Extra stress tensor in fiber suspensions: Mechanics and thermodynamics

Abstract

Formulas expressing the extra stress tensor, σ, in fiber suspensions in terms of microstructural state variables are derived by using two types of arguments: mechanical and thermodynamical. Results are compared for the distribution function, ψ, and the orientation tensor, A, playing the role of state variables. The main results are the following: (i) In the thermodynamical analysis the formulas arise as compatibility conditions between the time evolution of the fluid velocity and the time evolution of the internal structure. (ii) A complete agreement among the formulas arising in mechanics and thermodynamics is seen only in kinetic theory (i.e., with ψ as the state variable) and only with the Chan–Terentjev mechanical formula. (iii) Theoretical arguments as well as numerical illustrations indicate that the larger is the role of the reversible part of the time evolution of the microstructure, the larger is the difference in predicted stresses (i.e., the formulas for σ evaluated at solutions of the microstructural equations) calculated with the thermodynamic and the Dinh–Armstrong mechanical formulas.