Elementary Thermodynamic and Stochastic Arguments on Non-Newtonian Fluid
- Müller, Ingo
- Wilmański, Krzysztof
2010 Mathematics Subject Classification
- 76A05 76A10
- Non-Newtonian fluids
A solution of dumbbell in a Newtonian solvent is a convenient molecular model for a non-Newtonian or visco-elastic fluid. The distribution of Hookean dumbbells obeys a continuity equation on which a hierarchy of moment equations may be erected. The closure of this hierarchy is effected by the observation that the dumbbell solution attempts to minimize its free energy, a combination of elastic energy, potential energy of the Stokes friction and entropy. The minimization provides an expression for the equilibrium distribution.
In this paper the hierarchy is closed after the second moment - the dumbbell stress tensor - by use of the equilibrium distribution. A rheological equation of state results from the closed system of equations. That rheological equation of state is simultaneously of "rate-type" and of "grade-type", in the jargon of continuum mechanics, and it satisfies all natural stability criteria. If the rheological equation of state is forcefitted into an equation of grade-type the stability is lost. The conclusion from these considerations is that constitutive equations of grade-type do not represent viscoelastic properties of fuids well.