Question

Let v = [−y, x, pi] be the velocity vector of a steady  fluid  flow. Is the  flow irrotational? Incompressible?

Answer 1

Let V = [−y, x, pi] be the velocity vector of a steady fluid flow. Is the flow irrotational? Incompressible?

Since δV/δt = 0, then the flow is steady

The flow is irrotational if the curl of V is zero.

Curl V = (δV3/δy – δV2/δz, δV1/δz – δV3/δx, δV2/δx – δV1/δy),

where V1 = -y, V2 = x and V3 = pi

curl V = ((0 – 0), (0 – 0), (1 + 1)) = (0, 0, 2) not equal to zero!

Since curl V is not equal to zero, the flow is not irrotational.

 

If the fluid is incompressible, then div V is zero.

Div (V) = δV1/δx + δV2/δy + δV3/δz

Dic(V) = 0 + 0 + 0 = 0

Since div V = 0, then the fluid is incompressible.

Comments

I knew that curl was zero for irrotationality but I didn't know that div was zero for incompressibility. Thanks.

---

This really was just a google search on vector flow irrotatoinal incompresssible

I found a wikipedia page that filled the bill.

https://en.wikipedia.org/wiki/Flow_velocity

 

---

I Googled, too, but missed that one! Thanks. Still, I found some good sites for lectures on vectors. Brings back memories of those school and university years!

---

To Mathical,

It looks as though you have made a couple of comments to this post. I've gotten one email from the website telling my this.

I haven't answered your question because I don't see any comment on here by yourself. Only Rod and myself.

Have you hidden your comment? I wouldn't see it then.

If not, could you try to add another comment, then when I get the email. your comment will be included on that.

Thanks,

Fermat

---

It's OK, Mathical left a comment for me, commenting on my comment. I hid and deleted it so as not confuse you, because I thought you would assume the comment was for you! But of course, the system would send you an e-mail as soon as the comment was posted, before it was hidden and deleted, which explains why it disappeared.

---

Ah, thanks Rod, for clearing that up. We get there in the end.