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arxiv: 2512.05787 · v1 · pith:ENV6ECPCnew · submitted 2025-12-05 · ⚛️ physics.flu-dyn · cond-mat.soft

Nature of continuous spectra in wall-bounded shearing flows of FENE-P fluids

Pratyush Kumar Mohanty , P. S. D. Surya Phani Tej , Ganesh Subramanian , V. Shankar This is my paper

Pith reviewed 2026-05-22 13:04 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn cond-mat.soft
keywords continuous spectraFENE-P fluidsshearing flowsviscoelastic flowsOldroyd-Bfinite extensibilitylinear stabilitywall-bounded flows
0
0 comments X

The pith

Wall-bounded shearing flows of FENE-P fluids exhibit up to six distinct continuous spectra.

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

The paper shows that the linearized equations for FENE-P fluids in rectilinear and curvilinear shear flows produce up to six continuous spectra. This arises because the constitutive relation introduces singular points whose locations depend on the local base velocity and the model parameters. A reader would care because these spectra must be separated from discrete eigenvalues to determine whether a flow is physically unstable. When the extensibility parameter exceeds 50, three spectra become nearly identical and independent of the viscosity ratio while the other three remain dependent on it, one of them recovering the known viscous spectrum from Oldroyd-B fluids and the other two being new features that can lie outside the base velocity range.

Core claim

The authors demonstrate analytically that shearing flows of FENE-P fluids possess up to six distinct continuous spectra. For finite extensibility parameter L greater than 50, three of these spectra are nearly identical and independent of the solvent-to-solution viscosity ratio β. The remaining three depend on β, one matching the viscous continuous spectrum of Oldroyd-B fluids, while the other two are new and can feature phase speeds outside the base velocity range, even negative values. This added complexity is expected to appear in other nonlinear viscoelastic models that show shear-thinning.

What carries the argument

Singular points in the linearized differential operators created by the spatially local FENE-P constitutive relation, with locations fixed by the local base-state velocity together with the parameters L and β.

Load-bearing premise

The FENE-P constitutive relation remains spatially local after linearization around the base shear profile.

What would settle it

A numerical computation of the full eigenspectrum for a specific FENE-P rectilinear shear flow with L greater than 50 that finds a number of continuous-spectrum branches other than six would test the analytical count.

Figures

Figures reproduced from arXiv: 2512.05787 by Ganesh Subramanian, Pratyush Kumar Mohanty, P. S. D. Surya Phani Tej, V. Shankar.

Figure 1
Figure 1. Figure 1: FIG. 1: Eigenspectra for viscoelastic Dean flow for [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Numerical eigenspectra for plane Couette flow showing the independence of CS with respect to the spanwise [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Schematic location of the various CS for plane Couette flow for [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Numerical eigenspectra for plane Couette flow at different [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Effect of wavenumber [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Analytical predictions for the variation of the imaginary part ( [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Bifurcation of CS3a,b with varying [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Variation of the different CS with [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Numerical and analytical CS for plane Couette flow. Data for [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Schematic location of the various CS for pressure-driven channel flow for [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Analytical CS and numerical spectra for pressure-driven channel flow. Data for [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Numerical and analytical CS for pressure-driven channel flow. Data for [PITH_FULL_IMAGE:figures/full_fig_p016_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: Analytical CS and numerical spectra for pressure-driven pipe flow for [PITH_FULL_IMAGE:figures/full_fig_p017_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14: Numerically computed spectra showing the independence of CS for two different [PITH_FULL_IMAGE:figures/full_fig_p018_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15: Schematics of the various CS for Dean flow. [PITH_FULL_IMAGE:figures/full_fig_p018_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16: Analytical CS and numerical spectra for Dean flow subjected to axisymmetric disturbances ( [PITH_FULL_IMAGE:figures/full_fig_p019_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17: Effect of [PITH_FULL_IMAGE:figures/full_fig_p021_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18: Analytical CS and numerical spectra for Dean flow. Data for [PITH_FULL_IMAGE:figures/full_fig_p022_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19: Collapse of analytical CS for various ( [PITH_FULL_IMAGE:figures/full_fig_p023_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20: Analytical predictions for CS3a, CS3b for different [PITH_FULL_IMAGE:figures/full_fig_p023_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21: Schematics of the various CS for Taylor Couette flow. [PITH_FULL_IMAGE:figures/full_fig_p023_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: FIG. 22: Analytical CS and numerical spectra for Taylor-Couette flow for [PITH_FULL_IMAGE:figures/full_fig_p024_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: FIG. 23: Analytical CS and numerical spectra of Taylor-Couette flow. Data for [PITH_FULL_IMAGE:figures/full_fig_p025_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: FIG. 24: Effect of [PITH_FULL_IMAGE:figures/full_fig_p025_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: FIG. 25: Analytical CS and numerical spectra for Taylor-Couette flow for [PITH_FULL_IMAGE:figures/full_fig_p026_25.png] view at source ↗
read the original abstract

Owing to the spatially local nature of the constitutive equations typically used to model polymeric stresses, the differential operators governing the linearized dynamics of bounded viscoelastic shearing flows have singular points. As a result, the eigenspectra of such shearing flows contain, in addition to discrete eigenvalues, continuous spectra (CS) comprising singular eigenfunctions. A clear understanding of the theoretical CS loci is crucial in discriminating physically genuine (discrete) eigenvalues from the poorly approximated numerical CS. For rectilinear shear flows of Oldroyd-B fluids, the CS are a pair of line segments, with lengths equal to the base-state range of velocities. In this study, we provide the first comprehensive account of the nature of the CS for both rectilinear and curvilinear shearing flows of the FENE-P fluid. In stark contrast to the CS for the Oldroyd-B fluid mentioned above, we show analytically that there are up to six distinct continuous spectra for shearing flows of FENE-P fluids. When the finite extensibility parameter $L > 50$, as appropriate for large molecular weight polymers used in experiments, three of the CS are nearly identical, and independent of the solvent-to-solution viscosity ratio ($\beta$). The other three CS are $\beta$-dependent, with one of them being the analogue of the solvent (viscous) continuous spectrum in the Oldroyd-B fluid. The remaining two $\beta$-dependent CS are novel features of the FENE-P spectrum, and can have phase speeds outside the base range of velocities, including negative ones. The complexity of the CS predicted here for shearing flows of FENE-P fluids is expected to carry over to other nonlinear viscoelastic models that exhibit a shear-thinning rheology.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims to provide the first comprehensive analytical account of the continuous spectra (CS) arising from singular points in the linearized differential operators for wall-bounded rectilinear and curvilinear shearing flows of FENE-P fluids. In contrast to the two velocity-range line segments known for Oldroyd-B fluids, it derives up to six distinct CS branches: for L > 50 three are nearly identical and independent of the solvent-to-solution viscosity ratio β, while the remaining three are β-dependent, one being the analogue of the Oldroyd-B viscous spectrum and two novel branches whose phase speeds can lie outside the base-flow velocity range (including negative values). The result is asserted to hold for both rectilinear and curvilinear base profiles and is expected to generalize to other shear-thinning nonlinear viscoelastic models.

Significance. If the analytical loci are correct, the work supplies a much-needed theoretical benchmark for distinguishing discrete eigenvalues from numerical artifacts in simulations of FENE-P flows, which are the standard model for finite-extensibility polymers in experiments. The separation into β-independent and β-dependent branches, together with the prediction of out-of-range phase speeds, offers concrete guidance for interpreting stability and transient-growth calculations in both channel and curvilinear geometries.

major comments (2)
  1. [Section 3] Section 3 (linearization of the FENE-P constitutive equation): the central claim that the six singularity loci are determined solely by the local base velocity U(y) together with L and β must be shown explicitly. Because the base conformation A0(y) is a nonlinear function of the local shear rate through the Peterlin function, the coefficients of the linearized 6-component system contain y-dependent factors involving A0 and its derivatives. It is not demonstrated that these factors produce no additional roots for the phase speed c at fixed y, nor that they leave the claimed β-independence of three branches intact for curvilinear profiles where shear rate varies.
  2. [§4.2] §4.2 (curvilinear-flow results): the assertion that the same six-branch structure and the same separation into β-independent and β-dependent spectra hold for both rectilinear and curvilinear base states rests on the locality assumption. A concrete verification that the determinant condition for the singular points remains independent of the explicit A0(y) dependence (beyond its implicit effect through U(y)) is required to support the claim for non-uniform shear.
minor comments (2)
  1. [Abstract] The abstract states 'up to six' but does not specify the parameter regimes (small L, extreme β) in which the number of distinct branches drops; a brief statement in the introduction would help readers.
  2. [Figure 2] Figure 2 (or equivalent spectrum plot): the six branches should be labeled or color-coded to distinguish the three nearly identical β-independent loci from the three β-dependent ones.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of our work's significance and for the constructive comments that will improve the clarity of the derivations. We address each major comment in turn and will incorporate revisions to make the analysis fully explicit.

read point-by-point responses

  1. Referee: [Section 3] Section 3 (linearization of the FENE-P constitutive equation): the central claim that the six singularity loci are determined solely by the local base velocity U(y) together with L and β must be shown explicitly. Because the base conformation A0(y) is a nonlinear function of the local shear rate through the Peterlin function, the coefficients of the linearized 6-component system contain y-dependent factors involving A0 and its derivatives. It is not demonstrated that these factors produce no additional roots for the phase speed c at fixed y, nor that they leave the claimed β-independence of three branches intact for curvilinear profiles where shear rate varies.

    Authors: We agree that an explicit demonstration strengthens the central claim. The linearization in Section 3 yields a 6×6 system whose singular points are located by setting the determinant of the coefficient matrix to zero. Although the Peterlin function introduces A0(y)-dependent factors, these multiply entire rows or columns in a manner that factors out of the determinant without introducing new roots for c; the only c-dependent terms arise from the convective operators (U(y)−c) and the relaxation terms scaled by L and β. The three β-independent branches correspond to the solvent viscous mode and two polymer modes whose coefficients decouple from β after the determinant is formed. We will add this explicit determinant expansion (or its factored form) as a new subsection or appendix in the revised manuscript. The same algebraic structure holds for curvilinear base states because the local shear direction remains rectilinear and the base conformation enters only through the already-accounted-for U(y) and U'(y). revision: yes

  2. Referee: [§4.2] §4.2 (curvilinear-flow results): the assertion that the same six-branch structure and the same separation into β-independent and β-dependent spectra hold for both rectilinear and curvilinear base states rests on the locality assumption. A concrete verification that the determinant condition for the singular points remains independent of the explicit A0(y) dependence (beyond its implicit effect through U(y)) is required to support the claim for non-uniform shear.

    Authors: We accept that a concrete verification is desirable for non-uniform shear. In the revised manuscript we will supplement §4.2 with a brief symbolic or numerical check: for a representative curvilinear profile (e.g., circular Couette flow), we evaluate the determinant condition at several radial locations using the local U(y) and the corresponding A0(y) obtained from the base-state solution; the resulting loci coincide exactly with the six branches predicted by the local U(y), L, β expressions, confirming that explicit A0(y) derivatives cancel and introduce no additional c-roots. This verification will be presented both analytically (via the factored determinant) and numerically for a specific parameter set. revision: yes

Circularity Check

0 steps flagged

No circularity: loci of continuous spectra derived directly from linearized operator singularities

full rationale

The paper derives the up to six continuous spectra by direct algebraic examination of the singular points in the linearized FENE-P constitutive equations (6-component conformation tensor) after linearization around the base shear profile. The singularity condition is obtained from the vanishing of the coefficient matrix determinant, yielding loci determined by the local base velocity U(y) together with parameters L and β. This is a standard first-principles stability analysis step with no fitted inputs renamed as predictions, no self-definitional loops, and no load-bearing self-citations. The abstract explicitly states the result holds for both rectilinear and curvilinear cases via the spatially local constitutive model; the derivation chain remains self-contained and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard linearization of the incompressible Navier-Stokes equations coupled to the FENE-P constitutive model, plus the assumption that the base flow is a steady rectilinear or curvilinear shear whose velocity varies spatially between walls.

free parameters (2)
  • L
    Finite extensibility parameter; the claim that three spectra become nearly identical holds only for L>50, a regime chosen to match large-molecular-weight polymers.
  • β
    Solvent-to-total viscosity ratio; three of the six spectra are stated to depend on β while the other three do not.
axioms (2)
  • domain assumption Spatially local constitutive equations for polymeric stress
    Invoked in the first sentence to explain the origin of singular points in the differential operators.
  • standard math Linearization about a steady base shear flow
    Standard procedure for obtaining the eigenvalue problem whose continuous spectrum is analyzed.

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Reference graph

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