Clausius-Duhem inequality for quasi-1D transient flows in variable cross-section area deformable pipes

Abstract

This paper presents the derivation of the Clausius-Duhem inequality for quasi-1D transient flows of compressible fluids in radially deformable pipes of varying cross-section area. To do so, a decomposition of the normal stress in spherical and deviatoric components was considered, so that the power expended to radially deform the fluid could be properly accounted for in the first law of thermodynamics, generating a coherent Clausius-Duhem inequality. Based on the derived inequality for Newtonian-Fourier fluids, we focus on their implications and applications. As implications we show that the restrictions imposed on the dynamic and bulk viscosities are different from those retrieved from the 1D and 3D contexts. As applications, we present two distinct studies concerning pipe flows. The first shows by appealing to numerical simulations that the weighting-function and local-balance unsteady friction models violate the second law of thermodynamics since they present negative local rate of energy dissipation. The second presents an analysis to estimate upper bounds of the local rate of energy dissipation associated with shear, and volumetric and axial deformations for laminar and turbulent transient regimes, under flow conditions characterized by the Ghidaoui's dimensionless parameter P^, for liquids with different bulk to shear viscosity ratios. Although the dissipation is dominated by shear, that one due to volumetric deformation may become more relevant for laminar flows, when small numbers P^ and high bulk viscosities liquids are considered.