Vorticity and Irrotational motion

Question 1

What is the vorticity vector?

Answer 1

ω = curl u = ∇ × u

Question 2

Bernoulli’s Equation (vorticity)

Answer 2

1/2 u.u + Ω + integral dp/ρ = constant along a vortex line. A votex line is everywhere in the direction of vorticity ω.

Question 3

Angular velocity

Answer 3

v = a × r

Question 4

Velocity of a rigid body relative to a fixed origin

Answer 4

u = V + v = V + a × r

Question 5

Property of scalar triple products

Answer 5

c . d × e = c × d . e for any vectors c, d and e

Question 6

Definition of fluid angular velocity

Answer 6

a. n =
lim b→0 {1/2(area) closed integral (C) u. dr}
Component of velocity at r0.
A disc of radius b perpendicular to n, centres on r0.

Question 7

n.a =

Answer 7

n. a = 1/2 n . ∇ × u
The local angular velocity in the fluid is half the vorticity.

Question 8

Define irrotational

Answer 8

ω = ∇ × u = 0, fluid flow has no local rotation.

Question 9

What is the circulation?

Answer 9

closed integral (C) u . dr

Question 10

What is the velocity potential?

Answer 10

∇ × u = 0
u = ∇ϕ
ϕ(r, t) = integral (r0 - r) u . dr

Question 11

Laplace’s Equation

Answer 11

∇^2ϕ = 0

Question 12

Bernoulli’s Equation for Irrotational Motion

Answer 12

∂ϕ/∂t + 1/2 u.u + Ω + integral dp/ρ = 0
Fluid flow no longer has to be steady.