Stress boundary layers for the Phan-Thien–Tanner fluid at the static contact line in extrudate swell

• Published in: Journal of engineering mathematics Vol. 150; no. 1
• Main Authors: Evans, Jonathan D., Evans, Morgan L.
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.02.2025; Springer Nature B.V
• Subjects: Applications of Mathematics; Asymptotic methods; Asymptotic series; Boundary layers; Computational Mathematics and Numerical Analysis; Contact stresses; Die swell; Exact solutions; Free surfaces; Mathematical and Computational Engineering; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Shear thinning (liquids); Similarity solutions; Theoretical and Applied Mechanics; Viscoelastic fluids; Viscoelastic fluid; Stress singularity; Phan-Thien–Tanner; Matched asymptotics; Extrudate swell
• ISSN: 0022-0833; 1573-2703
• DOI: 10.1007/s10665-024-10384-w

The method of matched asymptotic expansions is used to examine the behaviour of the Phan-Thien–Tanner (PTT) fluid in the neighbourhood of the contact line singularity in extrudate swell. The solution structure for this shear-thinning viscoelastic fluid model has solvent stresses that dominate the polymer stresses, but requires stress boundary layers to fully resolve the solution at the die wall and free surface. The sizes and mechanism of the boundary layers at the two surfaces are different. We derive and present a similarity solution for the boundary layer at the die wall and an exact solution for the boundary layer at the free surface. An expression for the local behaviour of the free surface is also derived. The results for the boundary layers completes the preliminary asymptotic results for the PTT model presented in previous work.