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, 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