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 (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$N_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>N</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>) 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$Wi=100$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>W</mi> <mi>i</mi> <mo>=</mo> <mn>100</mn> </mrow> </math></EquationSource> </InlineEquation>, the internal conformation tensor trace grows to <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\approx 100$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>≈</mo> <mn>100</mn> </mrow> </math></EquationSource> </InlineEquation>, while the elastic stress trace remains small (<InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\approx 0.48$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>≈</mo> <mn>0.48</mn> </mrow> </math></EquationSource> </InlineEquation>). This study demonstrates that the NSF-gPTT formulation is a stable and powerful tool for probing complex viscoelastic phenomena in high-Wi regimes.

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

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2026年4月21日

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Fabiano Ruano Neto,
Juliana Bertoco,
Antonio Castelo Filho,
Cassio M. Oishi

摘要

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 ( $N_1$ $N_{1}$ ) 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 $Wi=100$ $W i = 100$ , the internal conformation tensor trace grows to $\approx 100$ $\approx 100$ , while the elastic stress trace remains small ( $\approx 0.48$ $\approx 0.48$ ). This study demonstrates that the NSF-gPTT formulation is a stable and powerful tool for probing complex viscoelastic phenomena in high-Wi regimes.