# Dissipative function

*dissipation function*

A function used to take into account the effect of the forces of viscous friction on the motion of a mechanical system. The dissipation function describes the rate of decrease of the mechanical energy of the system; it is also used, more generally, to allow for the transition of energy of ordered motion to energy of disordered motion (ultimately to thermal energy).

The dissipation function for an isotropic medium, referred to unit volume, has the form

$$ \Phi = \frac{M}{2} \sum _ {i = 1 } ^ { 3 } \sum _ {k = 1 } ^ { 3 } \left ( \frac{\partial v _ {i} }{\partial x _ {k} } + \frac{\partial v _ {k} }{\partial x _ {i} } \right ) ^ {2} + \left ( \eta - \frac{2}{3} \mu \right ) \sum _ {m = 1 } ^ { 3 } \left ( \frac{\partial v _ {m} }{\partial x _ {m} } \right ) ^ {2} , $$

where $ \partial v _ {i} / \partial x _ {k} $ are the components of the tensor of the deformation rates and $ \mu $ and $ \eta $ are the viscosity coefficients which describe the viscosity during motion and the viscosity during volume expansion, respectively.

The equation of change of entropy in a viscous medium has the form:

$$ \rho T \frac{dS}{dt} = \Phi , $$

where $ S $ is the specific entropy, $ \rho $ is the density and $ T $ is the temperature of the liquid.

The dissipation function is a characterization of the viscous forces during the motion of a continuous medium. The equation of motion of a viscous medium is

$$ \rho \frac{d v _ {i} }{dt } = \frac{\partial \sigma _ {ik} }{\partial x _ {k} } + \frac{\partial \sigma _ {ik} ^ \prime }{\partial x _ {k} } , $$

where the $ \sigma _ {ik} $ are the components of the friction-less part of the stress tensor, the

$$ \sigma _ {ik} ^ \prime = \mu \left \{ \left ( \frac{\partial v _ {i} }{\partial x _ {k} } + \frac{\partial v _ {k} }{\partial x _ {i} } \right ) - \frac{2}{3} \frac{\partial v _ {m} }{\partial x _ {m} } \delta _ {ik} \right \} + \eta \frac{\partial v _ {m} }{\partial x _ {m} } \delta _ {ik} $$

are the components of the "viscous" part of the stress tensor, and

$$ \Phi = \sigma _ {ik} \frac{\partial v _ {i} }{\partial x _ {k} } . $$

The dissipation function is employed to allow for the effect of the resistance to small vibrations of the system around its equilibrium position; to study the damping of vibrations in an elastic medium; to allow for heat losses during the damping of electric current vibrations in circuit systems; etc.

#### References

[1] | L.D. Landau, E.M. Lifshitz, "Fluid mechanics" , Pergamon (1959) (Translated from Russian) |

#### Comments

For a Newtonian fluid the stress tensor has components $ t _ {kl} = - \rho \delta _ {kl} + \sigma _ {kl} ^ \prime $, where $ - \rho \delta _ {kl} $ is the friction-less part (above called $ \sigma _ {kl} $) and $ \sigma _ {kl} ^ \prime $ is the viscous part.

#### References

[a1] | S.J. Pai, "Viscous flow theory" , 1. Laminar flow , v. Nostrand (1956) |

**How to Cite This Entry:**

Dissipative function.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Dissipative_function&oldid=46749