Two-fluid model

Equations

The two-fluid model can be described by a system of coupled inviscid and viscous fluid system, in the low velocity limit, the equations are given by

:\rho_{\rm n} \frac{\partial \mathbf v_{\rm n}}{\partial t}+\rho_{\rm n} (\mathbf v_{\rm n} \cdot \nabla) \mathbf v_{\rm n}=-\frac{\rho_{\rm n}}{\rho}\nabla p - \rho_{\rm s} \sigma \nabla T +\eta\nabla^2 \mathbf v_{\rm n} ;

:\rho_{\rm s} \frac{\partial \mathbf v_{\rm s}}{\partial t}+\rho_{\rm s} (\mathbf v_{\rm s} \cdot\nabla) \mathbf v_{\rm s}=-\frac{\rho_{\rm s}}{\rho}\nabla p + \rho_{\rm s} \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 per unit mass, and \rho=\rho_{\rm s}+\rho_{\rm n} is the density as the sum of the density of the two components such that it follows a continuity equation

:\frac{\partial \rho}{\partial t}+\nabla \cdot \mathbf J =0,

where the total flow is given by

:\mathbf J = \rho_{\rm s} \mathbf v_{\rm s} +\rho_{\rm n} \mathbf v_{\rm n} .

These corresponds to a coupled Navier-Stokes equations (normal component) to Euler equations (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 and Robert Herman. It was inspired by the superfluid model.

References

References

1. Khalatnikov, I.M.. (2000). "An Introduction to the Theory of Superfluidity.". Westview Press.
2. Balibar, S.. (2017). "Laszlo Tisza and the two-fluid model of superfluidity.". Comptes Rendus Physique.
3. (1990-01-01). "Superfluidity and Superconductivity". CRC Press.
4. (April 1979). "A Two-Fluid Approach to Town Traffic". [[Science (journal).