# Second Law Constraints on the Dynamics of a Mixture of Two Fluids at Different Temperatures

## Abstract

**:**

## 1. Introduction

- All properties of the mixture must be mathematical consequences of the properties of the constituents.
- So as to describe the motion of a constituent, we may in imagination isolate it from the rest of the mixture, provided we allow properly for the actions of the other constituents upon it.
- The motion of the mixture is governed by the same equations as is a single body.

In distinction to Truesdell’s principles, which stress generality, the later principle is a fundamental constraint on constitutive equations not unlike material frame indifference. As Green and Adkins did not provide guidance on implementation of this principle, it has had little impact in the development of constitutive theories of MCF to date.“Also, in the absence of one fluid the constitutive equation for the other must reduce to the usual one for that fluid alone.”

Based on this, he developed a general weighting function for the material properties in the constitutive equations for the viscous stresses with this property. The simplest form of the weighting function is“It indicates that the limiting case(s) or in the absence of one component, we expect the governing equations and the constitutive relations(s) to reduce to their appropriate form(s) for the single component.”

“The motion of the mixture is governed by the sum of the constituent balance equations. Furthermore, the equations of a single continuum must be recovered as a special case of the mixture relations."

- Under what conditions do the constitutive balance equations reduce to those of a single constituent?
- Is the third principle compatible with the Second Law in all cases?
- What are alternatives to the metaphysical principles?

## 2. Formulation

## 3. Constitutive Theory for a Mixture of Two Fluids

## 4. Entropy Production

## 5. Discussion

- The constituent balance equations for mass, linear and angular momentum, and internal energy must account for all interactions between the constituents in accordance with the general principles of physics. They must reduce to the balance equations for a pristine medium when all but one of the constituents vanishes.
- The evolution of MCF is determined by the simultaneous solution of the coupled balance equations for each of the constituents.
- All phenomenological coefficients must be constrained by the Second Law applied to the sum of all the constituents. Any coefficient not so constrained has no effect in the dynamical evolution of MCF.

## Acknowledgements

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Kirwan, A.D., Jr. Second Law Constraints on the Dynamics of a Mixture of Two Fluids at Different Temperatures. *Entropy* **2012**, *14*, 880-891.
https://doi.org/10.3390/e14050880

**AMA Style**

Kirwan AD Jr. Second Law Constraints on the Dynamics of a Mixture of Two Fluids at Different Temperatures. *Entropy*. 2012; 14(5):880-891.
https://doi.org/10.3390/e14050880

**Chicago/Turabian Style**

Kirwan, A. D., Jr. 2012. "Second Law Constraints on the Dynamics of a Mixture of Two Fluids at Different Temperatures" *Entropy* 14, no. 5: 880-891.
https://doi.org/10.3390/e14050880