3 Generalized Onsager Principle and It Applications
131
where C is a fourth order tensor known as the friction coefficient. If we choose
C i jkl = 2ηδ ik δ jl + νδ i j δ kl , the coefficient leads to the viscous stress, where η is the
shear viscosity and ν the volumetric viscosity. We write the elastic body force F e
using the Ericksen stress τ e defined by ∇ · τ e = F e .
We summarize the equations in the model as follows.
∇ · τ e = −−∇μ,
τ sym = τ e + 2η D + νtr(D)I − a/2[[m × R μm + mm × R μ],
τ antisym = −1/2[[m × R μm − mm × R μ], τ v = 2η D + νtr(D)I,
j x = −D s f · ∇μ,
j m = −D r f · R μ.
(3.146)
Then, the Smoluchowski equation for f takes a general, transparent, and conservative
form:
∂ f /∂t + (∇, R ) · (u f + −D s f · ∇μ, m × ˙
m f − D r f · R μ) = 0. (3.147)
In this way, we have “derived” a kinetic theory for a liquid crystal solution, which
is a two scale model. This generalized Onsager relation lays the foundation for
multiscale kinetic theories for complex fluids flows, including both solutions and
melts.
3.5 Conclusion
We have discussed the Onsager principle for irreversible nonequilibrium processes
and the generalized Onsager principle for both reversible and irreversible nonequilibrium processes. The constructive formulation of the Onsager principle applies
to general nonequilibrium processes while the Onsager maximum action potential
principle only applies to the irreversible processes. We demonstrate by examples that
the Onsager principle is an effective modeling tool to develop dynamical theories for
any nonequilibrium systems.
References
1. Baskaran, A., Marchetti, M.: Statistical mechanics and hydrodynamics of bacterial suspensions.
Proc. Natl. Acad. Sci. USA 106(37), 15567–15572 (2009)
2. Bird, B., Armstrong, R.C., Hassager, O.: Dynamics of polymeric liquids. Volume 1: Fluid
Mechanics. John Wiley and Sons, New York (1987)
131
where C is a fourth order tensor known as the friction coefficient. If we choose
C i jkl = 2ηδ ik δ jl + νδ i j δ kl , the coefficient leads to the viscous stress, where η is the
shear viscosity and ν the volumetric viscosity. We write the elastic body force F e
using the Ericksen stress τ e defined by ∇ · τ e = F e .
We summarize the equations in the model as follows.
∇ · τ e = −−∇μ,
τ sym = τ e + 2η D + νtr(D)I − a/2[[m × R μm + mm × R μ],
τ antisym = −1/2[[m × R μm − mm × R μ], τ v = 2η D + νtr(D)I,
j x = −D s f · ∇μ,
j m = −D r f · R μ.
(3.146)
Then, the Smoluchowski equation for f takes a general, transparent, and conservative
form:
∂ f /∂t + (∇, R ) · (u f + −D s f · ∇μ, m × ˙
m f − D r f · R μ) = 0. (3.147)
In this way, we have “derived” a kinetic theory for a liquid crystal solution, which
is a two scale model. This generalized Onsager relation lays the foundation for
multiscale kinetic theories for complex fluids flows, including both solutions and
melts.
3.5 Conclusion
We have discussed the Onsager principle for irreversible nonequilibrium processes
and the generalized Onsager principle for both reversible and irreversible nonequilibrium processes. The constructive formulation of the Onsager principle applies
to general nonequilibrium processes while the Onsager maximum action potential
principle only applies to the irreversible processes. We demonstrate by examples that
the Onsager principle is an effective modeling tool to develop dynamical theories for
any nonequilibrium systems.
References
1. Baskaran, A., Marchetti, M.: Statistical mechanics and hydrodynamics of bacterial suspensions.
Proc. Natl. Acad. Sci. USA 106(37), 15567–15572 (2009)
2. Bird, B., Armstrong, R.C., Hassager, O.: Dynamics of polymeric liquids. Volume 1: Fluid
Mechanics. John Wiley and Sons, New York (1987)