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Exacerbation of viscoelastic instability due to viscous heating

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Pith's one-line read Viscous heating-induced buoyancy lowers the critical Weissenberg number of the purely elastic channel instability from 1490 to 797.5.

desk verdict A numerically careful linear-stability study whose headline physical claim is undercut by an inconsistent temperature normalization and an unjustified independence of Gr and Br in the no-external-heating regime. read the letter →

arxiv 2506.04617 v1 pith:R2OR5PR4 submitted 2025-06-05 physics.flu-dyn

Ankush Kamboj , Ramkarn Patne , P. A. L. Narayana , Kirti Chandra Sahu This is my paper
classification physics.flu-dyn MSC 76E0576A1076E07 PACS 47.20.Ft47.50.-d47.57.Ng
keywords viscoelasticfluidviscousheatinglinearstabilityanalysispurelyelasticinstabilityOldroyd-BbuoyancyGrashofnumberWeissenberg
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper claims that the purely elastic instability found by Khalid et al. (2021b) in a pressure-driven Oldroyd-B channel sets in at a substantially lower Weissenberg number when viscous heating and the buoyancy it induces are included. Because the fluid is highly viscous, viscous dissipation warms the flow even with no external heating; the resulting temperature field feeds a buoyancy term $(Gr/Re)\theta$ in the momentum equation that becomes large at the very low Reynolds numbers where the elastic mode lives. A linear stability calculation puts the critical Weissenberg number at 797.5 for $Gr=0.1$, $Br=0.05$, compared with 1490 in the isothermal case, with the critical wavenumber rising from 1.05 to 2.65. If the prediction holds, experiments should observe the purely elastic instability at slower speeds than the isothermal theory says. The paper attributes the destabilization to viscous heating breaking the symmetry of the streamwise velocity eigenfunctions.

What carries the argument

The load-bearing object is the $(Gr/Re)\theta$ coupling in the linearized momentum equation, a Boussinesq buoyancy term proportional to the temperature disturbance, combined with the viscous-dissipation source in the energy equation. In an Oldroyd-B fluid whose stress follows the upper-convected derivative, the disturbance equations form a generalized eigenvalue problem $Ax=cBx$ that the paper solves by Chebyshev spectral collocation with $N=1000$ points. The buoyancy term is what breaks the centerline symmetry of the eigenfunctions; the base-state temperature gradient $\Theta_0'$ couples velocity and temperature disturbances through the energy equation, and the $(Gr/Re)\theta$ term feeds that temperature disturbance back into the momentum balance. This feedback loop is the mechanism that lowers the critical Weissenberg number.

What would settle it

Measure the centerline temperature rise in a pressure-driven Oldroyd-B channel with isothermal walls at the speeds corresponding to the predicted neutral curves; if the measured $\Delta T$ is close to $\eta U_m^2/(3\kappa)$, then $Br$ is fixed near 3 and the independent small-$Br$ neutral curves used in the paper are not physically realizable, so the reported $W_c$ shift should be re-evaluated with $Gr$ and $Br$ linked.

Watch

Extended reading notes

Core claim

Starting from the isothermal Oldroyd-B channel flow of Khalid et al. (2021b), the paper adds the thermal energy equation with viscous dissipation and a Boussinesq buoyancy force, so no external heating is imposed. For the steady base state, the velocity remains $U=1-y^2$ while the temperature profile becomes $\Theta_0 = (Br/3)(1-y^4)$. The linearized disturbance equations show that increasing $Br$ (viscous heating) or $Gr$ (buoyancy) widens the unstable band of wavenumbers, increases the maximum growth rate, and shifts the neutral curve so that $W_c$ falls from 1490 at $Gr=0$, $Br=0$ to 935 at $Gr=0.05$, $Br=0.05$ and to 797.5 at $Gr=0.1$, $Br=0.05$; the critical wavenumber rises from 1.05 to 2.65. The paper states this as a drastic decrease compared with Khalid et al. and concludes that viscous heating-induced buoyancy destabilizes the purely elastic instability, so the mode should be observable at a much lower Weissenberg number. The mechanism identified is the loss of symmetry of the streamwise velocity eigenfunctions about the channel centerline.

Load-bearing premise

The load-bearing premise is that $Gr$ and $Br$ can be chosen independently in a setup with no external heating, even though the temperature difference that defines both parameters is itself produced by the flow and is therefore fixed once the fluid and speed are chosen.

Editorial extensions

If this is right

  • Experiments seeking the purely elastic channel instability should expect onset near $W_c \approx 800$ in systems with viscous heating and buoyancy, not near the isothermal value of 1490.
  • The first unstable mode at onset will have a shorter wavelength, since $k_c$ increases from about 1.05 to 2.65 as $Gr$ and $Br$ grow.
  • Viscous heating is destabilizing at all wavenumbers in this model whenever buoyancy is included, in contrast to earlier creeping-flow viscoelastic analyses that omitted buoyancy.
  • Higher solvent-to-solution viscosity ratio $\beta$ remains stabilizing: raising $\beta$ from 0.993 to 0.997 increases $W_c$ from 358 to 797.5 even with heating present.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If $Gr$ and $Br$ are tied to the same physical temperature difference, with no external heating so that $\Delta T$ is set by the flow speed, then they cannot be varied independently and the reported threshold reduction may not survive a self-consistent parameter scan.
  • The predicted asymmetry concentrates disturbance activity in the lower half of the channel, so high-speed imaging or infrared thermography could look for that spatial bias as a signature of heating-driven destabilization.
  • The same $(Gr/Re)\theta$ coupling could lower the threshold for neighboring elastic instabilities, such as viscoelastic pipe flow or gravity-driven film flow, a transfer the paper does not make.
  • A direct numerical simulation with a fixed flow speed and real fluid properties would settle whether the linear mode saturates into elastic turbulence at the reduced threshold.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The manuscript presents a temporal linear stability analysis of pressure-driven plane Poiseuille flow of an Oldroyd-B fluid, with the energy equation including viscous dissipation and a Boussinesq buoyancy term. The aim is to show that the purely elastic instability identified by Khalid et al. (2021b) is exacerbated (i.e., the critical Weissenberg number is reduced) when viscous heating generates a temperature field that induces buoyancy. The governing equations, base-state solution, linearized perturbation equations, and Chebyshev spectral collocation solution are presented. The numerical implementation is validated against Schmid and Henningson (2001) and Sureshkumar and Beris (1995), and grid convergence is demonstrated. The central quantitative result is a decrease of Wc from 1490 (isothermal baseline) to 797.5 for Gr=0.1, Br=0.05, together with an increase in the critical wavenumber.

Significance. If the physical modelling were correct, the finding that viscous heating-induced buoyancy dramatically lowers the onset Weissenberg number of the purely elastic instability would be of substantial practical interest, as it would imply that such instabilities could be observed in experiments at significantly lower Weissenberg numbers. The manuscript is careful in its linear algebra and provides fully written perturbation equations, a documented numerical method, and validation against several published benchmarks. However, as detailed in the major comments, the parameterization of the temperature field is internally inconsistent and the treatment of Gr and Br as independent parameters is not compatible with the stated no-external-heating scenario. These issues affect the central claim, so the significance of the reported Wc reduction is currently not established.

major comments (2)
  1. [§2, Eq. (6)] The base-state temperature is given as Θ0 = (Br/3)(1−y^4), which gives Θ0(0)=Br/3. However, the dimensionless temperature is defined in §2 as θ=(T−T0)/(T0−Tc), which requires θ(0)=−1 at the centerline because T(0)=Tc. Even if one interprets ΔT=T0−Tc as a reference scale rather than the actual center-to-wall difference, the actual temperature difference obtained from Eq. (6) is (Br/3)ΔT_ref, so the physical Grashof number is (Br/3)Gr, not Gr. This normalization inconsistency is load-bearing because the reported reductions in Wc rely on assigning independent values to Gr and Br rather than to the product Gr·Br, which is the physically meaningful buoyancy parameter.
  2. [§1 and §5, Tables 5–6] The paper argues that because Gr is independent of velocity, the term (Gr/Re)θ becomes important as Re→0. In the stated setup there is no external heating, so the temperature rise is solely due to viscous dissipation; consequently the actual center-to-wall temperature difference scales as ηU_m²/κ. The physical Grashof number then scales as U_m² and Gr_phys/Re scales as U_m, which vanishes in the creeping limit. The computations set Re=10^-5 and Gr=0.1, Br=0.05, giving Gr/Re=10^4; such a value is not realizable with internally generated heating at that Reynolds number. The reported decrease of Wc from 1490 to 797.5 is therefore not established for the no-external-heating scenario claimed in the abstract and conclusions.
minor comments (4)
  1. [§2, Table 4] The text fixes Re=10^-5, but Table 4 lists the Reynolds number range as 10^-3 to 1; please reconcile these values or clarify the intended parameter space.
  2. [Throughout] The notation contains several typesetting artifacts (e.g., '∇.∇.∇.', 'eeey', 'θ=0aty=±1') that should be corrected.
  3. [§4, figure 3(a)] The text states that the wavenumber of the most-dangerous mode remains approximately constant, whereas Table 5 shows a substantial increase of the critical wavenumber with Gr; the difference between these two quantities should be stated explicitly.
  4. [§4] The claim that the asymmetry of the streamwise velocity eigenfunctions is the mechanism behind the destabilization is descriptive; an energy budget or a more direct causality analysis would strengthen the argument.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the critical Weissenberg numbers are direct eigenvalues of the linearized Oldroyd-B/Boussinesq system, and the Gr=Br=0 limit recovers the external Khalid et al. benchmark.

full rationale

The central result is obtained by solving a generalized eigenvalue problem, equations (9)-(15) with boundary conditions (16), discretized as Ax=cBx in equation (18). For chosen parameters (Re, Gr, Pe, Br, beta, W), the code computes eigenvalues c, and Wc is read off as the minimum W on the neutral-stability curve ci=0 (Figure 10, Tables 5-7). This is an output of a direct computation, not a parameter fitted to reproduce a target Wc. The paper explicitly notes that the equations reduce to those of Khalid et al. (2021b) when Gr, Br, and Pe are zero, and Tables 1 and 2 validate the solver against independent published eigenvalues (Schmid and Henningson 2001; Sureshkumar and Beris 1995; Khalid et al. 2021a). Thus the baseline is an external benchmark rather than a self-defined input. The cited works involving the present co-author (Sahu and Matar 2010; Sahu et al. 2010; Sahu 2011) supply the standard Boussinesq buoyancy term Gr/Re theta and background on viscous heating; they do not by themselves assert the Oldroyd-B viscoelastic instability threshold obtained here. A physical-consistency concern can be raised separately: the definition theta=(T-T0)/(T0-Tc) and the base-state solution Theta0=(Br/3)(1-y^4) imply Theta0(0)=Br/3 rather than -1, and treating Gr as independent of velocity in a purely internally heated, isothermal-wall flow is physically debatable. However, that is a modeling/correctness question, not a circular derivation. No equation in the paper defines the predicted decrease in Wc in terms of Gr and Br by construction, and no fitted quantity is relabeled as a prediction. The reported Wc values are therefore not circularly forced.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The model contains no fitted parameters in the statistical sense; the critical W is computed, not fit. However, the dimensionless inputs Gr, Br, Pe, Re, and β are chosen by hand and not tied together by the energy balance under no-external-heating, so the ledger shows five hand-set inputs, four domain assumptions, and one ad hoc setup statement.

free parameters (5)
  • Grashof number Gr = 0 to 0.1
    Varied independently in the stability runs; with no external heating, the actual Gr would be tied to U^2 through the dissipation-generated ΔT.
  • Brinkman number Br = 0 to 0.05
    Varied independently; under the natural scaling ΔT equal to the centerline-wall temperature difference from viscous heating, Br=3, not 0.05.
  • Péclet number Pe = 10^4
    Fixed at 10^4 for all reported neutral curves; no Pe sensitivity study is given, although Pe appears in the perturbation energy equation.
  • Reynolds number Re = 10^-5
    Fixed to isolate the elastic mode; the physical Gr/Re scaling in the no-external-heating scenario is not analyzed.
  • solvent-to-solution viscosity ratio β = 0.997 main case
    Chosen to match the ultra-dilute polymer regime of Khalid et al. (2021b); only one β scan is reported.
assumptions (5)
  • domain assumption Oldroyd-B constitutive equation with constant solution viscosity and no temperature-dependent rheology
    Invoked in Eq. (3); heat-induced changes in viscosity are neglected even though viscous heating is the subject.
  • domain assumption Boussinesq approximation with constant density except for the buoyancy term
    Invoked before Eq. (2); it requires small temperature differences, which conflicts with the large ΔT needed for measurable buoyancy at low Re.
  • domain assumption Base flow is parallel and unidirectional despite a y-dependent buoyancy force
    The base state in Eqs. (5)-(7) has no vertical velocity; the y-momentum balance must absorb the buoyancy term in the base pressure, which is not stated.
  • domain assumption Perturbations are two-dimensional
    Introduced with normal modes in Eq. (8); Squire's theorem is not established for this non-isothermal Oldroyd-B problem.
  • ad hoc to paper No external heating; only viscous dissipation creates the temperature field
    This is the stated setup in §1, but it is not used to constrain the independent choices of Gr and Br, which is the main physical flaw.

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Cite this review

Pith. "Pith review of Exacerbation of viscoelastic instability due to viscous heating." pith.science (2026). https://pith.science/paper/R2OR5PR4

@misc{pith2026250604617,
  author       = {Pith},
  title        = {Pith review of: Exacerbation of viscoelastic instability due to viscous heating},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R2OR5PR4}},
  note         = {Machine review of arXiv:2506.04617}
}
read the original abstract

The linear stability analysis of the pressure-driven flow of an Oldroyd-B fluid through a plane channel is performed to examine the effects of viscous heating-induced buoyancy on the ``purely elastic instability" predicted by \citet{khalid2021continuous} (Phys. Rev. Lett., 2021, 127, 134502). We do not impose any external heating; rather, the temperature increase in the system is solely due to the viscous heating generated by the flow of a highly viscous fluid, which induces buoyancy. This buoyancy effect adds an extra term to the momentum equation, proportional to the ratio of the Grashof and Reynolds numbers. Since the elastic instability manifests at very low Reynolds numbers, this buoyancy term is crucial for determining flow stability. Our analysis indicates a significant decrease in the critical Weissenberg number due to the viscous heating-induced buoyancy effect, implying that the elastic instability could potentially be observed experimentally at a significantly lower Weissenberg number than that predicted by Khalid et al. (2021b). The asymmetry in the streamwise velocity eigenfunctions, resulting from the presence of viscous heating, is found to be the mechanism behind the predicted destabilizing effect.

Figures

Figures reproduced from arXiv: 2506.04617 by the authors.

Figure 1
Figure 1. Schematic diagram of the plane-Poiseuille flow of an incompressible Oldroyd-B [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Grid convergence test of our numerical scheme showing the effect of the Chebyshev [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Dispersion curves for different values of [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Contours showing the spatial variation of the real [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]
Figure 5
Figure 5. Figure 5: Contours showing the spatial variations of the real [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: Contours showing the spatial variation of the real [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]
Figure 7
Figure 7. Figure 7: Contours showing the spatial variation of the real [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]
Figure 8
Figure 8. Figure 8: (a) the perturbation in streamwise velocity u and (b) the perturbation in temperature θ in the y-direction for the different values of Gr. The values of the rest of the parameters are P e = 104 , Br = 0.05, k = 0.6, β = 0.997, and W = 2500. 21 [PITH_FULL_IMAGE:figures…
Figure 9
Figure 9. Figure 9: Variations of the absolute values of (a) the perturbation in streamwise velocity u and (b) the perturbation in temperature θ in the y-direction for different values of the Brinkman number (Br). The values of the remaining parameters are P e = 104 , Gr = 0.1, k = 0.6, β…
Figure 10
Figure 10. Figure 10: Neutral stability curves in the W − k plane for different Grashof number (Gr), Brinkman number (Br), and the ratio of solvent to solution viscosity of the Oldroyd-B fluid (β). (a) Effect of the Grashof number (Gr) with Br = 0.05 and β = 0.997; (b) effect of the Brinkm…

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