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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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.

Rheology Viscoelasticity Rheometry Rheometer. All the variables are functions of a material point at at time.

From the balance of energy. In a real material, the dissipation is always greater than zero.

Laws Conservations Energy Mass Momentum. Now, using index notation with respect to a Cartesian coordinate system. Assume that is an arbitrary fixed control volume. This inequality is particularly useful in determining whether the constitutive relation of a material is thermodynamically allowable.

Using the divergence theoremwe get. Using the divergence theoremwe get. Since is arbitrary, we must have. The Clausius—Duhem inequality [1] [2] is a way of expressing the second law of thermodynamics that is used in continuum mechanics.

Surface tension Capillary action. In differential form the Clausius—Duhem inequality can be written as. The inequality dduhem be expressed in terms of the internal energy as.

Retrieved from ” https: Hence the Clausius—Duhem inequality is also called the dissipation inequality. Views Read Edit View history.

In differential form the Clausius—Duhem inequality can be written as. 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. Surface tension Capillary action. The Clausius—Duhem inequality can be expressed in integral form as. This inequality is a statement concerning the irreversibility of natural processes, especially when energy dissipation is involved.

### Clausius–Duhem inequality

Rheology Viscoelasticity Rheometry Rheometer. The Clausius—Duhem inequality [1] [2] is a way of expressing the second law of thermodynamics that is used in continuum mechanics.

Now, the material time derivatives of and are given by. From Wikipedia, the vuhem encyclopedia. Laws Conservations Energy Mass Momentum.

This inequality incorporates the balance of energy and the balance of linear and angular momentum into the expression for the Clausius—Duhem inequality. From the balance of energy.

## Clausius–Duhem inequality

The inequality can be expressed in terms of the internal energy as. By using this site, you agree to the Terms of Use and Privacy Policy. This inequality is particularly useful in determining whether the constitutive relation of a material is thermodynamically allowable.

This inequality is a statement concerning the irreversibility of natural processes, especially when energy dissipation is involved. In a real material, the dissipation is always greater than zero. Hence the Clausius—Duhem inequality is also called the dissipation inequality. This page was last inequalihy on 9 Augustat This inequality incorporates the balance of energy and the balance of linear and angular momentum into the expression for the Clausius—Duhem inequality.

The Clausius—Duhem inequality can clausuis expressed in integral form as. 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.