Euler equation in Polar coordinates

[stanley.st]

Hello.

I have 2D Euler equation for fluids. I can't 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

 

[eljose79]

for sharing your question with us. It seems like you are on the right track, but there may be a few errors in your calculations. Let's take a closer look at the steps you have taken so far.

First, you have correctly defined the functions u and v in polar coordinates. This is a good starting point. Next, you have correctly computed the partial derivatives of u' with respect to r and theta. However, when you express du/dx and du/dy in terms of these partial derivatives, there may be some mistakes. Make sure to use the chain rule when computing these derivatives.

Additionally, when inserting these expressions into the Euler equation, make sure to use the correct terms for the velocity components. In polar coordinates, the velocity components are u' and v', not u and v. This may be the reason why you are getting sin and cos terms in your result.

I would recommend double-checking your calculations and make sure to use the correct variables and terms in the Euler equation. If you are still having trouble, try reaching out to a colleague or mentor for assistance. Good luck!