Euler equation in Polar coordinates

  • Thread starter stanley.st
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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

[tex]\frac{\partial u'}{\partial r}=\cos\theta\frac{\partial u}{\partial x}+\sin\theta\frac{\partial u}{\partial y}[/tex]
and
[tex]\frac{\partial u'}{\partial \theta}=-r\sin\theta\frac{\partial u}{\partial x}+r\cos\theta\frac{\partial u}{\partial y}[/tex]

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

[tex]\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[/tex]
& similar for v'. Correct result does not contain sin & cos expressions.

Thanks
 

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