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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Laws Conservations Energy Mass Momentum. Views Read Edit View history. Laws Conservations Energy Mass Momentum. In differential form the Clausius—Duhem inequality can be written as. The inequality can be expressed in terms of the internal energy as.

### Clausius–Duhem inequality – Wikipedia

In a real material, the dissipation is inequaity greater than zero. 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.

Hence the Clausius—Duhem inequality is also called the dissipation inequality. Now, using index notation with respect to a Cartesian coordinate system. Using the divergence theoremwe get.

## Continuum mechanics/Clausius-Duhem inequality for thermoelasticity

In a real material, the dissipation is always greater than zero. From Wikipedia, the free encyclopedia.

The Clausius—Duhem inequality can inequwlity expressed in integral form as. This page was last edited on 9 Augustat This inequality is particularly useful in determining whether the constitutive relation of a material is thermodynamically allowable. 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 divergence theoremwe get.

The Clausius—Duhem inequality can be expressed in integral form as. Rheology Viscoelasticity Rheometry Rheometer. From the balance of energy.

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.

Assume that is calusius arbitrary fixed control volume. The Clausius—Duhem inequality [1] [2] is a way of expressing the second law of thermodynamics that is used in continuum mechanics. The inequality can be expressed in terms of the internal energy as. This inequality is a statement concerning the irreversibility of natural processes, especially when energy dissipation is involved.

Rheology Viscoelasticity Rheometry Rheometer. This inequality is a statement concerning the irreversibility of natural processes, especially when energy dissipation is involved. Retrieved from ” https: Hence the Clausius—Duhem inequality is also called the dissipation inequality.

This inequality is particularly useful in determining whether the constitutive relation of a material is thermodynamically allowable. All the variables are functions of a material point at at time. 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.

Since is arbitrary, we must have.

Surface tension Capillary action. Now, the material time derivatives of and are given by. By using this site, you agree to the Terms of Use and Privacy Policy.

### Clausius–Duhem inequality

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 duuem have Expanding out or, or, Now, the material time derivatives of and are given by Therefore, From the conservation of mass.

Clausius—Duhem inequality Continuum mechanics.