## anonymous 3 years ago Are velocity components Vr=5rcos(theta), Vtheta = -5rsin(theta) irrotational?

1. anonymous

Umm, irrotational? What do you mean?

2. anonymous

Irrotational flow

3. anonymous

LIke the curl is 0?

4. anonymous

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

5. anonymous

6. anonymous

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

7. anonymous

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

8. anonymous

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

9. anonymous

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

10. anonymous

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

11. anonymous

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

12. anonymous

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

13. anonymous

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

14. 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

15. 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)

16. anonymous

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

17. 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

18. anonymous

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

19. anonymous

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

20. anonymous

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

21. anonymous

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

22. 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

23. anonymous

and what he did was gradient x V

24. anonymous

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

25. anonymous

Whoops, did a bit of algebra wrong there.

26. anonymous

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

27. anonymous

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

28. anonymous

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

29. anonymous

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

30. anonymous

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

31. anonymous

= 0

32. 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.

33. anonymous

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

34. anonymous

$\frac{dx}{dt} = \frac{dx}{dr}\frac{dr}{dt}$

35. anonymous

anyway I'm going to bed.

36. anonymous