Numerical studies on the natural stress formulation applied to the gPTT model

Abstract Simulating viscoelastic fluids at high Weissenberg numbers is challenging due to numerical instabilities, especially in flows with singularities. The natural stress formulation (NSF) is a robust technique designed to overcome these issues. Separately, the generalized Phan-Thien and Tanner (gPTT) model offers enhanced rheological flexibility by using the Mittag-Leffler function. This work develops and validates an in-house, finite-difference NSF-gPTT solver. The method is first validated against the traditional HiG-Flow solver in channel flow and 1:4 sudden expansion geometries, showing excellent agreement for velocity/stress profiles and vortex reattachment lengths. We then apply the framework to the L-shaped channel benchmark, which features a re-entrant corner. The NSF-gPTT solver remains stable and accurate up to a Weissenberg number ( Wi ) of 100. The results reveal a counter-intuitive decrease in the peak of the first normal stress difference ( $$N_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>N</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> ) at the corner with increasing Wi , a direct result of the gPTT model’s shear-thinning. Furthermore, we demonstrate the NSF’s stability by showing that at $$Wi=100$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>W</mml:mi> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>100</mml:mn> </mml:mrow> </mml:math> , the internal conformation tensor trace grows to $$\approx 100$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>≈</mml:mo> <mml:mn>100</mml:mn> </mml:mrow> </mml:math> , while the elastic stress trace remains small ( $$\approx 0.48$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>≈</mml:mo> <mml:mn>0.48</mml:mn> </mml:mrow> </mml:math> ). This study demonstrates that the NSF-gPTT formulation is a stable and powerful tool for probing complex viscoelastic phenomena in high- Wi regimes.