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Start-up and cessation of steady shear and extensional flows: Exact analytical solutions for the affine linear Phan-Thien--Tanner fluid model

Exact analytical solutions for start-up and cessation flows are obtained for the affine linear Phan-Thien--Tanner fluid model. They include the results for start-up and cessation of steady shear flows, of steady uniaxial and biaxial extensional flows, and of steady planar extensional flows. The solutions obtained show that at start-up of steady shear flows, the stresses go through quasi-periodic exponentially damped oscillations while approaching their steady-flow values (so that stress overshoots are present); at start-up of steady extensional flows, the stresses grow monotonically, while at cessation of steady shear and extensional flows, the stresses decay quickly and non-exponentially. The steady-flow rheology of the fluid is also reviewed, the exact analytical solutions obtained in this work for steady shear and extensional flows being simpler than the alternative formulas found in the literature. The properties of steady and transient solutions, including their asymptotic behavior at low and high Weissenberg numbers, are investigated in detail. Generalization to the multimode version of the Phan-Thien--Tanner model is also discussed. Thus, this work provides a complete analytical description of the rheology of the affine linear Phan-Thien--Tanner fluid in start-up, cessation, and steady regimes of shear and extensional flows.