Stresses of PTT, Giesekus, and Oldroyd-B fluids in a Newtonian velocity field near the stick-slip singularity

Evans, Jonathan D.; Junior, Irineu L. Palhares; Oishi, Cassio M.

doi:10.1063/1.4993782

Cited by 14 publications

(12 citation statements)

References 38 publications

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“…As such, the flow field is locally Newtonian and it is this fact that we exploit to extract the solvent and polymer stress behaviours. The results presented here, extend those of stick-slip in [72,73,74] to more general separation angles relevant to extrudate swell.…”

Section: Introductionsupporting

confidence: 73%

The extrudate swell singularity of Phan-Thien–Tanner and Giesekus fluids

Evans

2019

Physics of Fluids

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The stress singularity for Phan-Thien-Tanner (PTT) and Giesekus viscoelastic fluids is determined for extrudate swell (also termed die swell). In the presence of a Newtonian solvent viscosity, the solvent stress dominates the polymer stresses local to the contact point between the solid (no-slip) surface inside the die and the free (slip) surface outside the die. The velocity field thus vanishes like r λ 0 , where r is the radial distance from the contact point and λ 0 is the smallest Newtonian eigenvalue (dependent upon the angle of separation between the solid and free surfaces). The solvent stress thus behaves like r −(1−λ 0 ) and dominates the polymer stresses, which are like r −4(1−λ 0 )/(5+λ 0 ) for PTT and r −(1−λ 0 )(3−λ 0 )/4 for Giesekus. The polymer stresses require boundary layers at both the solid and free surfaces, the thicknesses of which are derived. These results do not hold for the Oldroyd-B fluid.

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Section: Introductionsupporting

confidence: 73%

The extrudate swell singularity of Phan-Thien–Tanner and Giesekus fluids

Evans

2019

Physics of Fluids

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“…The geometry and governing equations relevant to this flow are described in Ref. 29. The flow is incompressible with the extra-stress tensor being rheologically decomposed into solvent and polymer components while the geometry used in all simulations is illustrated in Fig.…”

Section: Flow Equationsmentioning

confidence: 99%

“…In previous work, 29 we categorized the stress singularity for PTT and Giesekus fluids in planar stick-slip flow. Here, we continue the investigation by presenting a numerical scheme for solving the full flow equations.…”

Section: Introductionmentioning

confidence: 99%

Numerical study of the stress singularity in stick-slip flow of the Phan-Thien Tanner and Giesekus fluids

et al. 2019

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Stick-slip flow is a challenging viscoelastic benchmark problem due to the presence of a separation or transition point at the die exit where a sudden change in flow boundary conditions occurs. We present numerical simulations of transient planar stick-slip flow of the Phan-Thien-Tanner (PTT) and Giesekus fluids, investigating the polymer stress behavior around the stress singularity at the stick-slip point, confirming the asymptotic results presented by Evans et al. ["Stresses of the Oldroyd-B, PTT and Giesekus fluids in a Newtonian velocity field near the stickslip singularity," Phys. Fluids 29, 1-33 (2017)]. In order to improve the numerical knowledge about this viscoelastic benchmark problem, two distinct mathematical methodologies are used for comparison in the computational simulations: the Cartesian and natural stress formulations. The former is widely applied in computational rheology, while the latter is used for the first time in the context of this problem. The natural stress formulation gives improved convergence results both temporally and spatially near to the singularity while maintaining the same global flow characteristics as the Cartesian.

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“…However, as will be shown later, the strongest modes have exponents , so that gravity plays no part in forming the free surface very close to the point of separation. The asymptotic form of the velocity field and even the stresses of this simple flow are of great relevance to some viscoelastic fluid models and this is of interest because of the polluting effects that stress singularities can have on the computation of extrusion flows of polymer solutions and melts. For, example, Evans, Palhares Junior & Oishi (2017) and Evans & Evans (2019) have proved that for extrudate flow of both the Phan-Thien–Tanner (PTT) and Giesekus models in the presence of a solvent viscosity, the velocity field is dominated by the Newtonian contribution near the join of the die wall and free surface. A modified upper-convected Maxwell (MUCM) model was derived by Apelian, Armstrong & Brown (1988) from network theory and the fluid relaxation time made to decrease at increasing values of the trace of the stress tensor.…”

Section: Introductionmentioning

confidence: 99%

The separation angle of the free surface of a viscous fluid at a straight edge

Owens

2022

J. Fluid Mech.

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We rework some of the die-swell singularity analysis for Stokes flow, originally by Ramalingam (Ramalingam, 1994 Fiber spinning and rheology of liquid-crystalline polymers, PhD thesis, Massachusetts Institute of Technology) in appendix A of his PhD thesis, in an attempt to demonstrate that for capillary numbers in the range $(0,\infty )$ the curvature may enter into the normal stress balance on the free surface and lead to separation angles exceeding $180^{\circ }$ and infinite curvature at the separation point. The singular coefficients in the asymptotic solution and the free surface shape in a neighbourhood of the separation point cannot be determined by a local analysis of the Michael type (Michael, Mathematika, vol. 5, 1958, pp. 82–84) but must be found from matching with the solution valid away from the die edge. The numerical method that we use in the truncated die-swell domain is a boundary element method incorporating the singular solution near the separation point. Although there is some variation in the extrudate swell ratios at different capillary numbers reported in the numerical literature, our results for capillary numbers $Ca$ from $1$ to $1000$ are within the range of values published in earlier papers. The computed separation angles at different values of $Ca$ agree well with the range of separation angles to be found in experimental and numerical papers. The separation angle appears to converge to a value different from $180^{\circ }$ as $Ca$ increases, leading us to conclude that the case of zero surface tension ( $Ca=\infty$ , with corresponding separation angle of $180^{\circ }$ ), is a singular limit.

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Stresses of PTT, Giesekus, and Oldroyd-B fluids in a Newtonian velocity field near the stick-slip singularity

Cited by 14 publications

References 38 publications

The extrudate swell singularity of Phan-Thien–Tanner and Giesekus fluids

The extrudate swell singularity of Phan-Thien–Tanner and Giesekus fluids

Numerical study of the stress singularity in stick-slip flow of the Phan-Thien Tanner and Giesekus fluids

The separation angle of the free surface of a viscous fluid at a straight edge

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