Variational principles and entropy production in creeping flow of non-newtonian fluids

The extension of Helmholtz—Korteweg's theorem to purely viscous non-Newtonian fluids yields a variational principle for steady creeping flow for a rather wide class of fluids. Only for a restricted subclass (Newtonian and power-law fluids) does the quantity to be minimized coincide with the total rate of entropy production. It is argued that the Helmholtz—Korteweg theorem is a purely mechanical result which has no significant thermodynamic content.     

Generalized hydrodynamical equations obeying the entropy principle

Three generalizations of classical hydrodynamic theories that are compatible with equilibrium thermodynamics and that are suitable for an appropriate macroscopic dynamical theory of polymeric liquids are considered. The strain tensor, the stress tensor and the chain segment distribution function (introduced in the network theory of polymeric liquids) are accepted as new state variables. We find that the generalized hydrodynamic equations are compatible with equilibrium thermodynamics provided certain conditions restricting the freedom of choice of constitutive relations are satisfied. In some particular cases the conditions are known from other considerations. We say that a dynamical theory is compatible with equilibrium thermodynamics, or equivalently, that it obeys the entropy principle if the properties listed in section 2.1 are satisfied.     

The principle of minimal gravitational coupling

The principle of minimal gravitational coupling requires that the total Lagrangian for the field equations of general relativity consist of two additive parts, one part corresponding to the free gravitational Lagrangian and the other part to external source fields in curved spacetime. For source fields characterized by scalars, vectors and two-component spinors we prove that this additive decomposition of the total Lagrangian is an inevitable consequence of certain very general assumptions concerning the Euler-Lagrange expressions arising from the total Lagrangian.     

Galilean invariance and entropy principle for systems of balance laws

The Galilean invariance of a generic system of balance laws dictates a specific dependence of the densities and fluxes on velocity. Thus these quantities decompose in a unique manner into convective and non-convetive parts. Such a decomposition permits the elimination of velocity dependencies in the entropy principle, which becomes a constraint on the constitutive functions only. These results clarify the mathematical structure of extended thermodynamics. They also provide a connection between the equations of continuum thermodynamics and the Boltzmann equation.     

A feature in deriving the Gibbs distribution from the entropy maximum principle

It is proved that, in the general case, the Gibbs distribution may not provide an entropy maximum.     

A historical note on the entropy principle of Müller and Liu

Received July 12, 2001 / Published online February 28, 2002     

On the lubrication approximation for a class of viscoelastic fluids

We present a rigorous derivation of a dimensionally reduced Reynolds type equation for thin film flow lubrication of a class of viscoelastic fluids by employing a perturbation analysis on the upper-convected Maxwell model in natural orthogonal coordinates. This approximation accounts for the viscoelastic and curvature corrections to the classical Reynolds lubrication approximation. Comparison of our approximation with the classical Reynolds approximation suggests that viscoelasticity can have a significant influence on the lubrication characteristics, at least for certain values of the film thickness and of the eccentricity ratios of the journal bearing.     

On the criteria of the absolute convective stability for compressible fluids

The conditions under which natural convection is absent from compressible fluids are investigated. It is shown that in the parameter “Rayleigh number-given temperature difference” plane there is a domain in which convection occurs for neither Rayleigh numbers. It is proposed to refer to this domain as the absolute convective stability region and to name the criterion determining the boundary of this region the absolute convective stability criterion. The necessary, sufficient, and necessary and sufficient conditions of the absolute convective stability for a viscous compressible fluid are derived. It is shown that in the particular case in which the thermal properties of the fluid and the adiabatic gradient are constant, these conditions coincide with the Schwarzschild criterion.