Editing Superfluid helium-4 (section)

=== Superfluid hydrodynamics  ===
The equation of motion for the superfluid component, in a somewhat simplified form,<ref>S. J. Putterman (1974), ''Superfluid Hydrodynamics'' (Amsterdam: North-Holland) {{ISBN|0-444-10681-2}}.</ref> is given by Newton's law

<math display="block">\vec F = M_4\frac{\mathrm{d}\vec v_s}{\mathrm{d}t}.</math>

The mass <math display="inline">M_4</math> is the molar mass of <sup>4</sup>He, and <math display="inline">\vec v_s</math> is the velocity of the superfluid component. The time derivative is the so-called hydrodynamic derivative, i.e. the rate of increase of the velocity when moving with the fluid. In the case of superfluid <sup>4</sup>He in the gravitational field the force is given by<ref>Landau, L. D. (1941), [http://e-heritage.ru/Book/10088665 "The theory of superfluidity of helium II"], ''Journal of Physics'', Vol. 5, Academy of Sciences of the USSR, p. 71.</ref><ref>Khalatnikov, I. M. (1965), ''An introduction to the theory of superfluidity'' (New York: W. A. Benjamin), {{ISBN|0-7382-0300-9}}.</ref>

<math display="block">\vec F = -\vec \nabla (\mu + M_4 gz).</math>

In this expression <math display="inline">\mu</math> is the molar chemical potential, <math display="inline">g</math> the gravitational acceleration, and <math display="inline">z</math> the vertical coordinate. Thus we get the equation which states that the thermodynamics of a certain constant will be amplified by the force of the natural gravitational acceleration

{{NumBlk2|:|<math display="block">M_4\frac{\mathrm{d}\vec v_s}{\mathrm{d}t}  = -\vec \nabla (\mu + M_4 gz).</math>|1}}

Eq.&nbsp;{{EquationNote|1|(1)}} only holds if <math display="inline">v_s</math> is below a certain critical value, which usually is determined by the diameter of the flow channel.<ref>{{cite journal |doi=10.1016/0031-9163(66)90958-9 |title=The dependence of the critical velocity of the superfluid on channel diameter and film thickness |year=1966 |last1=Van Alphen |first1=W. M. |last2=Van Haasteren |first2=G. J. |last3=De Bruyn Ouboter |first3=R. |last4=Taconis |first4=K. W. |journal=Physics Letters |volume=20 |issue=5 |page=474 |bibcode = 1966PhL....20..474V}}</ref><ref>{{cite book |doi=10.1016/S0079-6417(08)60052-9 |title=Thermodynamics and hydrodynamics of <sup>3</sup>He–<sup>4</sup>He mixtures |chapter=Chapter 3: Thermodynamics and Hydrodynamics of <sup>3</sup>He–<sup>4</sup>He Mixtures |series=Progress in Low Temperature Physics |date=1992 |last1=De Waele |first1=A. Th. A. M. |last2=Kuerten |first2=J. G. M. |isbn=978-0-444-89109-9 |volume=13 |page=167|chapter-url=https://research.utwente.nl/en/publications/thermodynamics-and-hydrodynamics-of-hehe-mixtures(17c67fdc-75ba-4c0f-9e13-c247b78ccf97).html}}</ref>

In classical mechanics the force is often the gradient of a potential energy. Eq.&nbsp;{{EquationNote|1|(1)}} shows that, in the case of the superfluid component, the force contains a term due to the gradient of the [[chemical potential]]. This is the origin of the remarkable properties of He-II such as the fountain effect.

[[File:Integration path in pT diagram 01.jpg|thumb|Fig. 6. Integration path for calculating <math display="inline">\mu</math> at arbitrary <math display="inline">p</math> and <math display="inline">T</math>.]]
[[File:Demo fountain pressure 01.jpg|thumb|Fig. 7. Demonstration of the fountain pressure. The two vessels are connected by a superleak through which only the superfluid component can pass.]]
[[File:Helium fountain 01.jpg|thumb|Fig. 8. Demonstration of the fountain effect. A capillary tube is "closed" at one end by a superleak and is placed into a bath of superfluid helium and then heated. The helium flows up through the tube and squirts like a fountain.]]