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arxiv: 2604.05596 · v1 · submitted 2026-04-07 · ⚛️ physics.flu-dyn

Aggregation Effects on Heat Transfer in Viscoplastic Nanofluid Entrance Flows

Deepa Madivalar , Vishwanath Kadaba Puttanna , A Kandasamy This is my paper

Pith reviewed 2026-05-10 19:21 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords viscoplastic nanofluidnanoparticle aggregationentrance region flowheat transfer enhancementyield stress effectsNusselt numberperformance evaluation criteriafinite difference method
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The pith

Aggregation of nanoparticles modifies friction, pressure drop, and heat transfer in viscoplastic nanofluid entrance flows, with performance criteria identifying an optimal volume fraction up to 5%.

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

The paper numerically examines laminar, incompressible flow of a viscoplastic nanofluid in the entrance region of a uniformly heated circular cylinder. It models non-aggregated cases with Brinkman and Maxwell relations and aggregated cases with Krieger-Dougherty and Maxwell-Bruggeman relations, while using the Bingham-Papanastasiou model to capture yield-stress behavior. Finite-difference solutions of the boundary-layer equations yield local friction, pressure drop, and Nusselt numbers across nanoparticle fractions from 0 to 5 percent and varying yield stresses. These quantities are then fed into performance evaluation criteria to locate the volume fraction that delivers the highest overall efficiency. A reader would care because entrance-region losses dominate in short heat-exchange devices and nanofluid properties can be tuned to reduce pumping power while raising heat transfer.

Core claim

Using finite-difference solutions of the boundary layer equations for a Bingham-Papanastasiou viscoplastic fluid with nanofluid properties modeled by Brinkman-Maxwell for non-aggregation and Krieger-Dougherty/Maxwell-Bruggeman for aggregation, the study quantifies how yield stress and nanoparticle volume fraction influence friction factor, pressure drop, and Nusselt number in the pipe entrance region, then applies performance evaluation criteria to identify the optimal volume fraction for maximum efficiency.

What carries the argument

Bingham-Papanastasiou regularization of yield stress combined with aggregation-aware effective viscosity and thermal-conductivity models, solved in the developing entrance boundary layer.

Load-bearing premise

The Brinkman/Maxwell and Krieger-Dougherty/Maxwell-Bruggeman models together with the Bingham-Papanastasiou regularization remain accurate representations of the nanofluid rheology and thermal conductivity throughout the entrance region.

What would settle it

Direct experimental measurements of pressure drop and Nusselt number in a short heated tube for a known viscoplastic nanofluid at 3% and 5% volume fraction, comparing aggregated versus non-aggregated particle states, would show whether the predicted optimal efficiency point is reached.

Figures

Figures reproduced from arXiv: 2604.05596 by A Kandasamy, Deepa Madivalar, Vishwanath Kadaba Puttanna.

Figure 1
Figure 1. Figure 1: Schematic of the viscoplastic nanofluid flow in a cylinder/pipe [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Finite Difference Scheme The Finite Difference Method (FDM), as adopted from [15], incorporates the necessary boundary conditions within its numerical formulation. In the present study, a central 10 [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Rheogram for the Bingham-Papanastasiou Model [ [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Grid independence study for different grid resolutions, [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Grid independent Study for axial velocity ( [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of friction factor at the wall, [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of Nusselt number, Nu, along the axial direction for different volume fractions and Pr=6.2 with constant heat flux boundary condition compared with Benkhedda et al [31]. 5. Results and discussion An investigation of a viscoplastic nanofluid flow in the entrance region of a circular cylinder has been conducted for the case of both non-aggregated and aggregated nanopar￾ticles. The dimensionless go… view at source ↗
Figure 8
Figure 8. Figure 8: Effective thermo-physical properties for both non-aggregation and aggregation models for [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Centerline velocity in axial direction for different volume fractions [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Variation of Pressure drop (Fig. 10a), Friction coefficient (Fig. 10b), Bulk temperature (Fig. 10c) and Nusselt number (Fig. 10d) along the axial direction for different values of Volume fraction considering both non-aggregation and aggregation models with Bingham number Bn = 10. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Variation of Nusselt Number along the axial direction for different values of Volume fraction [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Development of axial velocity along the radial direction at axial locations 0.001, 0.01, and 0.3 [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Variation of centerline axial velocity along the axial direction for various Bingham numbers [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Variation of Pressure drop (Fig. 14a), Friction coefficient (Fig. 14b), Bulk temperature (Fig. 14c) and Nusselt number (Fig. 14d) along the axial direction for different values of Bingham num￾bers considering both non-aggregation and aggregation models with volume fraction ϕ = 0.03. 5.3. Performance evaluation criteria(PEC): In summary, the heat transfer enhancement of the viscoplastic nanofluid with incr… view at source ↗
Figure 15
Figure 15. Figure 15: Variation of Nusselt Number along the axial direction for different values of Bingham numbers [PITH_FULL_IMAGE:figures/full_fig_p024_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Performance evaluation criteria for change in volume fraction at Bingham number, [PITH_FULL_IMAGE:figures/full_fig_p026_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Performance evaluation criteria for change in volume fraction at Bingham number, [PITH_FULL_IMAGE:figures/full_fig_p026_17.png] view at source ↗
read the original abstract

This study numerically investigates heat transfer enhancement in laminar, incompressible viscoplastic nanofluid flow through the entrance region of a circular cylinder with a uniformly heated wall, including the effects of both, non-aggregation and aggregation of nanoparticles. Nanofluid properties are modeled using Brinkman and Maxwell models in the case of non-aggregation, and Krieger-Dougherty, Maxwell-Bruggeman models in the case of aggregation, while the viscoplastic behavior is described by the Bingham-Papanastasiou model. The governing boundary layer equations are solved using a finite-difference method. The effects of yield stress and nanoparticle volume fraction (up to 5%) on friction, pressure drop, and Nusselt number are analyzed, and performance evaluation criteria are evaluated to identify the optimal volume fraction for maximum efficiency.

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 numerically investigates heat transfer enhancement in the entrance region of laminar incompressible viscoplastic nanofluid flow through a uniformly heated circular tube, considering both non-aggregated and aggregated nanoparticles. Nanofluid properties are modeled via Brinkman/Maxwell (non-aggregation) or Krieger-Dougherty/Maxwell-Bruggeman (aggregation) effective-medium relations, with viscoplastic rheology captured by the Bingham-Papanastasiou regularization. A finite-difference scheme solves the boundary-layer equations; the effects of yield stress and nanoparticle volume fraction (up to 5%) on friction, pressure drop, and Nusselt number are quantified, after which performance evaluation criteria (PEC) are applied to identify an optimal volume fraction.

Significance. If the numerical results prove reliable, the work supplies parametric information on how aggregation and yield stress jointly influence developing-flow friction and heat transfer, which may aid selection of nanofluid formulations for compact heat exchangers. The adoption of standard property models and a conventional discretization approach is a methodological strength, yet the absence of verification data reduces the immediate utility of the reported quantitative trends and optimal-fraction recommendation.

major comments (2)
  1. [Numerical Procedure] Numerical Procedure section: no grid-independence tests, Richardson extrapolation, or comparisons against known analytic limits (Newtonian entrance flow, fully developed Bingham flow, or published nanofluid benchmarks) are reported. Without these, the quantitative values of friction factor, pressure drop, and local Nusselt number that underpin the PEC-based optimal-volume-fraction claim cannot be trusted.
  2. [Results and Discussion] Results and Discussion: the identification of an optimal nanoparticle volume fraction rests on the PEC evaluated from the computed friction and Nusselt data, but no sensitivity study is performed with respect to the Papanastasiou regularization parameter or the aggregation-model constants. Because these parameters directly control the effective viscosity and conductivity used in the boundary-layer solution, the location of the reported optimum is not demonstrably robust.
minor comments (2)
  1. [Abstract] The precise algebraic definition of the performance evaluation criterion (PEC) employed to rank volume fractions is not stated in the abstract or early sections; a reference or explicit formula should be supplied.
  2. [Figures] Figure captions and axis labels should explicitly indicate whether results correspond to the non-aggregated or aggregated model set, and whether the plotted quantities are local or axially averaged.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help improve the rigor of our work. We address each major comment below and will revise the manuscript to incorporate the suggested verifications and analyses.

read point-by-point responses

  1. Referee: [Numerical Procedure] Numerical Procedure section: no grid-independence tests, Richardson extrapolation, or comparisons against known analytic limits (Newtonian entrance flow, fully developed Bingham flow, or published nanofluid benchmarks) are reported. Without these, the quantitative values of friction factor, pressure drop, and local Nusselt number that underpin the PEC-based optimal-volume-fraction claim cannot be trusted.

    Authors: We agree that explicit documentation of grid independence and validation is essential for trusting the quantitative results. In the revised manuscript, we will add a dedicated subsection to the Numerical Procedure section that reports a grid-convergence study (including Richardson extrapolation) for friction factor, pressure drop, and local Nusselt number. We will also include direct comparisons against analytic limits for Newtonian entrance flow (Shah & London), fully developed Bingham flow, and published nanofluid benchmarks. These additions will confirm the accuracy of the reported values and the PEC-based optimum. revision: yes

  2. Referee: [Results and Discussion] Results and Discussion: the identification of an optimal nanoparticle volume fraction rests on the PEC evaluated from the computed friction and Nusselt data, but no sensitivity study is performed with respect to the Papanastasiou regularization parameter or the aggregation-model constants. Because these parameters directly control the effective viscosity and conductivity used in the boundary-layer solution, the location of the reported optimum is not demonstrably robust.

    Authors: We concur that robustness with respect to the regularization parameter and aggregation constants must be demonstrated. In the revised Results and Discussion, we will add a sensitivity study varying the Papanastasiou parameter m over a literature-typical range (e.g., 100–1000) and the key constants in the Krieger-Dougherty and Maxwell-Bruggeman models. We will show that the identified optimal volume fraction remains stable, thereby supporting the reliability of the PEC conclusions. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper is a standard numerical parameter study: it solves the boundary-layer equations for developing flow in a circular duct using a finite-difference discretization, with nanofluid properties supplied by fixed, externally established constitutive models (Brinkman/Maxwell or Krieger-Dougherty/Maxwell-Bruggeman for viscosity and conductivity, Bingham-Papanastasiou regularization for yield stress). No quantity is fitted to the target data inside the paper and then re-labeled as a prediction; no uniqueness theorem or ansatz is imported via self-citation to force the modeling choices; the performance-evaluation criteria are applied after the simulations are complete. The derivation chain therefore consists of independent numerical integration of well-documented equations rather than any reduction to its own inputs by construction.

Axiom & Free-Parameter Ledger

2 free parameters · 3 axioms · 0 invented entities

The central results rest on standard rheological and thermal-property models plus the assumption that the boundary-layer approximation holds in the entrance region; no new entities are postulated.

free parameters (2)
  • nanoparticle volume fraction
    Swept from 0 to 5% as an input parameter whose effect is quantified.
  • yield stress
    Varied as a parameter in the Bingham-Papanastasiou model.
axioms (3)
  • domain assumption Brinkman and Maxwell models accurately capture non-aggregated nanofluid viscosity and conductivity.
    Invoked to close the property relations for the non-aggregation case.
  • domain assumption Krieger-Dougherty and Maxwell-Bruggeman models accurately capture aggregated nanofluid viscosity and conductivity.
    Invoked to close the property relations for the aggregation case.
  • domain assumption Bingham-Papanastasiou regularization correctly represents viscoplastic behavior.
    Used to describe the yield-stress fluid.

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