# 10. The motion of fluids#

Philip Lederer and Christoph Lehrenfeld

## 10.1. The system of PDEs#

We consider the velocity $$u : \Omega \rightarrow {\mathbb R}^d$$ and the pressure $$p : \Omega \rightarrow {\mathbb R}$$ in an Eulerian setting (i.e. we do not follow specific “fluid-particles”), a given kinematic viscosity $$\nu: \Omega \rightarrow {\mathbb R}$$ and a volume force $$u : \Omega \rightarrow {\mathbb R}^d$$. The motion of an incompressible fluid on a domain $$\Omega \subset \mathbb{R}^d$$ is given by the non-linear Navier-Stokes equations

\begin{alignat*}{2} \frac{\partial u}{\partial t} - \operatorname{div}( u \otimes u) - \operatorname{div} \sigma &=f \quad && \text{in } \Omega, \\ \operatorname{div} u &=0 \quad && \text{in } \Omega, \end{alignat*}

which follow by Newton’s laws, i.e., the conservation of momentum and the conservation of mass, respectively. Here, $$\sigma: \Omega \rightarrow {\mathbb R}^{d \times d}$$ is the stress tensor for which we have for a Newtonian fluid the relation

$\sigma = 2 \nu \varepsilon(u) - p I, \quad \text{with} \quad \varepsilon(u) = \frac{1}{2} ( \nabla u + \nabla u^T).$

For a constant viscosity and using the identity

$2\nu \operatorname{div} \varepsilon(u) = \nu \left(\Delta u + \nabla \operatorname{div} u \right) = \nu \Delta u,$

we have

\begin{alignat*}{2} \frac{\partial u}{\partial t} - \operatorname{div}( u \otimes u) - \nu \Delta u + \nabla p &=f \quad && \text{in } \Omega, \\ \operatorname{div} u &=0 \quad && \text{in } \Omega. \end{alignat*}

Note

Note, that above simplification of the diffusive part of the stress tensor only holds if the solution is smooth or when considering certain boundary conditions.

## 10.2. Boundary and initial conditions#

The Navier-Stokes equations need to be accomplished by a suitable initial condition $$u^0 = u(x, t = 0)$$ and boundary conditions. Let $$\partial \Omega = \Gamma = \Gamma_{wall} \cup \Gamma_{in} \cup \Gamma_{out}$$, then we consider solid walls (no slip, homogeneous Dirichlet)

$u = 0 \quad \text{on} \quad \Gamma_{wall},$

inflow boundary conditions (non homogenous Dirichlet)

$u = u_{in} \quad \text{on} \quad \Gamma_{in},$

and outflow (“do nothing” hom. Neumann, i.e. zero stress) conditions

$\sigma n = 0 \quad \text{on} \quad \Gamma_{out}.$

Note, that on a non penetrable wall one might also consider slip boundary conditions $$u \cdot n = 0, u \cdot t = u_{slip}$$ on a boundary $$\Gamma_{slip}$$.

In the next units we will treat the followin aspects of solving the Navier-Stokes equations, where a large first block will treat the Stokes equations (where the time derivative and the convective term are neglected):

• Conforming finite element discretizations for the Stokes equations

• iterative (and parallel) solvers for Stokes

• Normal-continuous Hybrid DG discretization (and solver) for Stokes

• Tunings and tricks for Hybrid DG and normal-continuous Hybrid DG discretizations

• Navier-Stokes solvers based on IMEX (implicit-explicit) time-stepping