condensed matter physics Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid State of matter, phases, that arise from electromagnetic forces between atoms and elec ...

, the two-fluid model is a macroscopic model to explain

superfluidity Superfluidity is the characteristic property of a fluid with zero viscosity which therefore flows without any loss of kinetic energy. When stirred, a superfluid forms vortices that continue to rotate indefinitely. Superfluidity occurs in two ...

. The idea was suggested by

László Tisza László Tisza (July 7, 1907 – April 15, 2009) was a Hungarian-born American physicist who was Professor of Physics Emeritus at MIT. He was a colleague of famed physicists Edward Teller, Lev Landau and Fritz London, and initiated the two ...

in 1938 and reformulated by

Lev Landau Lev Davidovich Landau (; 22 January 1908 – 1 April 1968) was a Soviet physicist who made fundamental contributions to many areas of theoretical physics. He was considered as one of the last scientists who were universally well-versed and ma ...

in 1941 to explain the behavior of superfluid

helium-4 Helium-4 () is a stable isotope of the element helium. It is by far the more abundant of the two naturally occurring isotopes of helium, making up about 99.99986% of the helium on Earth. Its nucleus is identical to an alpha particle, and consi ...

. This model states that there will be two components in liquid helium below its

lambda point The lambda point is the temperature at which normal fluid helium (helium I) makes the transition to superfluid state ( helium II). At pressure of 1 atmosphere, the transition occurs at approximately 2.17 K. The lowest pressure at which He-I and H ...

(the temperature where superfluid forms). These components are a normal fluid and a

ideal fluid In physics, a perfect fluid or ideal fluid is a fluid that can be completely characterized by its rest frame mass density \rho_m and ''isotropic'' pressure . Real fluids are viscous ("sticky") and contain (and conduct) heat. Perfect fluids are id ...

component. Each liquid has a different density and together their sum makes the total density, which remains constant. The ratio of superfluid density to the total density increases as the temperature approaches absolute zero.

Equations

The two-fluid model can be described by a system of coupled

inviscid Viscosity is a measure of a fluid's rate-dependent resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for example, syrup h ...

and

viscous Viscosity is a measure of a fluid's rate-dependent resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for example, syrup h ...

fluid system, in the low velocity limit, the equations are given by :

\rho_ \frac+\rho_ (\mathbf v_ \cdot \nabla) \mathbf v_=-\frac\nabla p - \rho_ \sigma \nabla T +\eta\nabla^2 \mathbf v_ ;

\rho_ \frac+\rho_ (\mathbf v_ \cdot\nabla) \mathbf v_=-\frac\nabla p + \rho_ \sigma \nabla T,

where the

P

is the pressure,

T

is the temperature,

\eta

is the viscosity of the normal component,

\sigma

is the

entropy Entropy is a scientific concept, most commonly associated with states of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the micros ...

per unit mass, and

\rho=\rho_+\rho_

is the density as the sum of the density of the two components such that it follows a continuity equation :

\frac+\nabla \cdot \mathbf J =0,

where the total flow is given by :

\mathbf J = \rho_ \mathbf v_ +\rho_ \mathbf v_ .

These corresponds to a coupled Navier-Stokes equations (normal component) to

Euler equations In mathematics and physics, many topics are eponym, named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include their own unique function, e ...

(ideal superfluid component).

Application to traffic

There is also a two-fluid model also refers to a macroscopic traffic flow model to represent traffic in a town/city or metropolitan area, put forward in the 1970s by

Ilya Prigogine Viscount Ilya Romanovich Prigogine (; ; 28 May 2003) was a Belgian physical chemist of Russian-Jewish origin, noted for his work on dissipative structures, complex systems, and irreversibility. Prigogine's work most notably earned him the 19 ...

and

Robert Herman Robert Herman (August 29, 1914 – February 13, 1997) was an American astronomer, best known for his work with Ralph Alpher in 1948–50, on estimating the temperature of cosmic microwave background radiation from the Big Bang explosion. Biogra ...

. It was inspired by the superluid model.

References

{{Reflist

External links

Two Fluid Model of Superfluid Helium Mathematical modeling Traffic flow Superfluidity Fluid dynamics Liquid helium