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Mathematics LibreTexts

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 \(P=-\chi_t/|\nabla_x\chi|\).

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