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REVIEW 4 major objections 5 minor 49 references

Vortex shedding and heat transfer from a heated circular cylinder in Bingham plastic fluids

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper claims that vortex shedding past a heated cylinder in a Bingham plastic fluid is a subcritical bifurcation: for $Re \geq 60$, steady and shedding wakes coexist over a range of Bingham numbers, so drag and heat transfer jump…

desk verdict Solid, well-validated numerics with a plausible but unproven subcritical-bifurcation claim that needs a stability analysis before it is taken as settled. read the letter →

arxiv 2411.16005 v1 pith:SEE4RVTV submitted 2024-11-24 physics.flu-dyn

Sai Peng , Xiang Li , Li Yu , Xiaoru Zhuang , Peng Yu This is my paper · ORCID
classification physics.flu-dyn
keywords BinghamplasticfluidvortexsheddingsubcriticalbifurcationhysteresisyieldstresscircularcylinderheattransferNusseltnumber
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

This paper tries to establish that vortex shedding behind a heated circular cylinder in a Bingham plastic fluid—a fluid that only deforms once the local shear stress exceeds a yield stress—is history-dependent. For $Re \geq 60$, the authors find two different critical Bingham numbers depending on whether the Bingham number is increased from a Newtonian shedding state or decreased from a steady yield-stress state, so steady and unsteady wakes coexist on an interval of $Bn$. If the claim is right, the transition is a subcritical bifurcation, and the mean drag coefficient and mean Nusselt number jump discontinuously at the transition, not smoothly. That matters because it means mixing and heat transfer in industrial yield-stress fluids can be controlled by operating history rather than only by the current flow conditions.

What carries the argument

The argument is carried by two coupled tools. The first is a regularized Bingham constitutive model (Papanastasiou regularization) that replaces the discontinuous yield-stress law with a smooth high-viscosity approximation, allowing a single finite-volume solver to represent both yielded and unyielded regions. The second is the pair of initialization protocols, IB and DB, which act as controlled disturbance levels: IB starts from the fully developed Newtonian vortex-shedding field and raises $Bn$, so the shedding perturbation is carried into the yield-stress regime; DB starts from a converged steady Bingham state and lowers $Bn$, so only intrinsic instability can seed new motion. The difference between the two critical values, $Bn_{cI}$ and $Bn_{cD}$, is the signature of the subcritical bifurcation, and the Carreau-Yasuda-like fit for $\overline{Nu}(Bn)$ is the engineering formula built on the same data.

What would settle it

Linearize the steady Bingham base flow around the cylinder at $Re \geq 60$ and compute its most unstable eigenmode. If the steady state is linearly unstable for $Bn$ below $Bn_{cI}$, then a fully converged steady-state initialization should spontaneously develop shedding, contradicting the claimed stable steady branch up to $Bn_{cI}$. Conversely, if the steady state is linearly stable down to $Bn_{cD}$ and shedding appears only after finite-amplitude disturbances, the subcritical claim is confirmed. An independent check would initialize the flow at a $Bn$ inside $(Bn_{cD}, Bn_{cI})$ with a controlled disturbance sweep and demonstrate two coexisting stable states separated by an unstable threshold.

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Extended reading notes

Core claim

The central discovery is that the onset of vortex shedding in a Bingham plastic flow around a cylinder is not a unique critical event but a subcritical bifurcation with a hysteresis loop. For $Re \geq 60$, increasing $Bn$ along the IB path suppresses shedding at $Bn_{cI}$, while decreasing $Bn$ along the DB path restores shedding only at the smaller value $Bn_{cD}$; between the two, the same parameters admit both a steady wake and a vortex-shedding wake. The root-mean-square lift coefficient jumps from a finite value to zero at $Bn_{cI}$ and from zero to a finite value at $Bn_{cD}$, and the mean drag coefficient and mean Nusselt number follow with abrupt jumps: for example, at $Re=180$ on the IB branch the mean Nusselt number drops from $7.4749$ to $7.3652$ as $Bn$ changes from $2.68$ to $2.7$. Away from the transition, drag and heat transfer vary smoothly with $Bn$, and the Nusselt-number data over the whole range are represented by a Carreau-Yasuda-like correlation (Eq. 37) that reduces to Newtonian and fully plastic limits.

Load-bearing premise

The conclusion rests on the assumption that the IB and DB initialization protocols differ only in the strength of the disturbance they carry, so the gap between $Bn_{cI}$ and $Bn_{cD}$ is a genuine physical hysteresis rather than an artifact of the regularization, time step, or finite-time statistics.

Editorial extensions

If this is right

  • For $Re \geq 60$, identical flow parameters can produce either a steady or a shedding wake, so numerical and experimental studies of Bingham flow around cylinders should report the initialization or operating history.
  • The abrupt jumps in $\overline{C_d}$ and $\overline{Nu}$ at $Bn_{cI}$ and $Bn_{cD}$ mean that correlations fitted to one branch will fail near the critical Bingham number, and a small change in yield stress or flow rate can switch heat-transfer performance sharply.
  • The approximately linear critical curve $Bn_{cI} \approx 0.0201 Re - 0.9993$ provides a quick rule of thumb for when yield stress suppresses vortex shedding.
  • The Carreau-Yasuda-like Nusselt correlation extends useful engineering prediction from $Bn=0$ to the fully plastic limit for steady flow, with the stated fitting error below 5%.

Reading between the lines

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

  • Editorial inference: the width of the hysteresis interval, $Bn_{cI}-Bn_{cD}$, should be proportional to the minimum finite-amplitude disturbance needed to trigger shedding; a controlled amplitude-continuation study would map the unstable threshold separating the two attractors.
  • Editorial inference: if the hysteresis is physical, a transient disturbance such as a short cylinder rotation or acoustic pulse could switch an otherwise steady Bingham wake into persistent vortex shedding, enhancing convective heat transfer without sustained energy input.
  • Editorial inference: applying the same IB/DB protocol to other yield-stress bluff-body geometries might show that some previously reported steady regimes are metastable, because earlier studies generally used a single start-up condition.
  • Editorial inference: the paper's discrete $Bn$ steps bracket, rather than resolve, the exact jumps; arc-length continuation or refined sweeps would reveal whether the jumps are true discontinuities or steep but continuous folds.
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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

4 major / 5 minor

Summary. The paper presents a numerical study of flow and heat transfer past a heated circular cylinder in a Bingham plastic fluid over Re = 10–180, Pr = 1–100, and Bn = 0–10^4, using the Papanastasiou regularization in a finite-volume solver. The authors validate their method against Newtonian cylinder data (Rec = 46.1, Stc = 0.1168, drag and Nusselt curves) and steady Bingham results from Nirmalkar & Chhabra and Mossaz et al. The central new claim is that, for Re ≥ 60, the onset of vortex shedding is subcritical: the critical Bingham number depends on the initialization protocol, with BncI (increasing Bn from a Newtonian shedding state) larger than BncD (decreasing Bn from a steady Bingham state), leading to hysteresis and abrupt jumps in the time-averaged drag coefficient and Nusselt number. The paper also reports a Carreau-Yasuda-like correlation for Nu as a function of Bn, and various local and global heat transfer characteristics.

Significance. If the subcritical-bifurcation claim is correct, it is a significant and non-obvious result for viscoplastic fluid mechanics, with practical consequences for heat-transfer control and for interpreting simulations of Bingham flows past bluff bodies. The manuscript's Newtonian validation is convincing, and the steady-state Bingham drag and Nusselt comparisons with published data are within 3–10%, giving reasonable confidence in the baseline solver. The paper also provides a broad parametric map of yielded/unyielded regions, shear-rate distributions, and local Nusselt numbers that could be useful for benchmarking. However, the central hysteresis claim is not yet backed by direct stability or amplitude-disturbance evidence; the present evidence is consistent with a subcritical bifurcation but also with finite-time and regularization artifacts. Because the central claim is load-bearing, the current evidence is not sufficient for acceptance without revision.

major comments (4)
  1. [Section 3.2.1, Fig. 6] The subcritical-bifurcation conclusion rests entirely on the observation that BncI and BncD differ for Re ≥ 60. The paper does not establish that the DB branch has reached the true asymptotic state. Near a bifurcation, perturbation growth rates are small, so a finite integration window can misclassify a slowly growing instability as steady in the DB branch and a slowly decaying disturbance as sustained shedding in the IB branch. The manuscript states only that 'the flow reaches statistical stationary state' (Section 2.3) without giving a quantitative stationarity criterion or reporting the transient duration at the transition points (e.g., Re = 180, Bn = 1.8 vs. 1.77). Without a linear stability analysis, a controlled finite-amplitude disturbance study, or at minimum a time-history check showing that the observed state is invariant over an extended window, the BncI–BncD gap does not prove subcritical bifurcation and may reflect finite-time statistics.
  2. [Section 2.3, Table 2] The Papanastasiou regularization parameter M U∞/D = 10^5 gives an effective viscosity in nominally unyielded regions of order 1 + Bn·M U∞/D, which for Bn ≈ 2–3 is O(10^5) times the plastic viscosity. This large artificial viscosity can artificially stabilize the steady branch and thereby shift BncD downward. The M-sensitivity check in Table 2 is performed at (Re, Bn) = (100, 5), a strongly unsteady shedding case, and does not constrain the behavior near BncI or BncD. A dedicated M-independence study at the transition points, or the use of an augmented-Lagrangian / unregularized method for at least a subset of cases, is needed to rule out a regularization-induced hysteresis interval.
  3. [Section 2.1] The mesh-independence of the reported hysteresis and jumps is not demonstrated. The mesh description (209,600 cells, first-cell spacing 0.0025D) is given, but no comparison with a finer or coarser mesh is presented. The Newtonian and steady Bingham validations at moderate parameters do not guarantee that the sharp transitions near Bnc are mesh-independent, especially because the yielded/unyielded boundary location is sensitive to the resolved velocity gradients in the regularization framework. A mesh-refinement study for a representative case such as Re = 180 near BncD and BncI is required to support the quantitative claims about abrupt jumps in Cd and Nu.
  4. [Section 3.2.2, Eqs. (37)–(38d)] The claim that the Nu–Bn data 'fits well with the Carreau-Yasuda-like non-Newtonian viscosity model' is a fitting statement, not a predictive validation. The parameters Nu0, Nu∞, n, and λ in Eqs. (38a)–(38d) are calibrated on the same simulation data that are then compared with Eq. (37) in Fig. 19. An error of less than 5% for the steady branch only confirms that the four-parameter function is sufficiently flexible to interpolate the data. The paper should state explicitly that Eq. (37) is a correlating equation and should give its intended range of applicability; it should not be presented as evidence for a physical analogy with the Carreau-Yasuda model without independent data.
minor comments (5)
  1. [Section 3.2.1] The yield criterion 'the flow yields when μ/μB < 10^5.4' is introduced without justification. The paper should explain how this threshold was chosen and whether the yielded-region morphology is sensitive to the choice within, say, one order of magnitude.
  2. [Section 2.3] The criterion for 'statistical stationary state' is qualitative. Please provide a quantitative definition, such as convergence of time-averaged Cd and Nu over a sliding window, and report the total simulation time in convective units for the reported cases.
  3. [Section 3.1, Eq. (28)] The piecewise Nu–Re correlation for Newtonian flow is discontinuous at Rec; the paper attributes this to the steady-to-unsteady transition. It would be helpful to state that the correlation is not intended to be continuous at Rec and to clarify whether the two branches are valid in the immediate vicinity of Rec.
  4. [Section 4] There is a typographical error: 'Bin gham' should be 'Bingham' in the first paragraph of the Conclusion.
  5. [Throughout] The paper contains several grammatical and punctuation errors (e.g., 'It should be note that', 'the flow is checked to be steady'). A careful language edit is recommended.

Circularity Check

1 steps flagged · score 6.0 of 10

Nu-Bn 'Carreau-Yasuda-like' agreement is an in-sample fit, but the central hysteresis claim is not circular.

  1. fitted input called prediction [Section 3.2.2 (Heat transfer feature), Eq. (37) and Eqs. (38a-d), Fig. 19; sentence after Eq. (38d)]
    "The best fitting results by Eq. (37) are also displayed in Fig. 19 for comparison. When the flow is steady (the right parts of the curves for a large Bn in Fig. 19), the error between the fitting data and the original data is less than 5%. ... Thus, Eq. (38a) can provide a better prediction for Nu0 in a Newtonian fluid when the flow is in a steady state."

    Eq. (37) is a four-parameter Carreau-Yasuda-like response surface (Nu0, Nu∞, n, λ), and Eqs. (38a-d) are correlations for those parameters in Re and Pr. The paper reports the 'agreement' of Eq. (37) with the original Nu-Bn data in Fig. 19 and measures it as 'the error between the fitting data and the original data.' Since the same simulation data were used to calibrate Nu0, Nu∞, n, λ (and then to fit Eqs. (38a-d) against those calibrated values), the small steady-flow error is a training-set residual, not an independent validation. Calling Eq. (38a) a 'prediction for Nu0' therefore presents a fitted parameter as a predicted quantity; the good agreement is forced by the fitting procedure rather than by an independent test of the Carreau-Yasuda analogy.

full rationale

The central subcritical-bifurcation claim is not circular: BncI and BncD are simulation outputs from two explicitly defined protocols (IB and DB), the gap between them is an observed hysteresis interval, and the IB critical line is compared with Mossaz et al.'s independent fit. The absence of a linear stability or disturbance-amplitude study, and the convergence checks performed only at (Re,Bn)=(100,5) rather than at the transition, are robustness and correctness concerns, not self-referential reductions. The Newtonian validation (Table 3, Figs. 3-4) and steady Bingham drag/Nu comparisons (Tables 4-5) provide external benchmarks for the numerical method. The only load-bearing step that reduces to its own inputs is the Nu-Bn 'Carreau-Yasuda-like' correlation in Sec. 3.2.2: parameters are calibrated on the same data whose agreement is then reported as support, and the one 'prediction' statement is an in-sample evaluation. This does not infect the hysteresis conclusion, but it does make the abstract's claim that Nu and Bn 'fits well' with the CY-like model a restatement of the fit rather than a validated prediction. Self-citations (refs 17, 23-26) are methodological and do not carry the physical conclusions, so they do not independently raise the score.

Assumptions & free parameters 7 free parameters · 7 assumptions · 0 invented entities

The load-bearing physics claim (subcritical bifurcation) rests on the interpretation of the IB/DB initialization protocols and on the Papanastasiou regularization being close enough to the ideal Bingham model. The heat transfer correlation rests on 14 fitted coefficients calibrated to the same dataset on which the fit is evaluated. No new physical entities are introduced; mu_eff and Re* are defined quantities, not new ontological claims.

free parameters (7)
  • Papanastasiou regularization parameter M = M U_infinity / D = 10^5
    Chosen as a numerical regularization; convergence checked at one state point (Table 2), but the ideal-Bingham limit is never reached. Affects yielded/unyielded morphology and transition thresholds.
  • Effective-drag prefactor X in Cd = X / Re* = 24.84
    Fitted to own steady simulation data for Re* < 0.5 (Eq. 35); close to Nirmalkar and Chhabra's 24.75.
  • Nu0 correlation coefficients (Eq. 38a) = 0.75505 Re^0.42779 Pr^0.322915
    Fitted to own Newtonian limiting Nusselt numbers; used as anchor for the Carreau-Yasuda-like correlation.
  • Nu-infinity correlation coefficients (Eq. 38b) = 1.2012 Re^0.40964 Pr^0.42006
    Fitted to own high-Bn (fully plastic) Nusselt numbers.
  • Power-law index n correlation (Eq. 38c) = 2.00299 - 0.03361 log(Pr) + Re[0.002835 - 0.0001028 log(Pr)]
    Fitted to slopes of (Nu-Nu0) versus Bn at small lambda*Bn (Eq. 40).
  • Lambda correlation (Eq. 38d) = [8.26512 - 1.16921 log(Pr)] Re^(-0.81331 - 0.01388 log(Pr))
    Fitted to curve shifts in log-log Nu-Bn data (Eq. 41).
  • Clrms scaling exponent and prefactor (Eq. 22) = 0.6554 and 56.9401
    Fitted to Newtonian lift fluctuation data; used to identify supercritical Hopf behavior in the Newtonian limit.
assumptions (7)
  • standard math Incompressible Navier-Stokes and thermal energy equations (Eqs. 1-3) with the Bingham constitutive relation are an adequate model for this flow.
    Used throughout; standard continuum assumptions for laminar viscoplastic flow.
  • domain assumption The Papanastasiou regularization with M U_infinity / D = 10^5 is close enough to the ideal Bingham fluid for the transition thresholds and heat transfer results.
    Invoked in Eq. (9) and Table 2; convergence is shown only at (Re, Bn) = (100, 5), not across the parameter space, and Bnc values may shift with stronger regularization.
  • ad hoc to paper The IB and DB initialization protocols differ only in disturbance intensity and therefore probe the same nonlinear bifurcation.
    Defined in Section 3.2.1; this is the premise underlying the subcritical-bifurcation inference from BncI != BncD.
  • ad hoc to paper A cell is considered yielded when mu / mu_B < 10^5.4.
    Bi-viscous criterion used to draw yielded/unyielded morphology in Section 3.2.1; the threshold is arbitrary.
  • domain assumption The flow is two-dimensional and thermophysical properties are independent of temperature, with negligible viscous dissipation.
    Stated in Section 2.2; decouples momentum and energy equations and restricts results to small Tw - T0.
  • domain assumption Ten cycles of statistically stationary data are sufficient for time-averaged drag and Nusselt numbers.
    Section 2.3 states data collected over more than 10 cycles; no cycle-count convergence test is reported.
  • domain assumption A 2% blockage with symmetric lateral boundaries approximates an unconfined cylinder.
    Section 2.1; no blockage correction is applied, so confinement may slightly shift Bnc and Nu.

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Pith. "Pith review of Vortex shedding and heat transfer from a heated circular cylinder in Bingham plastic fluids." pith.science (2026). https://pith.science/paper/SEE4RVTV

@misc{pith2026241116005,
  author       = {Pith},
  title        = {Pith review of: Vortex shedding and heat transfer from a heated circular cylinder in Bingham plastic fluids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SEE4RVTV}},
  note         = {Machine review of arXiv:2411.16005}
}
abstract

The present study numerically investigates the vortex shedding and heat transfer characteristics of a heated circular cylinder immersed in Bingham plastic fluids.The effects of three parameters, i.e., (i) plastic Reynolds number ($10 \leq Re \leq 180$), (ii) Prandtl number ($1\leq Pr \leq 100$), and (iii) the Bingham number ($0 \leq Bn \leq 10,000$), are evaluated. The Navier-Stokes and energy equations for flow and heat transfer are adopted, along with the incorporation of the Papanastasiou regularization to address the discontinuous-viscosity characteristics of Bingham plastic fluids. To illustrate the impact of fluid yield stress on the flow structure, the study provides comprehensive insights into flow transition, streamlines, shear rate and velocity distributions, the morphology of yielded/unyielded regions, and the drag coefficient ($C_d$). Additionally, the temperature distribution, the local Nusselt number ($\overline{Nu_{local}}$) along the cylinder, and the average Nusselt number on the cylinder ($\overline{Nu}$) are analyzed. The results indicate that the flow transition of Bingham fluids over a circular cylinder is dependent on external disturbances, exhibiting subcritical bifurcation behavior. This leads to abrupt jumps in the $\overline{Cd}$ - $Bn$ curve and the $\overline{Nu}$ - $Bn$ curve near the critical Bingham number $Bn_c$. Furthermore, the heat transfer performance is contingent upon the different distribution of shear strain rate in the boundary layer across various $Bn$ ranges. It is observed that $\overline{Nu}$ and $Bn$ fits well with the Carreau-Yasuda-like non-Newtonian viscosity model. This investigation enhances the understanding of the vortex shedding and heat transfer behaviors in Bingham plastic fluids.

Figures

Figures reproduced from arXiv: 2411.16005 by the authors.

Figure 1
Figure 1. (a) The relationship between shear stress (τ) and shear rate (𝛾̇) of a viscoplastic fluid. (b) The yielded (white) and unyielded (gray) regions of a viscoplastic fluid flow over a circular cylinder (the incoming flow is from left to right) at a vanishing Reynolds number, adapted from Nirmalkar & Chhabra4 . The Bingham or Herschel-Bulkley constitutive relation is commonly employed in numerical simulations of viscopla… view at source ↗
Figure 2
Figure 2. (a) Schematic of the computational domain. (b) The mesh around the cylinder. (c) The [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 5
Figure 5. Instantaneous streamlines for different Re and Bn [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figures from the paper (2 more)
Figure 7
Figure 7. Figure 7: Influence of Re and Bn on the morphology of the yielded (white) and unyielded (blue) regions. The regions are determined based on the time-averaged flow field. The presence of Zr3 is associated with the velocity stagnation points located upstream and downstream of the …
Figure 20
Figure 20. Figure 20: Variations of (a) 𝑁𝑢0, (b) 𝑁𝑢∞, (c) n, and (d) λ with Re at different Pr. 𝑑 log(𝑁𝑢̅̅̅̅−𝑁𝑢̅̅̅̅ 0 ) 𝑑 log(𝐵𝑛) = 𝑛−1 [1+(𝜆⋅𝐵𝑛) 𝑛−1 2 ] [PITH_FULL_IMAGE:figures/full_fig_p038_20.png]

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