Deriving Bernoulli's equation from Euler's equation

Question

I am hoping that some of you can point me in the right direction. I am doing a project regarding Navier-Stokes', Euler's and Bernoulli's equations. I am currently looking for source material that can help me understand the derivation of Bernoulli's equation from Euler's equation of motion. Ideally the source would cover the "transformation" from this version of Euler's equation:

$$\rho\left( \frac{\partial u}{\partial t} + u(u \cdot\nabla)\right)=-\nabla p + \rho g$$

to a version of Bernoulli's equation, eg. $$P_1 + \frac{1}{2} \rho v_1^2 +\rho g h_1 = P_2 + \frac{1}{2} \rho v_2^2 +\rho g h_2.$$ I have already looked around on the internet and in previous posts on this forum; however, I have not been able to find anything that describes this derivation in greater detail. Does anyone know of some material, where a derivation like this is described/explained.

Answer 1

Start from Euler (correctly written-- yours is not quite right) $$ \rho\left(\frac{\partial u}{\partial t}+ (u\cdot \nabla) u\right)= -\nabla (p+\rho g z) $$ and use $$ [(u\cdot \nabla) u]_j=(u_i\partial_i) u_j= u_i \partial_j u_i + u_i(\partial_i u_j-\partial_j u_i) $$ in the form of the vector identity $$ (u\cdot \nabla) u=- u\times (\nabla\times u)+\nabla \left(\frac 12 |u|^2\right) $$ to write it as $$ \rho\left(\frac{\partial u}{\partial t}- u\times (\nabla\times u)\right)= -\nabla \left(p+\rho g z+\frac 12\rho|u^2|\right) $$ Now consider the steady flow in which $\partial u/\partial t=0$ and take a dot product with $u$ on both sides. You get $$ (v\cdot \nabla)\left(p+\rho g z+\frac 12\rho|u^2|\right)=0 $$ which is Bernoulli --- i.e the quanity inside the parentheses is constant along a streamline.

For compressible flow you need to write $$ \frac 1 \rho \nabla p= \nabla h $$ where $h$ is the specific enthalpy ($H=E+PV$ per unit mass) and then Bernoulli becomes $$ h+ g z+\frac 12|v^2|= constant. $$