Are velocity components Vr=5rcos(theta), Vtheta = -5rsin(theta) irrotational?

- anonymous

Are velocity components Vr=5rcos(theta), Vtheta = -5rsin(theta) irrotational?

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- anonymous

Umm, irrotational? What do you mean?

- anonymous

Irrotational flow

- anonymous

LIke the curl is 0?

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## More answers

- anonymous

Ya....is that all I have to do?

- anonymous

gradient X V....?

- anonymous

Hold on, I'm not sure what irrotational is. Can you give me a definition?

- anonymous

Well my textbook it as flows for which no particle rotation occurs

- anonymous

Okay so curl is 0... meaning we need to find if it has a potential function.

- anonymous

This is impossible in reality because all fluids have viscosity, but flows can be assumed irrotational in certain cases

- anonymous

\[
\frac{\partial f}{\partial r} =5r\cos(\theta) \implies f = \frac{5}{2}r^2\cos(\theta) +g(\theta)
\]

- anonymous

\[
\frac{\partial f}{\partial \theta} = -\frac{5}{2}r^2\sin(\theta) +g'(\theta)=-5r\sin(\theta)
\]This let's us solve for \(g(\theta)\)

- anonymous

there might be another method, I'm not completely sure right now.

- anonymous

Okay. Well, I think it has to do with the gradient (del) cross velocity function (V)

- anonymous

The only thing I am confused about is whether there is a different between the irrotational flow and the incompressible flow. It seems like if I find that there is an incompressible flow, it's automatically going to give me an irrotational flow

- anonymous

My textbook defines the curl as (1/2)(del) X V.....but in rectangular coordinates it's simply defined as (1/2) (dv/dx - du/dx)

- anonymous

What's confusing me is if we take into consideration that it is in polar coordinates, and just treat it like Cartesian coordinates.

- anonymous

Ya. I mean every example I see in my book seems to look different ha....for example, for Vr = 0 and Vtheta = f(r) (1/r)* d/dr(r*Vtheta) = 0 for irrotational flow

- anonymous

I feel like I kind of understand what is going on, but just not quite able to piece it together.

- anonymous

\[
\frac{\partial f}{\partial \theta} = -\frac{5}{2}r^2\sin(\theta) +g'(\theta)=-5r\sin(\theta)\\
g'(\theta) =(5r^2/2+r) (-\sin\theta)\implies g = (5r^2/2+r) (\cos\theta)+C
\]

- anonymous

\[
\frac{\partial f}{\partial r} =5r\cos(\theta) \implies f = \frac{5}{2}r^2\cos(\theta) +g(\theta)\\
f = 5r^2\cos(\theta)+r\cos(\theta)
\]

- anonymous

It has a potential function, that means it's curl is 0, it's conservative, irrotational, etc.

- anonymous

Another example my prof. gave in class was this:
Given: velocity field V= \[\frac{ -q }{2 \Pi r } e _{r} + \frac{ K }{ 2 \Pi r } e _{}\] is it irrotational

- anonymous

and what he did was gradient x V

- anonymous

and got 0.....so I guess that is the same thing as you are saying essentially, right?

- anonymous

Whoops, did a bit of algebra wrong there.

- anonymous

\[
\frac{\partial f}{\partial \theta} = -\frac{5}{2}r^2\sin(\theta) +g'(\theta)=-5r\sin(\theta)\\
g'(\theta) =(5r^2/2-5r) (\sin\theta)\implies g(\theta) = -(5r^2/2+r) (\cos\theta)+C
\]

- anonymous

I keep messing up trying to find the damn potential function. Maybe it doesn't have one.

- anonymous

I think it is \[del = e _{_{r}} \frac{\partial}{\partial r} + e _{_{\theta}} \frac{ 1 }{ r } \frac{\partial}{\partial \theta} \]

- anonymous

and then that cross the the velocity vector of vr and vtheta

- anonymous

\[e _{_{r}} \frac{\partial}{\partial r} + e _{_{\theta}} \frac{ 1 }{ r } \frac{\partial}{\partial \theta} X [ e _{_{r}} 5\cos(\theta) + e _{_{\theta}} -5\sin(\theta)]\]

- anonymous

= 0

- anonymous

yeah, I don't know those formula unfortunately. The fact you bring them up makes me think we're dealing with polar coords though.

- anonymous

We are definitely dealing with polar coordinates, but I think it's still the curl

- anonymous

\[
\frac{dx}{dt} = \frac{dx}{dr}\frac{dr}{dt}
\]

- anonymous

anyway I'm going to bed.

- anonymous

Okay, thanks for your help

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