# What's the difference and relationship between perfect fluid and superfluid?

Sorry for this naive question, I just cannot find good references talking about the difference and relationship between perfect fluid and superfluid. Could someone explain this to me or point me to some references?

• What is a perfect fluid? – KF Gauss May 11 at 3:55
• @KFGauss He probably means ideal fluid. – Lith May 11 at 16:08

A perfect fluid is a mathematical idealization which assumes that a fluid has zero viscosity (and zero thermal conductivity). It is believed that perfect fluids do not exist in nature (other than zero $$T$$ superfluids). Any fluid made from elementary particles (or quantum fields) has some viscosity.

A superfluid (at finite temperature) is a mixture of two components, a superfluid component with zero viscosity, and a normal component with finite viscosity. The onset of superfluidity is a thermodynamic phase transition, characterized by some critical temperature $$T_c$$. At $$T_c$$ the normal mass fraction is 1, but as $$T\to 0$$ the normal mass fraction goes to zero.

The central feature of superfluidity is not inviscid flow, but irrotational flow. The superflow satisfies $$\nabla \times v_s = 0$$ so that the superfluid velocity is the gradient of a scalar field $$v_s = \frac{\hbar\nabla\phi}{m}$$ where $$\phi$$ is periodic with period $$2\pi$$. We can think of $$\phi$$ as the phase of the condensate wave function. Note that a perfect fluid is not assumed to be irrotational. The superfluid can carry rotation by forming line-like defects, called vortices, which carry quantized circulation $$\int\int (\nabla\times v)d\sigma = \oint v_s dl = \frac{2\pi\hbar}{m}$$ Quantized circulation means that a flow in a circular channel is persistent, the analog of a persistent current (and flux quantization) in superconducivity.

Here is the wiki link for "perfect fluid"

In physics, a perfect fluid is a fluid that can be completely characterized by its rest frame mass density $$ρ_m$$ and isotropic pressure p.

Real fluids are "sticky" and contain (and conduct) heat. Perfect fluids are idealized models in which these possibilities are neglected. Specifically, perfect fluids have no shear stresses, viscosity, or heat conduction.

Fluids are an emergent phenomenon from classical mechanics described with thermodynamic variables.

Here is superfluidity:

A remarkable transition occurs in the properties of liquid helium at the temperature 2.17K, called the "lambda point" for helium. Part of the liquid becomes a "superfluid", a zero viscosity fluid which will move rapidly through any pore in the apparatus.

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In 1938, F. London proposed a "two-fluid" model to explain the behavior of the liquid: normal liquid and the superfluid fraction consisting of those atoms which have "condensed" to the ground state and make no contribution to the entropy or heat capacity of the liquid. This condensed fraction is the standard example of Bose-Einstein condensation.

Ground state immediately refers to a quantum mechanical state, no longer classical.

So the basic difference between fluid and super fluid is that in superfluids a quantum mechanical model is necessary to model the behavior.

The same is true for the difference between conductivity and superconductivity. No classical model can fit the measurements of superconductive experiments.

The quantum mechanical Bose-Einstein model succeeds in describing the observations.

When helium is cooled to a critical temperature of 2.17 K, a remarkable discontinuity in heat capacity occurs, the liquid density drops, and a fraction of the liquid becomes a zero viscosity "superfluid". Superfluidity arises from the fraction of helium atoms which has condensed to the lowest possible energy.

A condensation effect is also credited with producing superconductivity. In the BCS Theory, pairs of electrons are coupled by lattice interactions, and the pairs (called Cooper pairs) act like bosons and can condense into a state of zero electrical resistance.

Once again, the "super" prefix indicates that the model is in the quantum mechanical framework, not the classical.

• Although an interesting point is that the mathematical model for a perfect fluid also seems to describe the flow of superfluids. Just like how Stokes flow streamlines look just like potential flow streamlines despite being opposite ends of the Reynolds number range. – tpg2114 May 12 at 14:24
• @tpg2114 well, the planetary model for the atom describes a number of things too, it is the necessity of invoking quantum states I wanted to stress – anna v May 12 at 14:42
• Definitely agree - I wasn't implying anything you said is insufficient. Just wanted to point out perhaps where the question might come from and why it is sometimes assumed that they are the same. – tpg2114 May 12 at 14:50