Abstract
We revisit some classical models for dilute polymeric fluids, and we show that thermodynamically consistent models for non-isothermal flows of these fluids can be derived in a very elementary manner. Our approach is based on the identification of energy storage mechanisms and entropy production mechanisms in the fluid of interest, which, in turn, leads to explicit formulae for the Cauchy stress tensor and for all of the fluxes involved. Having identified these mechanisms and derived the governing equations, we document the potential use of the thermodynamic basis of the model in a rudimentary stability analysis. In particular, we focus on finite amplitude (nonlinear) stability of a stationary spatially homogeneous state in a thermodynamically isolated system.
Highlights
Starting from the seminal work by Kramers [1] kinetic-type models have been widely used in the mathematical modelling of polymeric fluids, see the monographs by Bird et al [2], Beris and Edwards [3], Öttinger [4], Huilgol and Phan-Thien [5], Dressler et al [6], Öttinger [7]Kröger [8], and the review paper by Lozinski et al [9] to name a few
The temperature field is often tacitly assumed to be homogeneous in space, and an evolution equation describing the temporal and spatial variations of the temperature field is rarely formulated, albeit some approaches, such as the GENERIC formalism, see Öttinger [7], or Pavelka et al [14], allow one to do so
Following Coleman and Greenberg [44] and Coleman [19], see Gurtin [45,46], Grmela and Öttinger [47] and Bulíček et al [20] for further discussion, we can exploit the thermodynamic basis of the derived model in a rudimentary stability analysis of thermo-mechanical processes described by the corresponding governing equations
Summary
Starting from the seminal work by Kramers [1] kinetic-type models have been widely used in the mathematical modelling of polymeric fluids, see the monographs by Bird et al [2], Beris and Edwards [3], Öttinger [4], Huilgol and Phan-Thien [5], Dressler et al [6], Öttinger [7]. We provide a straightforward self-contained derivation of a simple kinetic-type model for non-isothermal flows of compressible dilute polymeric fluids. We do not model them individually, but we again follow a kinetic-type theory, and we instead formulate a general form of the Fokker–Planck equation for the configurational distribution function of the polymeric chains. We show that the system of governing equations has a “trivial” stationary solution, and using thermodynamic arguments, we explicitly construct a nonnegative functional that decays in time and vanishes if and only if the system reaches the spatially homogeneous stationary state The construction of such a functional is clearly a precursor for a rigorous stability analysis, which is, beyond the scope of the present contribution
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