7.1.1 Partial differential equations

Impulse balance and mass conservation of an incompressible (homogeneous) fluid result in the incompressible Navier-Stokes equations.
Find $u(x,t)\in [C^2(\Omega)\cup C(\overline{\Omega} \makebox[0pt]{})]\times [C^1((0,T])\cup C([0,T])] \enspace$, so that :

$$\boxed{ \begin{split}u_t - \nu\Delta u + \left( u\cdot \nabla \... ...0,T]\ u(x) \;=\; u_0(x) & \qquad \text{at} \; t=0 \end{split} }$$ (7.1)

We denote with :

$u$ - vector of velocity,
$p$ - pressure (exactly $p:=p/\varrho$),
$f$ - force field
$\nu$ - kinematic viscosity
$g$ - Dirichlet-B.C.
$u_0$ - initial values.

If velocity and pressure are time independent then one gets the stationary Navier-Stokes equations.
Find $u(x)\in C^2(\Omega)\cup C(\overline{\Omega} \makebox[0pt]{}) \enspace$, so that :

$$\boxed{ \begin{split}- \nu \Delta u + \left( u\cdot \nabla \rig... ... \ u \;=\; g & \qquad \text{on} \; \partial\Omega \end{split} }$$ (7.2)

A further simplification can be achieved by neglecting the convection $\left( u\cdot \nabla \right) u$. The remaining formula is called Stokes equations

$$\boxed{ \begin{split}- \nu \Delta u + \nabla p \;=\; f & \qquad... ... \ u \;=\; g & \qquad \text{on} \; \partial\Omega \end{split} }$$ (7.3)

The variational formulations of (7.1) - (7.3) with $u \in H^1$, $p \in L_2$ and proper test spaces can be easily derived (see [Joh97],...). The following discretization via FEM, FVM results in

• a series (7.1)
• of non-linear, non-symmetric (7.2)
• and indefinite (7.3) system(s) of equations.

To ensure stability of the discretization (discrete $\inf\sup$-condition), we use the finite elements introduced in section 7.1.3.