Thermodynamics and stability of fluids of third grade

Abstract

Today, even though the Clausius-Duhem inequality is widely considered to be of central importance in the subject of continuum thermomechanics, it is also believed to be a somewhat special interpretation of a more fundamental (second) law of thermodynamics. In this work, which is concerned with the relation between thermodynamics and stability for a class of non-Newtonian incompressible fluids of the differential type, we find it essential to introduce the additional thermodynamical restriction that the Helmholtz free energy density be at a minimum value when the fluid is locally at rest. As a background to our main considerations we begin by introducing the general class of Rivlin-Ericksen fluids of complexity n and obtain, for this class, a preliminary set of thermodynamical constitutive restrictions. We then give detailed attention to the special case of fluids of grade 3 and arrive at fundamental inequalities which restrict its (temperature dependent) material moduli. When the moduli are taken to be constant we find that these inequalities require that a body of such a fluid be stable in the sense that its total kinetic energy must tend to zero in time, no matter what its previous mechanical and thermal fields, provided it is both mechanically isolated and immersed in a thermally passive environment at constant temperature from some finite time onward. When the material constants of a fluid of grade 3 are such that the Clausius-Duhem inequality is satisfied but the free energy is not at a minimum in equilibrium, we show that for a broad class of reasonably posed problems the flows are necessarily asymptotically unbounded. Finally, we determine the stability character of non-trivial base flows for fluids of grade 3 with constant material moduli, and establish a uniqueness theorem for the initial-boundary value problem and a uniqueness theorem for problems involving sufficiently slow steady flows.

Entropy Analysis for a Nonlinear Fluid with a Nonlinear Heat Flux Vector

“…Finally, since our models for the stress tensor and the heat flux vector are nonlinear, it is important to do stability analyses and consider the uniqueness and existence of solutions. This is left for future studies (see [61][62][63]). …”

Weak and classical solutions of equations of motion for third grade fluids

“…More precisely (cf. [9]) the following result holds: if the Clausius-Duhem inequality is satisfied and the free energy is minimum at equilibrium then ν ≥ 0, β 1 = β 2 = 0, β≥ 0, α 1 ≥ 0.…”

“…We consider the matrices 3 × 3 as elements of L p (Ω) 9 and we define their norms L p by using (2.6) with N = 9. In the same way, we define the norms of tensors.…”

On steady third grade fluids equations

“…Above, u(x) denotes the velocity vector field, p is the fluid pressure, L denotes the gradient matrix of the velocity L = ∇u = (∂ j u i ) i,j , A = L + L t and ν, α 1 , α 2 , β are some material constants that, according to the thermodynamic study performed by Fosdick and Rajagopal [14], must verify the following assumptions:…”