & \nabla p & = & f \\ \operatorname{div} u & & & = & 0 \end{array} \end{split}\]

  It requires a well balanced combination of finite element spaces for velocity \(u\) and pressure \(p\).

• It contains a transport equation for moment (German: Impuls) density:

  \[ \frac{\partial u}{\partial t} + u \nabla u = \tilde f \]