[Paper Review] The Hamilton principle for fluid binary mixtures with two temperatures

Henri Gouin, Tommaso Ruggeri|ArXiv.org|May 2, 2009

Advanced Thermodynamics and Statistical Mechanics27 references3 citations

TL;DR

This paper extends the Hamilton principle to fluid binary mixtures with distinct component temperatures, deriving equations of motion and energy balance via a variational formulation that incorporates multiple entropies and relative velocities. The key contribution is the identification of a dynamical pressure term arising from temperature differences, which aligns with classical thermodynamic predictions and closes the system through the second law of thermodynamics.

ABSTRACT

For binary mixtures of fluids without chemical reactions, but with components having different temperatures, the Hamilton principle of least action is able to produce the equation of motion for each component and a balance equation of the total heat exchange between components. In this nonconservative case, a Gibbs dynamical identity connecting the equations of momenta, masses, energy and heat exchange allows to deduce the balance equation of energy of the mixture. Due to the unknown exchange of heat between components, the number of obtained equations is less than the number of field variables. The second law of thermodynamics constrains the possible expression of a supplementary constitutive equation closing the system of equations. The exchange of energy between components produces an increasing rate of entropy and creates a dynamical pressure term associated with the difference of temperature between components. This new dynamical pressure term fits with the results obtained by classical thermodynamical arguments and confirms that the Hamilton principle can afford to obtain the equations of motions for multi-temperature mixtures of fluids.

Motivation & Objective

• To formulate a variational principle for fluid binary mixtures where each component has its own temperature and entropy.
• To derive the equations of motion for each component and the total energy balance using Hamilton's principle in a non-conservative, multi-temperature setting.
• To close the system of equations by introducing a constitutive equation constrained by the second law of thermodynamics.
• To identify a dynamical pressure term due to temperature differences between components, consistent with classical thermodynamics.
• To demonstrate that the variational approach yields results matching established laws such as Fick’s and Fourier’s laws in the weakly dissipative limit.

Proposed method

• Formulate a generalized Hamilton action principle with a Lagrangian depending on densities, entropies, relative velocities, and external forces.
• Apply variations in Eulerian coordinates to derive the equations of motion for each component and the mass balance.
• Use time-translation invariance to derive a heat exchange balance equation between components.
• Derive a Gibbs dynamical identity connecting momentum, mass, energy, and heat exchange equations to obtain the total mixture energy equation.
• Introduce an average temperature for local equilibrium in weakly dissipative mixtures to compare real and equilibrium states.
• Apply the second law of thermodynamics to constrain the constitutive form of the heat exchange and momentum production terms.

Experimental results

Research questions

• RQ1 Can the Hamilton principle be extended to fluid mixtures with multiple temperatures to derive consistent equations of motion and energy balance?
• RQ2 How does the presence of distinct component temperatures affect the variational formulation and the resulting system of equations?
• RQ3 What is the role of the second law of thermodynamics in constraining the constitutive equation for heat and momentum exchange in such mixtures?
• RQ4 How does the dynamical pressure term arising from temperature differences relate to classical thermodynamic results?
• RQ5 In what way do the derived equations reduce to known laws (e.g., Fick’s, Fourier’s) in the weakly dissipative limit?

Key findings

• The variational formulation successfully produces the equations of motion for each component and a balance equation for heat exchange between components in a binary fluid mixture with two temperatures.
• A dynamical pressure term emerges from the temperature difference between components, which is consistent with classical thermodynamic predictions.
• The Gibbs dynamical identity allows the derivation of the total mixture energy equation from component-level balances, ensuring thermodynamic consistency.
• In the weakly dissipative limit, the model recovers Fick’s law of diffusion through a proportionality between momentum production and velocity difference.
• The entropy production term is shown to be non-negative, satisfying the second law of thermodynamics, and the system closes via a constitutive relation with a positive proportionality coefficient.
• The derived equations reduce to classical forms (e.g., Navier-Stokes, Fourier) in the first-order approximation, confirming consistency with established irreversible thermodynamics.