Full linear Phan-Thien--Tanner fluid model: Exact analytical solutions for steady, start-up, and cessation regimes of shear and extensional flows

TL;DR

This paper derives exact analytical solutions for the material functions of full linear Phan-Thien--Tanner fluids in various flow regimes, revealing detailed flow behavior and oscillatory stress dynamics.

Contribution

It provides the first explicit analytical expressions for these flows, simplifying the model with a unique scaling that reduces parameters from four to one.

Findings

Explicit solutions for steady, startup, and cessation flows

Conditions for oscillatory stress dynamics identified

Characterization of stress overshoots and undershoots

Abstract

Exact, fully explicit, purely real analytical expressions for the material functions describing steady, startup, and cessation regimes of shear flows and of planar, uniaxial, and biaxial extensional flows of full linear Phan-Thien--Tanner fluids are obtained. These expressions, which have no analogs in the literature, are formulated in compact, beautiful forms, partially due to the unique scaling procedure reducing the number of the model parameters from four to one. The properties of the material functions are investigated in detail. For steady extensional flows, the possible shapes of the extensional viscosity curves are described and the conditions for these shapes to occur are determined. For startup flows, it is found when exactly the stress dynamics is oscillatory, and, in this case, a detailed characterization of oscillations is given, which includes expressions for the position and magnitude of stress overshoots and undershoots.