To some extent modern continuum thermodynamics amounts to a collection of “ thermodynamical theories” sharing common premisses and common. sources on Ωt. Total entropy: units [J/K], defined up to a constant by. dS = dQ. T. Clausius-Duhem inequality: mathematical form of the 2nd law: DS. Dt. ≥. ∫. Ωt. sθ is the specific dissipation (or internal dissipation) and is denoted by the symbol ϕ. The Clausius-Duhem inequality can simply be written as.

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The Clausius—Duhem inequality can be expressed in integral form as. Since is arbitrary, we must have. Surface tension Capillary action. The Clausius—Duhem inequality [1] [2] is a way of expressing the second law of thermodynamics that is used in continuum mechanics. Then and the derivative can be taken inside the integral to give Using the divergence theoremwe get Since is arbitrary, we must have Expanding out or, or, Now, the material time derivatives of and are given by Therefore, From the conservation of mass.

In this equation is the time, represents a body and the integration is over the volume of the body, represents the surface of the body, is the mass density of the body, is the specific entropy entropy per unit massis the normal velocity ofis the velocity of particles insideis the unit normal to the surface, is the heat flux vector, is an energy source per unit mass, and is the absolute temperature. Hence the Clausius—Duhem inequality is also called the dissipation inequality.

Clausius—Duhem inequality Continuum mechanics. This inequality incorporates the balance of energy and the balance of linear and angular momentum into the expression for the Clausius—Duhem inequality. Retrieved from ” https: In differential form the Clausius—Duhem inequality can be written as.

In a real material, the dissipation is always greater than zero. Rheology Viscoelasticity Rheometry Rheometer. From the clusius of energy.

This inequality incorporates the balance of energy and the balance of linear and angular momentum into the expression for the Clausius—Duhem inequality. Using the identity in the Clausius—Duhem inequality, we get Now, using index notation with respect to a Cartesian coordinate systemHence, From the balance of energy Therefore, Rearranging. This page was last edited on 9 Augustat By using this site, you agree to the Terms of Use and Privacy Policy.

Using the divergence theoremwe get. duhrm

### Clausius–Duhem inequality – Wikipedia

This inequality is particularly useful in determining whether the constitutive relation of a material is thermodynamically allowable. The inequality can be expressed in terms of the internal energy as. All the variables are functions of a material point at at time. Using the divergence theoremwe get.

## Clausius–Duhem inequality

Surface tension Capillary action. Views Read Edit View history. Rheology Viscoelasticity Rheometry Rheometer. From the conservation of mass. In a real material, the dissipation is always greater than zero. Now, the material time derivatives of and are given by. From Wikipedia, the free encyclopedia. This inequality is a statement concerning the irreversibility of natural ineuqality, especially when energy dissipation is involved.

The inequality can be expressed in terms of the internal energy as.

### Clausius–Duhem inequality

From the balance of energy. Laws Conservations Energy Mass Momentum. Now, using index notation with respect to a Cartesian coordinate system. Laws Conservations Energy Mass Momentum. This inequality is a statement concerning the irreversibility of natural processes, especially when energy dissipation is involved.