Potential Flow

Question 1

What can we infer from the fact that flow is incompressible?

Answer 1

∇.v = 0, ∇.(∇Ф) = ∇^2Ф = 0

Question 2

What does Ф represent in a fluid?

Answer 2

The velocity potential.

Question 3

How can we find Ф for uniform flow: v = v i(hat)?

Answer 3

v = ∇Ф, dФ/dx = v -> Ф = v(x) + const

Question 4

How can we find Ф for a point source, where v = q/2πr r(hat) = q/2π(x^2+y^2) (x,y,0)?

Answer 4

• Any v = v(r) r(hat) with v(r) =/v(r)(z, θ) has ∇v = 0

- ∇.v = 1/2 * d/dr * r*v(r), v = (dФ/dx, dФ/dy, dФ/dz), so Ф = q/4π * ln(x^2 + y^2) = q/4π ln(r^2) = q/2π *ln(r)

Question 5

How do we define the streamfunction Ψ?

Answer 5

Define Ψ so that v(x) = dΨ/dy, v(y) = -dΨ/dx

Question 6

How can we write v in terms of the streamfunction Ψ?

Answer 6

v = v(x) i(hat) + v(y) j(hat) = dΨ/dy i(hat) - dΨ/dx j(hat)

Question 7

What do we find for the streamfunction if the flow is also irrotational?

Answer 7

∇v = 0, so dv(x)/dy - dv(y)/dx = d^2Ψ/dx^2 + d^2Ψ/dy^2 = 0 = Laplace’s equation again

Question 8

What is the equation for ΔΨ, small variation in Ψ?

Answer 8

ΔΨ = dΨ/dx *Δx + dΨ/dy *Δy = -v(y)Δx + v(x)Δy

- when Ψ=const, ΔΨ = 0

Question 9

What is the streamfunction for uniform flow: v = v i(hat)

Answer 9

dΨ/dy = v, -dΨ/dx = 0, so Ψ = vy

Question 10

How can we find the streamfunction Ψ for a point source, where v = q/2πr r(hat) = q/2π(x^2+y^2) (x,y,0)?

Answer 10

dΨ/dy = q/2π *x/(x^2+y^2), -dΨ/dx = q/2π *y/(x^2+y^2), so Ψ = q/2π *tan^-1(y/x)

Question 11

What is the equation for the streamfunction in 2D polar coordinates?

Answer 11

Ψ = q/2π *θ. Radial lines going out from origin are lines of constant Ψ, whereas the circle lines around the origin are lines of constant Ф