3.4.1: Examples

Example 3.4.1.1: Navier-Stokes equations

The Navier-Stokes system for a viscous incompressible liquid is
$$\begin{eqnarray*} v_t+(v\cdot\nabla_x)v&=&-\frac{1}{\rho}\nabla_x p+\gamma\triangle_x v\\ \text{div}_x\ v&=&0, \end{eqnarray*}$$
where
$\rho$ is the (constant and positive) density of liquid,
$\gamma$ is the (constant and positive) viscosity of liquid,
$v=v(x,t)$ velocity vector of liquid particles, $x\in\mathbb{R}^3$ or in $\mathbb{R}^2$,
$p=p(x,t)$ pressure.

The problem is to find solutions $v,\ p$ of the above system.

Example 3.4.2.1: Linear elasticity

Consider the system
$$\begin{equation} \label{elast} \rho\frac{\partial^2u}{\partial t^2}=\mu\triangle_x u+(\lambda+\mu)\nabla_x(\text{div}_x\ u)+f. \end{equation}$$
Here is, in the case of an elastic body in $\mathbb{R}^3$,
$u(x,t)=(u_1(x,t),u_2(x,t),u_3(x,t))$ displacement vector,
$f(x,t)$ density of external force,
$\rho$ (constant) density,
$\lambda,\ \mu$ (positive) Lamé constants.

The characteristic equation is $\det C=0$ where the entries of the matrix $C$ are given by
$$c_{ij}=(\lambda+\mu)\chi_{x_i}\chi_{x_j}+\delta_{ij}\left(\mu|\nabla_x\chi|^2-\rho\chi_t^2\right).$$
The characteristic equation is
$$\left((\lambda+2\mu)|\nabla_x\chi|^2-\rho\chi_t^2\right)\left(\mu|\nabla_x\chi|^2-\rho\chi_t^2\right)^2=0.$$
It follows that two different speeds $P$ of characteristic surfaces $\mathcal{S}(t)$, defined by
$\chi(x,t)=const.$, are possible, namely
$$P_1=\sqrt{\frac{\lambda+2\mu}{\rho}},\ \ \mbox{and}\ \ P_2=\sqrt{\frac{\mu}{\rho}}.$$
We recall that $P=-\chi_t/|\nabla_x\chi|$.

Contributors and Attributions

• Prof. Dr. Erich Miersemann (Universität Leipzig)

• Integrated by Justin Marshall.