3 Generalized Onsager Principle and It Applications
121
there is no problem when identifying the thermodynamic variables and their conjugate variables if we apply the generalized Onsager principle directly to arrive at
the kinetic equation. If one were to use the variational Onsager principle, one has to
justify what is the thermodynamical variable whose variation needs to be considered.
For the variational Onsager principle, regardless how the dissipation functional is
defined, one should always minimize the Rayleighian with respect to ˙
x.
3.4 Applications of the Generalized Onsager Principle
The generalized Onsager principle provides a mathematical description for any thermodynamical systems near equilibrium. Here, we demonstrate how to apply it to
derive thermodynamical models and hydrodynamical models for nonhomogeneous
systems that satisfy the second law of thermodynamics.
3.4.1 Dissipative Thermodynamical Models for
Nonequilibrium Systems
We consider a nonequilibrium thermodynamical system with N internal variables to
describe its state. The free energy of the system is given by
F =
f ({φ i }, {∇φ i }, {∇
2
φ i }, · · · )dx,
(3.94)
where φ i , i = 1, · · · , N are the internal variables. The energy dissipation rate of the
system is given by
d F
dt
=
(μ 1 , · · · , μ N ) · ( ˙
φ 1 , · · · , ˙
φ N )
T dx.
(3.95)
where μ i =
δ f
δφ i
, i = 1, · · · , N are chemical potentials. We also consider the generalized inertia in the system corresponding to the kinetic energy
1
2
N
i, j=1
φ i,t C i j φ j,t dx,
(3.96)
where C > 0 a constant “mass” matrix.
Generalized Onsager principle then implies
( ˙
φ 1 , · · · , ˙
φ N )
T
= −M · [(μ 1 , · · · , μ N )
T
+ C · (φ 1,tt , · · · , φ N ,tt )
T
], (3.97)