Continuity Equation in Polar Coordinates (Fluids)

apgraves1 · **Joined:** Mon, 23 Apr 2012 16:31:45 UTC **Posts:** 1

I have the continuity equation in the form div(u)=0. u=ui+vj. (Couldn't work out how to do vectors in latex)

In cartesian coordinates, this is:

$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0$

I need to be able to convert from this to polar coordinates, which should give:

$\frac{1}{r}\frac{\partial}{\partial r}(ru_{r}) + \frac{1}{r}\frac{\partial u}{\partial\theta}\theta$

I've tried using $x=rcos\theta$ and $y=rsin\theta$ , but I can't seem to get that result or figure out what I'm doing wrong. Most of the internet seems to use different notation to what I'm used to, which doesn't help either!

Thanks for any help.

Justin · **Joined:** Wed, 21 May 2003 04:27:18 UTC **Posts:** 992

apgraves1 wrote:

I have the continuity equation in the form div(u)=0. u=ui+vj. (Couldn't work out how to do vectors in latex)

In cartesian coordinates, this is:

$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0$

I need to be able to convert from this to polar coordinates, which should give:

$\frac{1}{r}\frac{\partial}{\partial r}(ru_{r}) + \frac{1}{r}\frac{\partial u}{\partial\theta}\theta$

I've tried using $x=rcos\theta$ and $y=rsin\theta$ , but I can't seem to get that result or figure out what I'm doing wrong. Most of the internet seems to use different notation to what I'm used to, which doesn't help either!

Thanks for any help.

Google yielded this.

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Continuity Equation in Polar Coordinates (Fluids)

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