Solution of Euler equations

Question

I have given $v$ as stationary, incompressible and irrotational flow and a constant density $\rho$.

Why $v$ is a solution of the Euler equations:

$$ \partial_t \rho + \nabla \cdot (\rho v ) = 0$$ $$ \rho \partial_t v + \rho (v \cdot \nabla ) v = - \nabla p $$

where $ p = - \frac{\rho}{2} |v|^2$ ?

How can I show this?

Answer 1

Since $\nu$ is stationary and incompressible, then that means $\partial_t \nu=0$ and $\nabla \cdot \nu=0$.

Since $\rho$ is constant, then it follows \begin{align} \partial_t \rho +\nabla\cdot(\rho \nu)=0+\rho\nabla\cdot\nu = 0. \end{align}

Next, we have \begin{align} \rho\partial_t\nu +\rho(\nu\cdot \nabla)\nu= \rho\cdot 0+\rho \nu\cdot \nabla\nu =\frac{1}{2}\rho\nabla|\nu|^2 =-\nabla\left(-\frac{1}{2}\rho |\nu|^2\right)=-\nabla p. \end{align}