# Vortex Nucleation Critical velocity using the Uncertainty Principle

• Ted55
In summary: Keep in mind that this is just a rough approximation and there may be other factors at play. In summary, to calculate the vortex nucleation from the potential gradient, we need to solve for the critical velocity, v_c, using the Heisenberg Uncertainty Principle and the expression for ΔE given.
Ted55
Homework Statement
Calculate the critical velocity for vortex nucleation for HeII passing through a capillary of inner diameter 0.1mm at T=0.

The energy of the ring is E = 1/2 * ρ * k^2 r ( ln(8r/a) -7/4)
You may estimate the rate of vortex nucleation from the uncertainty principle.
Where ρ is the superfluid density, a is the radius of the normal-fluid core of the vortex ring, and k is the circulation quantum.
Relevant Equations
ΔEΔt≥hbar/2
Hi there, I'm very stuck on this problem when approaching it like this. I know I could use the Landau Criterion for rotons but that's not accepted here, it wants the approach to come from the uncertainty principle.

My thinking is along these lines:
There will be a change in chemical potential of the superfluid flowing through the capillary, we can call it Δμ.
Then we can say that we have two energies due to this potential gradient, one at one end of the capillary, and one at the other end, we can say that the energy at the end of the capillary is: E_2 = 1/2 * ρ * k^2 r ( ln(8r/a) -7/4) * Δμ.
The energy at the 'start' of the capillary is simply E_1 = 1/2 * ρ * k^2 r ( ln(8r/a) -7/4), such that we have a change in energy of:

ΔE = E_2 -E_1 = 1/2 * ρ * k^2 r ( ln(8r/a) -7/4)(Δμ -1).

The critical velocity would imply a minima condition and so we would change the inequality above to: ΔEΔt=hbar/2.
I can sub my expression for ΔE in but it doesn't seem to get me anywhere, how do I calculate the vortex nucleation from this potential gradient? I fear I am approaching it all wrong!

I think you are on the right track. The way to approach this is to solve for the critical velocity, v_c, at which a vortex will nucleate in the superfluid. To do this, we need to use the Heisenberg Uncertainty Principle, which states that the product of the position uncertainty, Δx, and momentum uncertainty, Δp, must be greater than or equal to Planck's Constant, h.Now, the position uncertainty can be related to the velocity of the superfluid. Since the velocity of the superfluid is proportional to the gradient of the chemical potential, Δμ, we can say that Δx = v_c/Δμ. Also, the momentum uncertainty can be related to the energy uncertainty, ΔE, such that Δp = ΔE/v_c. Substituting these two equations into the Heisenberg Uncertainty Principle and rearranging for v_c, we get:v_c = h/(Δx*Δp) = h/(1/Δμ * ΔE/v_c) = h*Δμ/ΔE.Finally, substituting your expression for ΔE into this equation should give you an expression for the critical velocity at which a vortex will nucleate in the superfluid.

## What is vortex nucleation?

Vortex nucleation is a phenomenon in which a fluid begins to rotate and form a swirling pattern, often caused by a sudden change in velocity or pressure. This can occur in various natural and man-made systems, such as in weather patterns, ocean currents, and even aircraft wings.

## What is the critical velocity for vortex nucleation?

The critical velocity for vortex nucleation is the minimum velocity at which a fluid will begin to form a vortex. It is dependent on various factors, such as the viscosity and density of the fluid, as well as the geometry and surface conditions of the system in which the vortex is forming.

## What is the Uncertainty Principle?

The Uncertainty Principle, also known as the Heisenberg Uncertainty Principle, is a principle in quantum mechanics that states that it is impossible to know both the position and momentum of a particle with absolute certainty. This is due to the inherent uncertainty and randomness of quantum particles at a microscopic level.

## How does the Uncertainty Principle relate to vortex nucleation critical velocity?

In the study of vortex nucleation, the Uncertainty Principle is used to understand and predict the critical velocity at which a vortex will form. This is because the uncertainty in the position and momentum of the fluid particles can affect the fluid's behavior and lead to the formation of vortices.

## What are the applications of studying vortex nucleation critical velocity using the Uncertainty Principle?

The study of vortex nucleation and its critical velocity using the Uncertainty Principle has various applications in engineering, meteorology, and fluid dynamics. Understanding and predicting vortex formation can help improve the design and efficiency of various systems, such as aircraft wings, turbines, and pumps. It can also aid in weather forecasting and predicting natural phenomena involving vortex formation, such as tornadoes and hurricanes.

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