Clausius–Duhem inequality

Clausius–Duhem inequality in terms of the specific entropy

The Clausius–Duhem inequality can be expressed in integral form as : \frac{d}{dt}\left(\int_\Omega \rho \eta , dV\right) \ge \int_{\partial \Omega} \rho \eta \left(u_n - \mathbf{v}\cdot\mathbf{n}\right) dA - \int_{\partial \Omega} \frac{\mathbf{q}\cdot\mathbf{n}}{T}~ dA + \int_\Omega \frac{\rho s}{T}~dV. In this equation t is the time, \Omega represents a body and the integration is over the volume of the body, \partial \Omega represents the surface of the body, \rho is the mass density of the body, \eta is the specific entropy (entropy per unit mass), u_n is the normal velocity of \partial \Omega, \mathbf{v} is the velocity of particles inside \Omega, \mathbf{n} is the unit normal to the surface, \mathbf{q} is the heat flux vector, s is an energy source per unit mass, and T is the absolute temperature. All the variables are functions of a material point at \mathbf{x} at time t.

In differential form the Clausius–Duhem inequality can be written as : \rho \dot{\eta} \ge - \boldsymbol{\nabla} \cdot \left(\frac{\mathbf{q}}{T}\right)

• \frac{\rho~s}{T} where \dot{\eta} is the time derivative of \eta and \boldsymbol{\nabla} \cdot (\mathbf{a}) is the divergence of the vector \mathbf{a}.

Assume that \Omega is an arbitrary fixed control volume. Then u_n = 0 and the derivative can be taken inside the integral to give : \int_\Omega \frac{\partial }{\partial t}(\rho~\eta)dV \ge -\int_{\partial \Omega} \rho\eta~(\mathbf{v}\cdot\mathbf{n})dA - \int_{\partial \Omega} \cfrac{\mathbf{q}\cdot\mathbf{n}}{T}dA + \int_\Omega \cfrac{\rho~s}{T}dV. Using the divergence theorem, we get : \int_\Omega \frac{\partial }{\partial t} (\rho\eta) dV \ge -\int_\Omega \boldsymbol{\nabla} \cdot (\rho\eta~\mathbf{v})dV - \int_\Omega \boldsymbol{\nabla} \cdot \left(\frac{\mathbf{q}}{T}\right)dV + \int_\Omega \frac{\rho s}{T}dV. Since \Omega is arbitrary, we must have : \frac{\partial }{\partial t}(\rho\eta) \ge -\boldsymbol{\nabla} \cdot (\rho\eta\mathbf{v}) - \boldsymbol{\nabla} \cdot \left(\frac{\mathbf{q}}{T}\right) + \frac{\rho s}{T}. Expanding out : \frac{\partial \rho}{\partial t}\eta + \rho\frac{\partial \eta}{\partial t} \ge -\boldsymbol{\nabla} (\rho~\eta) \cdot \mathbf{v} - \rho~\eta~(\boldsymbol{\nabla} \cdot \mathbf{v}) - \boldsymbol{\nabla} \cdot \left(\frac{\mathbf{q}}{T}\right) + \frac{\rho~s}{T} or, : \frac{\partial \rho}{\partial t}\eta + \rho\frac{\partial \eta}{\partial t} \ge -\eta~\boldsymbol{\nabla} \rho\cdot\mathbf{v} - \rho~\boldsymbol{\nabla} \eta\cdot\mathbf{v} - \rho~\eta~(\boldsymbol{\nabla} \cdot \mathbf{v}) - \boldsymbol{\nabla} \cdot \left(\cfrac{\mathbf{q}}{T}\right) + \cfrac{\rho~s}{T} or, : \left(\frac{\partial \rho}{\partial t} + \boldsymbol{\nabla} \rho\cdot\mathbf{v} + \rho~\boldsymbol{\nabla} \cdot \mathbf{v}\right) \eta + \rho\left(\frac{\partial \eta}{\partial t} + \boldsymbol{\nabla} \eta\cdot\mathbf{v}\right) \ge -\boldsymbol{\nabla} \cdot \left(\cfrac{\mathbf{q}}{T}\right) + \cfrac{\rho~s}{T}. Now, the material time derivatives of \rho and \eta are given by : \dot{\rho} = \frac{\partial \rho}{\partial t} + \boldsymbol{\nabla} \rho\cdot\mathbf{v} ~;~~ \dot{\eta} = \frac{\partial \eta}{\partial t} + \boldsymbol{\nabla} \eta\cdot\mathbf{v}. Therefore, : \left(\dot{\rho} + \rho~\boldsymbol{\nabla} \cdot \mathbf{v}\right)\eta + \rho\dot{\eta} \ge -\boldsymbol{\nabla} \cdot \left(\cfrac{\mathbf{q}}{T}\right) + \cfrac{\rho~s}{T}. From the conservation of mass \dot{\rho} + \rho~\boldsymbol{\nabla} \cdot \mathbf{v} = 0. Hence, : \rho~\dot{\eta} \ge -\boldsymbol{\nabla} \cdot \left(\cfrac{\mathbf{q}}{T}\right) + \cfrac{\rho~s}{T}.

Clausius–Duhem inequality in terms of specific internal energy

The inequality can be expressed in terms of the internal energy as : \rho~(\dot{e} - T~\dot{\eta}) - \boldsymbol{\sigma}:\boldsymbol{\nabla}\mathbf{v} \le

• \cfrac{\mathbf{q}\cdot\boldsymbol{\nabla} T}{T} where \dot{e} is the time derivative of the specific internal energy e (the internal energy per unit mass), \boldsymbol{\sigma} is the Cauchy stress, and \boldsymbol{\nabla}\mathbf{v} is the gradient of the velocity. 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 \boldsymbol{\nabla} \cdot (\varphi~\mathbf{v}) = \varphi~\boldsymbol{\nabla} \cdot \mathbf{v} + \mathbf{v}\cdot\boldsymbol{\nabla} \varphi in the Clausius–Duhem inequality, we get : \rho~\dot{\eta} \ge - \boldsymbol{\nabla} \cdot \left(\frac{\mathbf{q}}{T}\right)

• \frac{\rho~s}{T} \qquad \text{or} \qquad \rho~\dot{\eta} \ge - \frac{1}{T} ~ \boldsymbol{\nabla} \cdot \mathbf{q} - \mathbf{q}\cdot\boldsymbol{\nabla} \left(\frac{1}{T}\right)
• \frac{\rho~s}{T}. Now, using index notation with respect to a Cartesian coordinate system \mathbf{e}_j, : \boldsymbol{\nabla} \left(\cfrac{1}{T}\right) = \frac{\partial }{\partial x_j} \left(T^{-1}\right) \mathbf{e}_j = -\left(T^{-2}\right)\frac{\partial T}{\partial x_j}\mathbf{e}_j = -\frac{1}{T^2}\boldsymbol{\nabla} T. Hence, : \rho~\dot{\eta} \ge - \cfrac{1}{T}\boldsymbol{\nabla} \cdot \mathbf{q} + \cfrac{1}{T^2}\mathbf{q}\cdot\boldsymbol{\nabla} T
• \frac{\rho~s}{T} \qquad\text{or}\qquad \rho~\dot{\eta} \ge -\cfrac{1}{T}\left(\boldsymbol{\nabla} \cdot \mathbf{q} - \rho~s\right) + \frac{1}{T^2}\mathbf{q}\cdot\boldsymbol{\nabla} T. From the balance of energy : \rho\dot{e} - \boldsymbol{\sigma}:\boldsymbol{\nabla}\mathbf{v} + \boldsymbol{\nabla} \cdot \mathbf{q} - \rho~s = 0 \qquad \implies \qquad \rho~\dot{e} - \boldsymbol{\sigma}:\boldsymbol{\nabla}\mathbf{v} = - (\boldsymbol{\nabla} \cdot \mathbf{q} - \rho~s). Therefore, : \rho~\dot{\eta} \ge \frac{1}{T} \left(\rho~\dot{e}-\boldsymbol{\sigma}:\boldsymbol{\nabla}\mathbf{v}\right) + \frac{1}{T^2}\mathbf{q}\cdot\boldsymbol{\nabla} T \qquad \implies \qquad \rho\dot{\eta}T \ge \rho\dot{e}-\boldsymbol{\sigma}:\boldsymbol{\nabla}\mathbf{v} + \frac{\mathbf{q}\cdot\boldsymbol{\nabla} T}{T}. Rearranging, : \rho~(\dot{e} - T~\dot{\eta}) - \boldsymbol{\sigma}:\boldsymbol{\nabla}\mathbf{v} \le

• \frac{\mathbf{q}\cdot \boldsymbol{\nabla} T}{T} Q.E.D.

Dissipation

The quantity : \mathcal{D} = \rho~(T~\dot{\eta}-\dot{e}) + \boldsymbol{\sigma}:\boldsymbol{\nabla}\mathbf{v}

• \cfrac{\mathbf{q}\cdot\boldsymbol{\nabla} T}{T} \ge 0 is called the dissipation which is defined as the rate of internal entropy production per unit volume times the absolute temperature. Hence the Clausius–Duhem inequality is also called the dissipation inequality. In a real material, the dissipation is always greater than zero.

References

References

1. Truesdell, Clifford. (1952). "The Mechanical foundations of elasticity and fluid dynamics". Journal of Rational Mechanics and Analysis.
2. Truesdell, Clifford. (1960). "Handbuch der Physik". Springer.
3. Frémond, M.. (2006). "Nonsmooth Mechanics and Analysis". Springer.