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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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 [1] [2] is a way of expressing the second law of thermodynamics claussius is used in continuum mechanics. This inequality is a statement concerning the irreversibility of natural processes, especially when energy dissipation is involved.

This inequality is particularly useful in determining whether the constitutive relation of a material is thermodynamically allowable.

## Clausius–Duhem inequality

Now, using index notation with respect to a Cartesian coordinate system. Now, the material time derivatives of and are given by. This inequality is particularly useful in determining whether the constitutive relation of a material is thermodynamically allowable.

Surface tension Capillary action. Clausius—Duhem inequality Continuum mechanics.

### Continuum mechanics/Clausius-Duhem inequality for thermoelasticity – Wikiversity

Using the divergence theoremwe get. Hence the Clausius—Duhem inequality is also called the dissipation inequality. In duhme form the Clausius—Duhem inequality can be written as.

The Clausius—Duhem inequality [1] [2] is a way of expressing the second law of thermodynamics that is used in continuum mechanics. The Clausius—Duhem inequality can be expressed in integral form 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.

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From the balance of energy. From the conservation of mass. Hence the Clausius—Duhem inequality is also called the dissipation inequality.

The Clausius—Duhem inequality can be expressed in integral form as. Views Read Edit View history. Laws Conservations Energy Mass Momentum. This inequality is a statement concerning the irreversibility of natural processes, especially when energy dissipation is involved. 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.

From the balance of energy. In a real material, the dissipation is always greater than zero. Retrieved from ” https: From Wikipedia, the free encyclopedia. The inequality can be expressed in terms of the internal energy as.

Assume that is an arbitrary fixed control volume. In a real material, the dissipation is always 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. Surface tension Capillary action.

This inequality incorporates the balance of energy and the balance of linear and angular momentum into the expression for the Clausius—Duhem inequality. All the variables are functions of a clausuus point at at time.

Laws Conservations Energy Mass Momentum. Using the divergence theoremwe get. The inequality can be expressed in terms of the internal energy as. This page duuem last edited on 9 Augustat Since is arbitrary, we must have. In differential form the Clausius—Duhem inequality can be written as.