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REVIEW 3 major objections 5 minor 49 references

The paper shows that in a log-strain viscoelastic model, the polymer-to-solvent viscosity ratio independently controls how strongly non-Newtonian flow profiles develop with increasing Weissenberg number, and that the model cannot be reduced

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2026-08-04 18:05 UTC pith:6K5GLNPG

load-bearing objection First FE benchmarks for the log-strain model with a credible ξ–Wi story, but the claimed enforcement of positive definiteness isn't in the equations. the 3 major comments →

arxiv 2608.01925 v1 pith:6K5GLNPG submitted 2026-08-03 cond-mat.soft

A numerical study to analyze the interplay of Weissenberg number and viscosity ratio in a log-strain tensorial model for viscoelastic fluids

Nehal Dash , Ramon Codina , Giulio G. Giusteri This is my paper
classification cond-mat.soft MSC 76A1065M6076M10 PACS 47.50.-d83.60.Bc
keywords viscoelastic fluidslog-strain tensorial modelWeissenberg numberviscosity ratioshear-thinning flowsgeneralized Newtonian fluidflow-type dependencefinite element simulation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper studies a recently proposed viscoelastic model in which the elastic stress is a logarithmic function of the elastic strain tensor, the log-strain model. Using a stabilized finite-element method, it simulates pressure-driven flow in a straight channel, around a cylinder, and through a 4:1 contraction. It finds that the polymer-to-solvent viscosity ratio, not the Weissenberg number alone, controls how far the steady velocity profiles depart from the Newtonian parabolic shape: at small ratios the profiles stay nearly parabolic for all Wi, while at large ratios they become plug-like and shear-thinning. A second result is that a Generalized Newtonian Fluid fitted to the model's simple-shear viscosity reproduces the channel flow but not the cylinder flow, indicating flow-type dependence that a rate-dependent viscosity cannot describe. The paper concludes that the viscosity ratio must be treated as an independent parameter alongside the Weissenberg number in any study of this model.

Core claim

The central claim, on the paper's own terms, is that the Weissenberg number and the viscosity ratio xi = kappa*tau_r/eta play independent roles in the log-strain model, with xi determining the extent to which raising Wi produces observably non-Newtonian behavior. In straight-channel flow the profiles interpolate between the solvent-only and zero-shear Newtonian limits, developing a plug-like central region at intermediate Wi that shrinks as Wi grows; this plug-like regime requires sufficiently large xi. In flow past a cylinder the drag coefficient transitions from the zero-shear to the solvent baseline as the effective strain rate rises. In the 4:1 contraction the corner vortex size varies n

What carries the argument

The central object is the elastic strain tensor Bel, a symmetric positive definite tensor with unit determinant that plays the role of a conformation tensor; its evolution is the upper-convected derivative of Bel balanced by the relaxation term -(1/tau_r) Bel*log(Bel), and the elastic stress is Tel = kappa*log(Bel). The independent dimensionless parameter xi = kappa*tau_r/eta, the ratio of polymeric to solvent viscosity contributions, is the knob that controls how much shear thinning the model displays. The computational machinery is a staggered semi-implicit time-stepping scheme using a generalized Lie derivative with semi-Lagrangian advection for the strain update, a Variational Multiscale

Load-bearing premise

The simulations assume that the computed elastic strain tensor Bel stays symmetric positive definite with unit determinant; the code's projection enforces only symmetry and unit determinant, not positive eigenvalues, so the physical validity of the reported solutions depends on an unverified positivity property.

What would settle it

Check the computed Bel field in any of the reported runs for points where the symmetrized tensor has a non-positive determinant or a negative eigenvalue. The existence of even one such point would mean the asserted preservation of positive definiteness does not hold in the discretization, and the steady states reported at high Wi may reflect the projection's repair rather than the log-strain model's dynamics.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • In straight-channel pressure-driven flow, increasing xi at fixed Wi turns the parabolic velocity profile into a plug-like shear-thinning profile with a central plug region that shrinks as Wi grows; at small xi the profile stays nearly parabolic for all Wi.
  • In flow past a cylinder, the drag coefficient drops from the zero-shear Newtonian baseline toward the solvent-only baseline as the effective strain rate increases, with the magnitude of the reduction set by xi.
  • In the 4:1 planar contraction at large xi, the corner vortex size changes non-monotonically with Wi and no secondary lip vortex appears, in line with strongly shear-thinning behavior.
  • A Generalized Newtonian Fluid fitted to the simple-shear viscosity of the log-strain model reproduces the channel-flow profile to within a few percent but fails around the cylinder, where extensional deformation makes the log-strain model more dissipative; the model is therefore not equivalent to a shear-rate-dependent viscosity.
  • The stabilized formulation sustains steady-state computations up to Wi around 35 in these benchmark geometries, beyond the typical limit of many unstabilized viscoelastic formulations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If this xi-dependence persists in other geometries, then reporting the Weissenberg number alone is insufficient: experiments and simulations should quote the viscosity ratio as well, or the same Wi can correspond to qualitatively different flow fields.
  • The mismatch between the log-strain and Generalized Newtonian solutions in mixed shear/extension flows could serve as a quantitative probe of flow-type-dependent rheology, potentially distinguishing this model from purely shear-thinning constitutive equations in microfluidic experiments.
  • A direct way to test the numerical enforcement claim is to record the eigenvalues of Bel during the simulations; if positivity is ever lost locally, the high-Wi results would be artifacts of the projection step rather than predictions of the model.
  • The predicted absence of secondary lip vortices in the 4:1 contraction could be used experimentally to distinguish strongly shear-thinning log-strain behavior from constant-viscosity viscoelastic models that typically produce lip vortices.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript reports a numerical study of the log-strain viscoelastic model of Alrashdi and Giusteri, focusing on the separate roles of the Weissenberg number Wi and the viscosity ratio ξ = κτ_r/η. A stabilized mixed finite element method (equal-order P1-P1 with VMS stabilization and a DG0 strain field) is combined with a backward-Euler/semi-Lagrangian treatment of the upper-convected derivative. The method is applied to three benchmark flows: straight channel, flow past a cylinder, and 4:1 planar contraction. The authors find that ξ controls the degree of shear-thinning and hence the appearance of plug-like profiles, that drag reduction occurs in cylinder flow, that no lip vortices appear in the contraction, and that the model cannot be reduced to a Generalized Newtonian Fluid: comparing the log-strain model with a GNF model fitted to the same simple-shear effective viscosity, they report significant profile differences in the cylinder flow that they attribute to flow-type dependence.

Significance. If the numerical results are reliable, the paper makes a useful contribution by demonstrating that Wi and ξ are independent control parameters for the log-strain model and that the model's behavior in non-viscometric flows is not captured by a shear-rate-dependent viscosity alone. The GNF comparison is a good way to isolate flow-type effects, and reaching Wi ~ 35 with a relatively new tensorial model is a practical achievement. However, the paper's central numerical-verification claim is not supported: the projection used in the discrete update does not enforce positive definiteness, and mesh/time-convergence evidence is only qualitative. The physical conclusions are therefore conditional on additional verification.

major comments (3)
  1. [Section 3.1, Eq. (18)] The abstract and Section 3.1 state that the computational method enforces symmetry, unit determinant, and positive definiteness of B_el. The projection in Eq. (18) only symmetrizes the tensor and rescales it to unit determinant. A symmetric 2x2 matrix with eigenvalues (-1,-1) has determinant 1 but is negative definite, and Eq. (18) leaves it unchanged. If det(sym(B)) ≤ 0 the projection is undefined. Since Eq. (17) requires log B_el^n, loss of positive definiteness makes the evolution ill-defined and the subsequent results non-physical. The manuscript reports no check on the minimum eigenvalue of B_el. This is load-bearing because all reported flow fields and the GNF comparison depend on the computed B_el being admissible. The authors should either prove that the discrete update plus projection preserves positive definiteness under the conditions used, or add numerical monitoring of the s
  2. [Section 4 (general numerical verification)] Mesh convergence is asserted qualitatively: Section 4.1.1 states a 64x64 mesh is 'sufficient', and Sections 4.2.1 and 4.3.1 state that convergence was 'verified through successive refinements' without quantitative tables or error norms. Similarly, steady state is defined by increment norms decaying to 'machine precision', but no tolerance, history, or time-step dependence is reported. For a numerical study that reaches Wi ~ 35 and includes a re-entrant corner singularity, quantitative mesh-convergence and time-step studies are necessary to support the reliability of the reported steady-state fields. Please add tables or figures with grid sizes, L2 error norms or Richardson extrapolation, and the convergence history of the increment norms.
  3. [Section 4.4, Figures 11-12] The central claim that flow-type dependence is significant in the cylinder flow rests on comparing the log-strain model with a GNF model whose viscosity is an approximate Cross fit (β = 0.788, n = 1.458). The authors acknowledge a mismatch in the intermediate shear-rate region and attribute the 2.63% channel profile discrepancy to it. In the cylinder flow, they argue that the profile differences are not a mere scaling and hence cannot be due to fitting error. This is suggestive but not conclusive: a local shear-rate error in the GNF model could produce shape changes in a non-unidirectional flow. To make the identification robust, the comparison should be repeated using the exact tabulated effective viscosity from the log-strain model in simple shear, or the sensitivity of the cylinder profiles to plausible variations of β and n should be quantified.
minor comments (5)
  1. [Figure 7] The axis/panel labels appear garbled ('b=25' and '8=...' instead of ξ and Wi). Please check the figure rendering.
  2. [Figure 9] The legend in panel (b) repeats the entry 'Wi=0.0, C=10.0' twice; one of the entries should be a different Newtonian reference (e.g., C=15.0).
  3. [Section 3.1] The hat notation is used in Eq. (14) before it is defined after Eq. (20). Define the notation earlier.
  4. [Section 4.4.1] The Cross fit parameters β and n are given without confidence intervals or a goodness-of-fit measure. Given the importance of the fit to the GNF comparison, report at least one measure such as R² or the mean relative error.
  5. [Data availability] The code and data are only available 'on request'. For a numerical study of this type, a public repository with the FEniCSx scripts would substantially strengthen reproducibility.

Circularity Check

0 steps flagged

No significant circularity: the GNF comparison is a fitted surrogate used only as a baseline, and the main numerical conclusions are independent simulations of the stated model.

full rationale

The paper is a numerical study of a constitutive model introduced in [23] by one of the authors. The central claims — that the viscosity ratio ξ modulates the extent to which Wi produces non-Newtonian profiles, and that the log-strain model differs from a Generalized Newtonian Fluid surrogate in a flow with extensional character — are obtained by solving the stated model equations (6)–(8) for specified parameters. No parameter appearing in the reported velocity profiles or drag coefficients is fitted to those target outputs: ξ is set by κτr/η (Eq. 9), Wi by Uτr/L, and the Cross parameters β,n in Eq. (44) are fitted only to the log-strain model's own steady simple-shear effective viscosity (Eq. 43) to build the comparison GNF model. That is a controlled surrogate construction, not a hidden prediction of the target results. The differences between log-strain and GNF in cylinder flow are argued from non-scaling profile shapes, not from the fitted values. Citations to the authors' earlier paper [23] introduce the model and its linearized limits, but the quantitative results here are independent simulations and do not reduce to those citations. The projection in Eq. (18) may fail to enforce positive definiteness, but that is a numerical correctness concern, not circularity: it does not make an output equal to an input. No definitional reduction, fitted-input-as-prediction, or self-citation chain forces the conclusions.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

The central claims rest entirely on the log-strain constitutive model taken from prior work [23] and on the adequacy of the numerical scheme. No new physical entities (particles, forces, dimensions) are introduced. The only numbers fitted to data are the Cross-model parameters β and n used to build the GNF comparison baseline, which is disclosed and does not enter the log-strain simulations themselves.

free parameters (3)
  • Cross model time constant β = 0.788
    Fitted by nonlinear regression to the log-strain model's steadystate simple shear effective viscosity for ξ=25 (Fig. 10, Eq. 44). Used only in the GNF surrogate model, not in the log-strain simulations.
  • Cross model power-law exponent n = 1.458
    Same fit as β; governs the slope of the shear-thinning transition in the GNF model. Used only for the comparison baseline.
  • VMS stabilization constants c1, c2 = c1 = 4k^4, c2 = 2k^2 (k = 1 for P1)
    Chosen by hand following the established VMS literature (Eq. 29). They are algorithmic constants that affect the discrete solution but are not fitted to physical data.
axioms (5)
  • domain assumption The log-strain constitutive equations (6)-(8) correctly describe the viscoelastic fluid and preserve det Bel = 1, symmetry, and positive definiteness.
    The paper adopts the model from Alrashdi and Giusteri [23] without re-derivation; Section 2 repeats the properties. These properties motivate the numerical projection and the physical interpretation of the results.
  • domain assumption A steady state exists for each simulated parameter set and is reached by time marching to machine-precision increment norms.
    Section 4 states that solutions are considered steady when increment norms decay to machine precision. This assumes the time-marching sequence converges to a true steady solution for all reported parameters.
  • domain assumption The semi-Lagrangian characteristic mapping (Eq. 17) is a consistent and sufficiently stable discretization of the upper-convected Lie derivative.
    This is the core of the constitutive update. No rigorous error analysis is given in this paper; the authors cite related work [39-43] and rely on it as a stable discretization.
  • domain assumption The VMS stabilization with parameters α1 and α2 defined in Eq. (29), using c1 = 4k^4 and c2 = 2k^2, yields a stable and converged discretization for the tested regimes.
    The stabilization is standard in the cited literature [30-32,45]. The paper relies on these choices without an a priori stability analysis for the log-strain model.
  • domain assumption The discrete update plus projection (18) keeps Bel in the physically admissible set of symmetric positive definite tensors with determinant 1.
    This is the load-bearing numerical premise. The projection as written only enforces symmetry and unit determinant; positivity is assumed rather than guaranteed. The paper does not report eigenvalues or other checks that all computed Bel remain positive definite.

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read the original abstract

We present a computational study aimed at exploring the different and independent roles of the Weissenberg number and of the ratio between the polymeric and solvent viscosity contributions in a viscoelastic fluid model. The tensorial model under investigation, recently proposed, is based on a logarithmic relation between the elastic (or recoverable) strain and the elastic stress. In this model, the elastic strain plays the role of a conformation tensor and its evolution equation inherently preserves its determinant and positive definiteness. These properties are also enforced in the computational method employed in the study. A finite-difference discretization in time is combined with a stabilized mixed finite element formulation based on the Variational Multiscale method for the spatial discretization and with a generalized Lie derivative approach for the advection terms. The behavior of the model is analyzed in paradigmatic pressure-driven flows and we find that the value of the viscosity ratio is crucial in determining to which extent non-Newtonian flow profiles are observed upon increasing the Weissenberg number. By comparing the solutions of the log-strain tensorial model with those of a suitable Generalized Newtonian Fluid model, we show that flow-type dependence plays a significant role even in the simple planar flow past a cylinder.

Figures

Figures reproduced from arXiv: 2608.01925 by Giulio G. Giusteri, Nehal Dash, Ramon Codina.

Figure 1
Figure 1. Figure 1: Geometry of the straight channel domain. [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The viscosity ratio ξ modulates the influence of Wi on the velocity profile. Horizontal normalized velocity profiles 8uxηeff /C, at steady state for different pressure gradients C and corresponding Wi. Results are compared with Newtonian profiles with viscosity η (red dotted lines) and ηeff = η + κτr = η(1 + ξ) (black dashed lines). The comparison is shown for three values of the viscosity ratio: (a) ξ = 1… view at source ↗
Figure 3
Figure 3. Figure 3: Geometry of the channel with a cylindrical obstacle. [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The log-strain model displays an enhanced drag reduction compared to the Newtonian case. Variation of drag coefficient CD with effective strain rate γ˙eff for (a) ξ = 1 and (b) ξ = 25 (red circles), compared with Newtonian results with viscosity η = 1 (blue squares) and ηeff = η(1 + ξ) (green diamonds). 13 [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Geometry of the planar 4:1 contraction. CD = F¯ x/(U 2R). The same procedure is applied to two Newtonian limiting cases and the resulting drag coefficient values are compared to quantify the drag reduction achieved by the present model as a function of the effective strain rate ( [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Computational mesh used in the 4:1 contraction problem. [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Steady streamline patterns in the 4:1 planar contraction for different values of [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Horizontal velocity ux at steady state for pressure gradient C = 1.0 and C = 5.0 along the lines at (a, b) x = 19, and (c, d) x = 30. The values are normalized with respect to the maximum velocity umax,0 for ξ = 0. to a larger energy dissipation. The profile of the vertical velocity along the line y = 1 (top wall of the narrow region) for the present model and the Newtonian limit with viscosity ηeff is sho… view at source ↗
Figure 9
Figure 9. Figure 9: Vertical velocity uy at steady state along the line at y = 1 (top wall of the narrow region) for two different values of viscosity ratio (a) ξ = 1, and (b) ξ = 25, including a comparison with a Newtonian case (Wi = 0, dashed lines) with viscosity ηeff . 16 [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Normalized effective viscosity ηeff /η as a function of Wi = ˙γτr. Circles report the data from the log-strain model. The red solid line indicates the fitted Cross law for the effective viscosity used in the GNF model. The fitting parameters are β = 0.788, and n = 1.458. 4.4.1. Viscosity mapping and model parameterization To compute the effective shear viscosity predicted by the log-strain model, we need … view at source ↗
Figure 11
Figure 11. Figure 11: Horizontal velocity ux at steady state in a straight channel for the log-strain model (blue solid line) and the GNF model (orange dashed line). The viscosity ratio is ξ = 25 and the driving pressure gradient is given by C = 50. differences are balanced out and the local flow type is everywhere that of a simple shear, we expect to observe almost identical profiles for the log-strain model and the GNF model… view at source ↗
Figure 12
Figure 12. Figure 12: Horizontal velocity ux profiles at steady state in a flow past a cylinder of radius R and centered at (cx, cy) for the log-strain model (blue solid lines) and the GNF model (orange dashed lines). The viscosity ratio is ξ = 25 and the driving pressure gradient is given by C = 50. The data are taken for two cross sections at (a) x = cx − 2R and (b) x = cx + R. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗

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Reference graph

Works this paper leans on

49 extracted references

  1. [1]

    M. D. Graham, Drag reduction in turbulent flow of polymer solutions, Rheology reviews 2 (2004) 143–170

    2004

  2. [2]

    Groisman, V

    A. Groisman, V. Steinberg, Elastic turbulence in a polymer solution flow, Nature 405 (2000) 53–55

    2000

  3. [3]

    S. S. Datta, A. M. Ardekani, P. E. Arratia, A. N. Beris, I. Bischofberger, G. H. McKinley, J. G. Eggers, J. E. López-Aguilar, S. M. Fielding, A. Frishman, et al., Perspectives on viscoelastic flow instabilities and elastic turbulence, Physical Review Fluids 7 (2022) 080701

    2022

  4. [4]

    H. A. C. Sánchez, M. R. Jovanović, S. Kumar, A. Morozov, V. Shankar, G. Subramanian, H. J. Wilson, Understanding viscoelastic flow instabilities: Oldroyd-B and beyond, Journal of Non- Newtonian Fluid Mechanics 302 (2022) 104742

    2022

  5. [5]

    J. G. Oldroyd, On the formulation of rheological equations of state, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 200 (1950) 523–541

    1950

  6. [6]

    Oldroyd, Non-newtonian effects in steady motion of some idealized elastico-viscous liquids, Proceedings of the Royal Society of London

    J. Oldroyd, Non-newtonian effects in steady motion of some idealized elastico-viscous liquids, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 245 (1958) 278–297

    1958

  7. [7]

    Giesekus, Die elastizität von flüssigkeiten, Rheologica Acta 5 (1966) 29–35

    H. Giesekus, Die elastizität von flüssigkeiten, Rheologica Acta 5 (1966) 29–35. 21

    1966

  8. [8]

    H. Giesekus, A simple constitutive equation for polymer fluids based on the concept of deformation-dependent tensorial mobility, Journal of Non-Newtonian Fluid Mechanics 11 (1982) 69–109

    1982

  9. [9]

    Phan-Thien, R

    N. Phan-Thien, R. I. Tanner, A new constitutive equation derived from network theory, Journal of Non-Newtonian Fluid Mechanics 2 (1977) 353–365

    1977

  10. [10]

    Fattal, R

    R. Fattal, R. Kupferman, Constitutive laws for the matrix-logarithm of the conformation tensor, Journal of Non-Newtonian Fluid Mechanics 123 (2004) 281–285

    2004

  11. [11]

    Balci, B

    N. Balci, B. Thomases, M. Renardy, C. R. Doering, Symmetric factorization of the conformation tensor in viscoelastic fluid models, Journal of Non-Newtonian Fluid Mechanics 166 (2011) 546– 553

    2011

  12. [12]

    Afonso, F

    A. Afonso, F. Pinho, M. Alves, The kernel-conformation constitutive laws, Journal of Non- Newtonian Fluid Mechanics 167 (2012) 30–37

    2012

  13. [13]

    R. B. Bird, R. C. Armstrong, O. Hassager, Dynamics of polymeric liquids. Vol. 1: Fluid mechanics, John Wiley and Sons Inc., New York, NY, 1986

    1986

  14. [14]

    Phan-Thien, Understanding viscoelasticity: An introduction to rheology, Applied Rheology 13 (2003) 240–241

    N. Phan-Thien, Understanding viscoelasticity: An introduction to rheology, Applied Rheology 13 (2003) 240–241

    2003

  15. [15]

    Alves, P

    M. Alves, P. Oliveira, F. Pinho, Numerical methods for viscoelastic fluid flows, Annual Review of Fluid Mechanics 53 (2021) 509–541

    2021

  16. [16]

    M.V.Shitikova, A.I.Krusser, Modelsofviscoelasticmaterials: areviewonhistoricaldevelopment and formulation, Theoretical Analyses, Computations, and Experiments of Multiscale Materials: A Tribute to Francesco dell’Isola (2022) 285–326

    2022

  17. [17]

    Renardy, Recent advances in the mathematical theory of steady flow of viscoelastic fluids, Journal of Non-Newtonian Fluid Mechanics 29 (1988) 11–24

    M. Renardy, Recent advances in the mathematical theory of steady flow of viscoelastic fluids, Journal of Non-Newtonian Fluid Mechanics 29 (1988) 11–24

    1988

  18. [18]

    J. Zhou, I. Papautsky, Viscoelastic microfluidics: Progress and challenges, Microsystems & Nanoengineering 6 (2020) 113

    2020

  19. [19]

    Renardy, B

    M. Renardy, B. Thomases, A mathematician’s perspective on the Oldroyd-B model: progress and future challenges, Journal of Non-Newtonian Fluid Mechanics 293 (2021) 104573

    2021

  20. [20]

    A. N. Beris, Continuum mechanics modeling of complex fluid systems following Oldroyd’s seminal 1950 work, Journal of Non-Newtonian Fluid Mechanics 298 (2021) 104677

    1950

  21. [21]

    P. S. Stephanou, V. G. Mavrantzas, G. C. Georgiou, Continuum model for the phase behavior, microstructure, and rheology of unentangled polymer nanocomposite melts, Macromolecules 47 (2014) 4493–4513

    2014

  22. [22]

    Ferrás, M

    L. Ferrás, M. Morgado, M. Rebelo, G. H. McKinley, A. Afonso, A generalised Phan– Thien—Tanner model, Journal of Non-Newtonian Fluid Mechanics 269 (2019) 88–99

    2019

  23. [23]

    M.A.H.Alrashdi, G.G.Giusteri, Evolutionoflocalrelaxedstatesandthemodelingofviscoelastic fluids, Physics of Fluids 36 (2024)

    2024

  24. [24]

    R. G. Owens, T. N. Phillips, Computational rheology, World Scientific, 2002

    2002

  25. [25]

    Hulsen, A

    M. Hulsen, A. Van Heel, B. Van Den Brule, Simulation of viscoelastic flows using brownian configuration fields, Journal of Non-Newtonian Fluid Mechanics 70 (1997) 79–101. 22

    1997

  26. [26]

    Fattal, R

    R. Fattal, R. Kupferman, Time-dependent simulation of viscoelastic flows at high weissenberg number using the log-conformation representation, Journal of Non-Newtonian Fluid Mechanics 126 (2005) 23–37

    2005

  27. [27]

    M. A. Hulsen, R. Fattal, R. Kupferman, Flow of viscoelastic fluids past a cylinder at high weissenberg number: stabilized simulations using matrix logarithms, Journal of Non-Newtonian Fluid Mechanics 127 (2005) 27–39

    2005

  28. [28]

    O. M. Coronado, D. Arora, M. Behr, M. Pasquali, A simple method for simulating general viscoelastic fluid flows with an alternate log-conformation formulation, Journal of Non-Newtonian Fluid Mechanics 147 (2007) 189–199

    2007

  29. [29]

    Advances in Stabilized Methods in Computational Mechanics

    T.J.Hughes, G.R.Feijóo, L.Mazzei, J.-B.Quincy, Thevariationalmultiscalemethod–aparadigm for computational mechanics, Computer Methods in Applied Mechanics and Engineering 166 (1998) 3–24. Advances in Stabilized Methods in Computational Mechanics

    1998

  30. [30]

    R. Codina, Stabilization of incompressibility and convection through orthogonal sub-scales in finite element methods, Computer methods in applied mechanics and engineering 190 (2000) 1579–1599

    2000

  31. [31]

    R. Codina, Stabilized finite element approximation of transient incompressible flows using orthog- onal subscales, Computer methods in applied mechanics and engineering 191 (2002) 4295–4321

    2002

  32. [32]

    Codina, Analysis of a stabilized finite element approximation of the oseen equations using orthogonal subscales, Applied Numerical Mathematics 58 (2008) 264–283

    R. Codina, Analysis of a stabilized finite element approximation of the oseen equations using orthogonal subscales, Applied Numerical Mathematics 58 (2008) 264–283

    2008

  33. [33]

    Castillo, R

    E. Castillo, R. Codina, Stabilized stress–velocity–pressure finite element formulations of the Navier–Stokes problem for fluids with non-linear viscosity, Computer methods in applied me- chanics and engineering 279 (2014) 554–578

    2014

  34. [34]

    Castillo, R

    E. Castillo, R. Codina, Variational multi-scale stabilized formulations for the stationary three- field incompressible viscoelastic flow problem, Computer Methods in Applied Mechanics and Engineering 279 (2014) 579–605

    2014

  35. [35]

    Castillo, R

    E. Castillo, R. Codina, Numerical analysis of a stabilized finite element approximation for the three-field linearized viscoelastic fluid problem using arbitrary interpolations, ESAIM: Mathe- matical Modelling and Numerical Analysis 51 (2017) 1407–1427

    2017

  36. [36]

    G. R. Barrenechea, E. Castillo, R. Codina, Time-dependent semidiscrete analysis of the vis- coelastic fluid flow problem using a variational multiscale stabilized formulation, IMA Journal of Numerical Analysis 39 (2019) 792–819

    2019

  37. [37]

    Kwack, A

    J. Kwack, A. Masud, A three-field formulation for incompressible viscoelastic fluids, International Journal of Engineering Science 48 (2010) 1413–1432

    2010

  38. [38]

    Kwack, A

    J. Kwack, A. Masud, K. Rajagopal, Stabilized mixed three-field formulation for a generalized incompressible Oldroyd-B model, International Journal for Numerical Methods in Fluids 83 (2017) 704–734

    2017

  39. [39]

    D. D. Medeiros, H. Notsu, C. M. Oishi, Second-order finite difference approximations of the upper-convected time derivative, SIAM Journal on Numerical Analysis 59 (2021) 2955–2988

    2021

  40. [40]

    B. S. Ashby, T. Pryer, Discretisation of an Oldroyd-B viscoelastic fluid flow using a lie derivative formulation, Advances in Computational Mathematics 51 (2025) 1

    2025

  41. [41]

    Y.-J. Lee, J. Xu, C.-S. Zhang, Global existence, uniqueness and optimal solvers of discretized viscoelastic flow models, Mathematical Models and Methods in Applied Sciences 21 (2011) 1713– 1732. 23

    2011

  42. [42]

    Y.-J. Lee, J. Xu, New formulations, positivity preserving discretizations and stability analysis for non-Newtonian flow models, Computer methods in applied mechanics and engineering 195 (2006) 1180–1206

    2006

  43. [43]

    Y.-J. Lee, J. Xu, C.-S. Zhang, Stable finite element discretizations for viscoelastic flow models, in: Handbook of numerical analysis, volume 16, Elsevier, 2011, pp. 371–432

    2011

  44. [44]

    T. J. Hughes, G. R. Feijóo, L. Mazzei, J.-B. Quincy, The variational multiscale method—a paradigm for computational mechanics, Computer methods in applied mechanics and engineering 166 (1998) 3–24

    1998

  45. [45]

    Moreno, R

    L. Moreno, R. Codina, J. Baiges, E. Castillo, Logarithmic conformation reformulation in viscoelas- tic flow problems approximated by a VMS-type stabilized finite element formulation, Computer Methods in Applied Mechanics and Engineering 354 (2019) 706–731

    2019

  46. [46]

    Castillo, R

    E. Castillo, R. Codina, Finite element approximation of the viscoelastic flow problem: A non- residual based stabilized formulation, Computers & Fluids 142 (2017) 72–78

    2017

  47. [47]

    I. A. Baratta, J. P. Dean, J. S. Dokken, M. Habera, J. HALE, C. N. Richardson, M. E. Rognes, M. W. Scroggs, N. Sime, G. N. Wells, DOLFINx: the next generation FEniCS problem solving environment (2023)

    2023

  48. [48]

    M. A. Alves, P. J. Oliveira, F. T. Pinho, Benchmark solutions for the flow of Oldroyd-B and PTT fluids in planar contractions, Journal of Non-Newtonian Fluid Mechanics 110 (2003) 45–75

    2003

  49. [49]

    H. C. Choi, J. H. Song, J. Y. Yoo, Numerical simulation of the planar contraction flow of a giesekus fluid, Journal of Non-Newtonian Fluid Mechanics 29 (1988) 347–379. 24

    1988

This paper was first reviewed by deepseek-v4-flash on August 4, 2026.