### The behavior of superfluids in gravitational field

Superfluids apparently defy gravitational force. In an arrangement involving a vessel of superfluid inside another one such that the levels of superfluids are different in the two vessels, the superfluid flows spontaneously along the walls of the vessels as a superfluid film. The flow is from the vessel in which the level of superfluid is higher until the heights are equal or

*all*fluid has left the other container. For illustrations see the pictures of the article "Why does superfluid helium leak out of an open container?".

What is strange that all the fluid flows from the vessel to another one it the height of vessel is high enough. According to the prevailing wisdom superfluid actually consists of ordinary fluid and genuine superfluid. The fluid flows from the vessel as a genuine superfluid so that the the process must involve a phase transition transforming the ordinary fluid component present in the fluid to superfluid keeping superfluid fraction constant. A further strange feature is that the superfluid flows as a film covering the inner (and also outer) surface of entire container so that return flow is not possible. This suggests an interpretation as a macroscopic quantum phenomenon.

According to the article of Golovko the existing wisdom about flow is that it corresponds to wetting. This would however predict that the phenomenon takes place also above the critical point (λ point) for the ordinary fluid. This is not the case. Secondly, the force responsible for the sucking the superfluid from the container would act only at the boundary of the film. As the film covers both the interior and exterior walls of the container the boundary vanishes, and therefore also the force so that the flow of the superfluid to another container should stop. The amount of the superfluid leaving the container should be small and equal to the amount of super-fluid in the film: this is not the case. Hence the conventional explanation does not seem to work.

**TGD inspired model for the fountain effect**

What could be the TGD explanation for fountain effect?

- Macroscopic quantum coherence in the scale of the film is suggestive and hierarchy of Planck constants h
_{eff}=n× h and magnetic flux quanta suggest themselves. Whether this notion is relevant also for the description of super-fluid itself is not of course obvious and one might argue that standard description is enough.

Just for fun, we can however for a moment assume that the super-fluid fraction could correspond to dark phase of 4

^{H}e located at flux quanta. The natural candidates for the flux quanta is flux sheet connecting the vessel to the external world or smaller vessel and larger vessel to each other. Flux sheet would accompany the film covering the inside and outside walls.

- The notion of gravitational Planck constant

h

_{gr}= GMm/v_{0}=r_{S}m/2β_{0}, β_{0}=v_{0}/c .

In the expression of h

_{gr}M is the "large" mass - naturally Earth's mass M_{E}. m would be the mass of^{4}He atom. r_{S}= 2GM/c denotes Schwartschild radius of Earth, which from M_{E}= 3× 10^{-6}M_{Sun}and from r_{S}(Sun)= 3 km is 4.5 mm. v_{0}would be some characteristic velocity for Earth-superfluid system and the rotation velocity v_{0}= 465.1 m/s of Earth is a good candidate in this respect. Also the radius of Earth R_{E}= 6.38× 10^{6}meters will be needed.

- In TGD inspired biology the hypothesis h
_{gr}=h_{eff}=n× h. One of the basic implications is that the energies of cyclotron photons associated with magnetic flux tubes have universal energy spectrum since the dependence on the mass of the charged particle disappears. Also the gravitational Compton length. The gravitational Compton length λ_{gr}=h_{gr}/m does not depend on the mass of the particle and equals to λ = Gm/v_{0}≈ 645 meters in the recent case. The scale of the superfluid system is thus much smaller than the coherence length.

- The fact that the flow seems to defy gravitational force suggests that macroscopic quantum coherence is involved in these degrees of freedom and that one should describe the situation in terms of wave function for super-fluid particles in the gravitational potential of Earth. For ordinary value of Planck constant one cannot of course expect macroscopic quantum coherence since coherence length is not expected to be much larger than Compton length. Now the coherence length of 645 meters justifies the application of Schrödinger equation.

A simple model for the situation would be based on Schrödinger equation at the flux quantum which is locally a thin hollow cylinder turning around at the top of the wall of the container.

- One obtains 1-dimensional Schrödinger equation

(-ℏ

^{2}∂_{z}^{2}/2m +mgz)Ψ= EΨ .

- By introducing dimensionless variable

u= [z-(E/mg)/z

_{0}, z_{0}=[2m^{2}g/ℏ^{2}]^{-1/3}

one can cast the equation to the standard form of the equation for Airy functions encountered in WKB approximation

-d

^{2}Ψ/du^{2}+uΨ=0 .

- The interesting solutions correspond to Airy functions Ai(u), which approach rapidly zero for the values of u>1 and oscillate for negative values

of u. These functions Ai(u +u_{1}) are orthogonal for different values of u_{1}. The values of u_{1}correspond to different initial kinetic energies for the motion in vertical direction. In the recent situation these energies correspond to the initial vertical velocities of the super-fluid in the film. u=u_{0}=1 defines a convenient estimate for the value of z coordinate above which wave function approaches rapidly to zero. The corresponding value of z is just the length z_{0}already defined:

z

_{0}= [r_{S}(E)R_{E}^{2}]^{1/3}/[4π(v_{0}/c)^{2}]^{1/3}.

- By feeding in the values of various parameters one obtains z= 2.85× 10
^{7}meters. This corresponds to a time scale of .1 seconds in good approximation and this in turn defines a fundamental bio-rhythm and secondary p-adic time scale for electron. The value of z is somewhat smaller than the circumference of Earth which corresponds to Schumann resonance 7.8 Hz. This co-incidence is not trivial and together with many other similar "co-incidences" provides further support for the deep interconnections between gravitation and biology suggested by TGD.

To sum up, from the large value of z

_{0}it is clear that the quantum motion of the^{4}He is essentially free motion in the scales considered so that one can understand why it apparently defies gravitation.

** What about Sun?**

Just for interest one can also look what one obtains in the case of Sun.

- This requires scaling r
_{S}by a factor 10^{6}/3, the scaling of R_{E}by factor about 110, and scaling of v_{0}/c by factor 4.3 if v_{0}is identified as solar rotation velocity. The resulting value of z_{0}is 1.7× 10^{10}m whereas the distance of Earth from Sun is R=1.5×

10^{11}m, roughly 10 times larger than z_{0}.

- On the other hand, if one uses the value v
_{0}/c≈ 2^{-11}needed in the model of inner planetary orbits as Bohr orbits, one obtains z_{0}=7.3× 10^{8}m to be compared with the value of solar radius R_{S}= 6.96× 10^{8}meters. For this value of v_{0}the gravitational Compton length is λ_{gr}=6× 10^{6}m.

- The challenge is to predict the value of the parameter v
_{0}. The above observation suggests that one could pose the consistency consistency condition R= z_{0}to fix the value of v_{0}. This would give the formula

β

_{0}= (r_{s}4π R)^{1/2}.

This scales up β

_{0}from 1.6× 10^{-6}to 2.3× 10^{-6}by a factor 1.41≈ 2^{1/2}. For Sun one obtains β_{0}= 5.85× 10^{-4}consistent with the value required by Bohr quantization.

For details see the article Criticality and dark matter.

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