# Euler equation in Polar coordinates

1. Sep 26, 2012

### stanley.st

Hello.

I have 2D Euler equation for fluids. I cant derive it in polar coordinates. I defined functions u(x,y,t) = u'(r, theta, t) and v(x,y,t) = v'(r, theta, t). I started by computing derivatives

$$\frac{\partial u'}{\partial r}=\cos\theta\frac{\partial u}{\partial x}+\sin\theta\frac{\partial u}{\partial y}$$
and
$$\frac{\partial u'}{\partial \theta}=-r\sin\theta\frac{\partial u}{\partial x}+r\cos\theta\frac{\partial u}{\partial y}$$

If I express du/dx and du/dy and insert that expressions into Euler Eq. I didnt obtain the right result. my result is

$$\frac{\partial u'}{\partial t}+u'\left(\cos\theta\frac{\partial u'}{\partial r}-\frac{\sin\theta}{r}\frac{\partial u'}{\partial\theta}\right)+v'\left(\sin\theta\frac{\partial u'}{\partial r}+\frac{\cos\theta}{r}\frac{\partial u'}{\partial\theta}\right)=pressure$$
& similar for v'. Correct result does not contain sin & cos expressions.

Thanks