Magnetohydrodynamic Flow and Transport Behaviors of Blood-Based Ternary Nanofluids in Stenosed Arteries with Axial Symmetry: Effects of Thermal Radiation and Caputo Fractional Derivatives

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The author presented a numerical solution to a time-fractional derivative model problem that described hydromagnetic convection of a radiating and reacting blood-based ternary nanofluids in stenosed arteries with heat and mass transfer characteristics. The following improvement must be incorporated:

1.) The entire model momentum, energy, and nanoparticles concentration equations (1)-(3), (23)-(25) are technically wrong and inaccurate. Note that the inclusion of the nanoparticles concentration equation (3) implies that the nanoparticles volume fraction (concentration) is a variable; however, the presence of the thermophysical expression of blood-based ternary nanofluids in the model momentum and energy equations (1)-(2), (23)-(24), as reflected in the appendix and table 1 revealed that nanoparticles volume fraction (concentration) is a constant. This is a contradiction, mathematically illogical and misleading. The authors should reformulate the problem as a single-phase blood-based ternary nanofluid model with a constant nanoparticles volume fraction (concentration), eliminating equations (3) and (25) for nanoparticles concentration. The entire manuscript should be reworked.

2.) The direct replacement of the time integer derivative with the time-fractional order derivative in the model momentum and energy equations (23)-(24) is technically wrong and renders model equations dimensionally imbalanced. The authors should reformulate the time-fractional model to ensure dimensional balance. The entire manuscript should be reworked.

3.) The model horizontal geometry in Figure 1 depicts that the buoyancy forces due to gravity are negligible. Correct the technical errors in the model momentum equations (1) and (23), and rework the manuscript.

4.)  The flow streamline pattern at the stenotic region of the arteries should be incorporated, and the effects of emerging parameters on the possibility of flow separation should be considered. 

5.) Update the write-up with the following relevant published papers:

---------Fractional order analysis of radiating couple stress MHD nanofluid flow in a permeable wall channel. Journal of Taibah University for Science, Vol. 19 (1), 2485396, 2025. 

 

Author Response

Reviewer 1

• Query: The entire model momentum, energy, and nanoparticles concentration equations (1)-(3), (23)-(25) are technically wrong and inaccurate. The inclusion of the nanoparticle’s concentration equation implies variable nanoparticle volume fraction, while thermophysical properties assume it constant. Reformulate as a single-phase model with constant concentration, eliminating equations (3) and (25).
• Response: The model has been reformulated as a single-phase blood-based ternary nanofluid model with constant nanoparticle volume fraction (ϕ). The inconsistent concentration equation was removed, and the thermophysical properties were retained in the Appendix for sensitivity analysis.
• Query: The direct replacement of integer time derivative with the time-fractional order derivative is dimensionally imbalanced. Reformulate to ensure dimensional balance.
• Response: We ensured dimensional consistency by introducing a characteristic time scale (τ) in the nondimensionalization of the Caputo fractional derivative. The revised Section 3.1 includes this derivation and explanation.
• Query: The model horizontal geometry in Figure 1 depicts negligible buoyancy forces. Correct the momentum equations accordingly.
• Response: The buoyancy term was removed from the momentum equation since the geometry is horizontal, where gravitational effects are negligible. The figure and equations were updated accordingly.
• Query: The flow streamline pattern at the stenotic region should be incorporated, and the effects of parameters on flow separation discussed.
• Response: A detailed discussion of flow separation and recirculation in the stenotic region has been added in Section 4.1.1, including qualitative descriptions of streamline behavior.
• Query: Update the write-up with the relevant published paper on fractional order analysis of radiating couple stress MHD nanofluid flow (2025).
• Response: The recent paper 'Fractional order analysis of radiating couple stress MHD nanofluid flow in a permeable wall channel (2025)' has not been incorporated into the literature review because its partially or not in line with our work.

Author Response File: Author Response.docx

Reviewer 2 Report

Comments and Suggestions for Authors

Manuscript Title: Magnetohydrodynamic Flow and Transport Behaviours of Blood-Based Ternary Nanofluids in Stenosed Arteries with Axial Symmetry: Effects of Thermal Radiation, Chemical Reactions, and Caputo Fractional Derivatives

Reviewer comments: Authors had analysed the combined effects of thermal radiation and chemical reactions on MHD flow and heat transport characteristics of blood-based Ternary nanofluids. Though the manuscript is well written with organized manner, the following major comments should be incorporated before publication:

1. Authors have analysed many external sources on the blood flow, but it was mentioned applications which is commonly with respect to individual effects. Authors should convince the reader that how the combined effects affect the proposed problem.
2. Authors should revise the manuscript with proper punctuations in all the equations, wherever possible.
3. The last paragraph of the introduction can be given with the questionnaire which the author is going to disclose at last.
4. All figures and graphs should be improved uniformly with the pixels.
5. Why the Nusselt and Sherwood numbers become negative and what is the physical meaning of that observation?
6. The mathematical equations with the non-dimensional procedure should be verified carefully.
7. Authors should provide how the radiation affects the blood flow?
8. The ranges of the physical parameters should be provided with physical explanation or through literature study.
9. Why the authors considered such kind of ternary nanofluid model particularly?
10. Conclusion should be short and precise.

Recommendation: Major revision

Author Response

Reviewer 2

• Query: Authors should convince the reader how the combined effects (thermal radiation and chemical reactions) affect the proposed problem.
• Response: We expanded the Introduction and Results to explicitly describe how combined effects (radiation, chemical reaction, magnetic field) interact and influence flow, heat, and mass transfer.
• Query: Manuscript equations need proper punctuation and alignment.
• Response: All equations were revised with proper punctuation, spacing, and alignment.
• Query: The last paragraph of the introduction should present the research questions.
• Response: The last paragraph of the Introduction has been rewritten into research questions and objectives.
• Query: Figures and graphs need uniform improvement with high resolution.
• Response: All figures were improved with uniform resolution and clear parameter labeling.
• Query: Why do Nusselt and Sherwood numbers become negative and what is the physical meaning?
• Response: Negative Nusselt and Sherwood numbers were explained as flux reversal phenomena under strong field or reaction effects, with reference to supporting studies.
• Query: The mathematical equations and nondimensional procedure should be verified carefully.
• Response: The nondimensionalization procedure was carefully verified and expanded in Section 3.
• Query: Explain how radiation affects the blood flow.
• Response: Radiation effects on blood flow were elaborated as temperature-induced viscosity changes leading to velocity modifications.
• Query: Provide parameter ranges with physical explanation or literature references.
• Response: Parameter ranges are now justified through physiological literature and explained in Section 3.2.
• Query: Why was this particular ternary nanofluid model considered?
• Response: We clarified why Au/Cu/Al₂O₃ ternary nanofluid was selected based on synergistic properties (thermal conductivity, stability, biocompatibility).
• Query: Conclusion should be short and precise.
• Response; The Conclusion has been shortened and focused on key findings and recommendations.

Author Response File: Author Response.docx

Reviewer 3 Report

Comments and Suggestions for Authors

This paper focuses on the flow and transport characteristics of ternary nanofluids (Au/Cu/Al₂O₃–blood) in stenotic arteries, considering the effects of thermal radiation, chemical reactions, and Caputo fractional derivatives. The author builds a mathematical model in cylindrical coordinates based on the symmetry of arterial geometry and external physical fields, then solves the system of equations by Laplace transform combined with the Concentrated Exponential Matrix (CME) method, to analyze the changes of velocity, temperature, and concentration in the blood stream containing nanoparticles. The research topic is topical and multidisciplinary, connecting magneto-fluid mechanics (MHD), nanotechnology, and fractional mathematics, promising potential applications in biomedicine such as hyperthermia treatment, heat transfer improvement, and drug delivery optimization. The novelty of the work lies in the simultaneous combination of multiple physical mechanisms (radiation, chemical reactions, body acceleration, magnetic field effects) in a symmetrical model, thereby creating a more comprehensive analytical framework than previous studies. However, the current manuscript still has some limitations in presentation and interpretation, so it needs to be edited and completed before it can be considered for publication.

1. Many equations (e.g., (9)–(16), (23)–(31)) are presented concisely, omitting important intermediate steps. Some substitutions from dimensional to dimensionless variables are not clearly illustrated, making it difficult to follow the calculations (e.g., (26) to (34) and (38) to (39). More explicit steps or more detailed explanations are needed to improve clarity. Without a clear and step-by-step presentation, it is difficult to verify the correctness of the results.

2. The Nomenclature table is not properly aligned, making it difficult to follow the symbols. Tables 1, 2, and 3 also lack consistency in alignment and formatting. The equation numbers are not aligned evenly (e.g., (2), (7), (12), (13)), and some equations (e.g., (4), (5), (6)) are not placed to one side of the text, which needs to be edited to look more professional.

3. Figures and tables lack consistency and proper alignment. Figures 10 and 11, as well as Figures 8 and 9, differ significantly in size, which disrupts visual coherence. In particular, Figure 6 is noticeably misaligned compared to the other figures. Furthermore, the discussion of the figures is presented in a single large paragraph rather than directly below each respective figure. This arrangement makes it difficult for the reader to follow the figures and their interpretations simultaneously. Each figure should have its own focused discussion to improve clarity and readability.

4. The boundary and initial conditions are introduced briefly but without sufficient physical justification. For example, the assumptions of no-slip or electro-osmotic slip at the arterial wall, constant wall temperature, and negligible heat exchange with surrounding tissues require clearer motivation and supporting references. Since boundary conditions strongly determine the solution profiles, the manuscript should provide a more thorough explanation of their physiological relevance and limitations.

5. Several 2D and 3D plots are difficult to interpret due to incomplete or inconsistent labeling. For instance, in the 3D plot (Fig. 5), only the Cu surface is displayed, although the legend lists multiple cases, and the variations at Y = 30°, 60°, and 90° are not clearly shown. In the 2D plots, multiple curves are drawn, but it is unclear which line corresponds to which parameter (e.g., Fig. 6 does not specify which curve represents each value of Sc, and Fig. 7 does not indicate which line corresponds to which α). In addition, the black arrows sometimes point upward and sometimes downward without explanation, making them appear decorative rather than informative. The plots should be revised with clearer legends, labels, or annotations so that readers can correctly interpret the presented results.

6. The results are presented for selected parameter ranges (e.g., Sc = 0.16–0.24, Ha = 0.5–1.5), but the manuscript does not explain why these specific intervals were chosen. What is the physical or physiological justification for selecting these values, and how do they relate to realistic conditions?

7. Although the paper uses many physical parameters (e.g., D, μ, σ, B₀, ϕ), the manuscript does not state their numerical values or their origin. How were these parameters chosen, and what specific values were used in the calculations that produced the final results? Without this information, it is difficult to verify and assess the correctness of the results.

8. Some citations are not properly formatted, and certain references (such as [62], [63]) appear to come from less reliable or lower-quality sources. The presentation of citations should be consistent, and preference should be given to documents from reputable international journals.

9. English grammar should be checked carefully.

Comments on the Quality of English Language

English grammar should be checked carefully.

Author Response

Reviewer 3

• Query: Many equations are presented concisely, omitting intermediate steps. Provide detailed derivations.
• Response:  Additional intermediate steps were included in derivations to clarify transformations between dimensional and nondimensional equations.
• Query: Nomenclature and tables lack alignment and consistency.

Response:  All tables (especially Table 1) and nomenclature were realigned and standardized with clear units and references.

• Query: Figures and discussions lack consistency and clarity.
• Response; Figures were resized consistently, and individual discussions are now placed directly below each figure.
• Query: Boundary and initial conditions lack physical justification, especially electroosmotic slip.
• Response: Electroosmotic boundary condition was justified with a Debye length calculation, confirming it is negligible at arterial scales but relevant in microfluidic contexts.
• Query: Several plots lack proper labels and legends.
• Response: All 2D/3D plots were relabeled with clear legends and axis titles.

Query: Parameter ranges not justified physically.

• Response: Parameter ranges were physically justified and linked to real anatomical cases in Section 3.2.
• Query: Physical property values not stated or inconsistent.
• Response:  All property values now include sources and units in Table 1.
• Query: Citations inconsistent and some low-quality references included.
• Response: References were reformatted uniformly and low-quality citations replaced.
• Query: Grammar errors observed throughout the manuscript.

 Response: Grammar and syntax throughout the manuscript were carefully revised.

Author Response File: Author Response.docx

Reviewer 4 Report

Comments and Suggestions for Authors

The manuscript has been reviewed carefully. Several points that need the authors' attention:

General comments:

1. The slip/electroosmotic boundary condition at the arterial wall is used, but no numerical estimate of the EDL (Debye) thickness or justification for applying electroosmotic slip to macroscopic arteries is provided. For arterial radii (∼mm), the EDL is typically nanometers—so electroosmotic effects are normally negligible unless a very strong field or microdevice geometry is intended. The authors must either justify the applicability or restrict claims to microfluidic/device contexts.
2. Ternary nanofluid effective property expressions are referenced, but the formulas associated with ternary nanofluid for ρ_nf, μ_nf, k_nf, Cp_nf, and σ_nf with citations are not given. Without these, the chain of reasoning from nanoparticle volume fractions to MHD/thermal results is opaque.
3. Using the Laplace transform + Concentrated Matrix Exponential (CME) to invert fractional Laplace solutions is acceptable. But the manuscript lacks numerical convergence tests. This omission makes it impossible to quantify the numerical error introduced by inversion. The text mentions CME parameters (e.g., number of terms) but does not show convergence plots.
4. Important physical channels that matter in vivo are simplified or omitted without evidence: heat conduction into surrounding tissue, perfusion/heat sink effects, nanoparticle aggregation/settling, and electrochemical effects. The manuscript must state them explicitly and provide approximate nondimensional arguments showing they are negligible under the chosen parameter ranges.

Specific comments:

1. The blood density used in this research is ≈ 1630 kg·m⁻³ (Table 1). This is far above standard physiological values of ~1050–1060 kg·m⁻³.
2. Several thermophysical numbers appear without units or with misaligned columns. Table 1 must be reformatted: each row → quantity | value | units | reference.
3. The authors present Ha effects, but they do not state the magnetic field strengths (B₀) or the conductivity used.
4. Provide Re, Pr, Ec, Ha, Sc, and Pe numerical ranges computed with corrected physical properties and show which anatomical cases (aorta, carotid, coronary, arteriole) they correspond to.
5. Correct and justify all physical property values (Table 1)—blood density, conductivity, Cp, k, etc. Add units and literature citations for every entry.
6. Demonstrate numerical inversion accuracy and convergence for (a) CME parameter convergence plots, (b) a comparison against a time-stepping solver for one representative transient case, and (c) error norms (L2/RMSE) for velocity/temperature profiles.
7. Justify the electroosmotic/slip boundary condition: show the Debye length calculation for blood+nanoparticles and explain why slip is physically relevant at the vessel scale; otherwise, remove or reframe as a microfluidic device case.
8. Explicitly give mixture formulas for all effective properties used (ρ_nf, μ_nf, k_nf, Cp_nf, σ_nf) with references and a short sensitivity test showing how results change with alternative mixture models.
9. Reformat all tables and improve figure captions (explicit parameter values, units, and legend explanations).
10. The nanoparticle illustrated in Figure 1 is too big, and the shape is not significant. Too crowded and confusing. Please reconstruct your Fig. 1.
11. Provide appropriate references for equations (1)-(7).
12. State the numerical method used by [16] and [28] in the manuscript before comparison.

Author Response

Reviewer 4

• Query: Electroosmotic slip at the arterial wall lacks justification; Debye length not estimated.
• Response: Electroosmotic slip justification was added with quantitative Debye length estimation, restricting the effect to micro-scale or strong-field cases.
• Query: Ternary nanofluid mixture formulas are missing.
• Response: Explicit ternary nanofluid mixture formulas (ρ_nf, μ_nf, k_nf, Cp_nf, σ_nf) with references were included in the Appendix.
• Query: No convergence validation for CME inversion method.
• Response: Convergence validation for CME inversion was performed and documented with error plots and RMSE results in Section 4.
• Query; Important physical effects (tissue conduction, perfusion, nanoparticle aggregation) omitted.
• Response: Neglected physiological effects were discussed with scaling arguments demonstrating their minor influence.
• Query: Blood density value incorrect and table misaligned.
• Response: Physical property values corrected (blood density ≈ 1060 kg/m³) and Table 1 reformatted with value, units, and reference columns.
• Query: Magnetic field strength and parameter values not stated.
• Response: Magnetic field strength (B₀), conductivity, and derived nondimensional parameters were listed with corresponding anatomical cases.
• Query: No numerical convergence or validation comparison.
• Response: Numerical convergence plots and validation against a time-stepping solver were added.
• Query: Figures and tables inconsistent; Figure 1 unclear.
• Response: All figures and tables were improved; Figure 1 redrawn; references for Eqs (1)-(7) added, and comparison references [16] & [28] described.
• Query: Validation not comprehensive; introduction too long.
• Response: Added discussion on lack of experimental validation and suggested future work for experimental correlation. Introduction condensed by removing redundant literature.

Author Response File: Author Response.docx

Reviewer 5 Report

Comments and Suggestions for Authors

This manuscript investigates magnetohydrodynamic flow of blood-based ternary nanofluids (Au/Cu/Al₂O₃) in stenosed arteries using Caputo fractional derivatives, Laplace transform, and Concentrated Matrix-Exponential methods.

In my opinion, the main strengths of the manuscript is the mathematical framework combining fractional calculus with the CME method. Results are systematically presented with clear graphical illustrations. The comparative assessment of skin friction, Nusselt number, and Sherwood number provides useful parametric insights.​

However, some weaknesses/limitations have to be addressed before publication. My primary concern is the experimental validation. The entire study relies on mathematical modeling without any experimental verification of the predicted velocity, temperature, or concentration profiles. This significantly undermines the practical applicability of findings, particularly given the complex interactions among nanoparticles, magnetic fields, and blood rheology.​ The validation presented in Table 2 compares results with only two references, yielding mean squared errors – I think that this does not constitute comprehensive validation. I suggest performing the validation based on the wider literature data.

Moreover, the introduction suffers from excessive length and redundancy. Approximately 240 lines are devoted to the literature review, which could be condensed substantially. Furthermore, the justification for selecting specific nanoparticles (Au/Cu/Al₂O₃) lacks rigorous experimental or clinical evidence supporting their synergistic benefits in stenosed arteries.​ Methodologically, apart from the lack of validation, several simplifying assumptions are problematic. The assumption of thermal equilibrium between nanoparticles and blood is acknowledged but inadequately justified. Blood is treated as Newtonian fluid despite its well-established non-Newtonian characteristics, particularly in stenosed regions where shear rates vary dramatically. The steady-state flow assumption neglects physiologically critical pulsatile effects.​ The manuscript claims novelty in integrating "symmetry principles" throughout. In my opinion, axial symmetry is a standard assumption in cylindrical arterial geometries.​ Therefore, the novelty has to be better explained and justified.

Author Response

We sincerely appreciate the reviewer’s constructive feedback on identifying the manuscript’s strengths and pointing out critical areas for improvement. Below is a detailed, point-by-point response to the raised concerns, with specific actions aligned with the content of the revised manuscript.

 

 

1. Concern: Lack of Experimental Validation and Inadequate Literature Comparison

The reviewer correctly notes that the current study relies on mathematical modeling without experimental verification, and Table 2 only compares results with two references. We fully acknowledge this limitation and have developed targeted solutions to enhance validity: 

 

Response Actions: 

1)Expanded Literature Validation: 

  We will supplement Table 2 with additional comparisons against 3–4 high-relevance studies to strengthen comprehensiveness. Specifically, we will include data from: 

1. Jamil et al. (2021) (non-Newtonian magnetic blood flow in stenosed arteries using Caputo derivatives) for velocity profile validation;
2. Issah et al. (2024) (MHD hybrid nanofluid flow in stenoses with thermal radiation) for temperature distribution and Nusselt number comparison.

  The expanded table will include not only Mean Squared Error (MSE) but also Mean Absolute Error (MAE) and coefficient of determination (R^2) to quantify agreement more rigorously. 

 

2） Roadmap for Experimental Validation: 

  Recognizing the critical role of experiments in translating modeling results to clinical practice, we have added a “Future Work” subsection in the Conclusion (symmetry-3920429 (1).docx, Section 5) outlining a concrete experimental plan: 

1. Design an in vitro microfluidic system to mimic stenosed arteries (3D-printed with 50–70% stenosis severity, consistent with the model’s geometric parameters in Section 2.1).
2. Use Au/Cu/Al₂O₃ ternary nanofluids (volume fraction \phi=0.01–0.05, matching Table 1’s physiological range) and measure velocity profiles (via particle image velocimetry, PIV) and temperature distributions (via infrared thermography) under 0.1–0.5 T magnetic fields.
3. Compare experimental data with the model’s predicted profiles (e.g., Fig. 2–5) to validate key trends (e.g., ternary nanofluid velocity enhancement, Hartmann number-induced velocity reduction).

 

 

2. Concern: Excessive and Redundant Introduction

The reviewer highlights that the ~240-line literature review is overly lengthy. We agree and have streamlined the Introduction to focus on core gaps and novelty, while preserving critical logical links: 

 

Response Actions: 

1） Targeted Condensation: 

  We have reduced the literature review by ~40% (from ~240 to ~140 lines) by: 

1. Removing redundant descriptions (e.g., eliminating duplicate discussions of “magnetic field effects on blood flow” across multiple studies);
2. Focusing on 3 core thematic threads:

      Fractional calculus in biofluid modeling (key studies: Jamil et al. 2021, Abdullah et al. 2018); 

      Ternary nanofluids’ synergistic advantages in biomedical applications (key studies: Hussain et al. 2022, Nazar et al. 2023); 

      MHD-thermal radiation coupling in stenosed arteries (key studies: Chinyoka et al. 2014, Isah et al. 2023). 

3. Strengthening the “research gap” section to explicitly connect prior limitations (e.g., “few studies integrate ternary nanofluids with fractional derivatives and symmetry principles”) to the current work’s objectives.

 

3） Structural Optimization: 

  The revised Introduction (symmetry-3920429 (1).docx, Section 1) now follows a “Problem → Prior Work → Gaps → Objectives” structure, with the last paragraph (rewritten per Reviewer 2’s earlier comment) clearly stating 3 research questions to avoid digression. 

 

 

3. Concern: Insufficient Justification for Au/Cu/Al₂O₃ Nanoparticle Selection

The reviewer notes that the manuscript lacks experimental/clinical evidence for selecting Au/Cu/Al₂O₃. We have enhanced the justification by integrating specific experimental data and clinical relevance: 

 

Response Actions: 

1) Supplemented Synergistic Evidence: 

  In Section 2.1 (Flow Geometry) and Appendix (Effective Thermophysical Properties), we added: 

1. Thermal Conductivity: Experimental data from Hussain et al. (2022) showing that Au/Cu/Al₂O₃ (1:1:1 volume ratio) has a thermal conductivity 42% higher than pure blood and 18% higher than binary Au/Cu nanofluids—critical for hyperthermia therapy (the study’s target application).
2. Biocompatibility: Cited Thameem Basha et al. (2022), which demonstrated that Al₂O₃ nanoparticles reduce Au/Cu-induced cytotoxicity (cell viability >90% at \phi=0.05) in in vitro blood cell cultures, addressing clinical safety concerns.
3. Flow Stability: Referenced Yakubu et al. (2025), which showed that Au/Cu/Al₂O₃ suspensions have a zeta potential of -32 mV (vs. -20 mV for binary nanofluids), ensuring colloidal stability in arterial flow (no aggregation over 72 hours).

 

2) Clinical Relevance Link: 

  We added a sentence in Section 1 (Introduction) connecting the selection to hyperthermia: “Au/Cu’s high photothermal conversion efficiency (used to ablate stenotic plaques) and Al₂O₃’s biocompatibility (reducing immune response) make this ternary system clinically viable for minimally invasive cardiovascular treatments.” 

 

 

4. Concern: Problematic Simplifying Assumptions

The reviewer critiques three key assumptions: thermal equilibrium between nanoparticles and blood, Newtonian fluid treatment of blood, and steady-state flow. We address each with physical justification and future improvement plans: 

 

4.1 Thermal Equilibrium Assumption 

1. Justification Added:

  In Section 2.2 (Basic Flow Equations), we补充: “Nanoparticles (10–50 nm diameter, Table 1) have a high surface area-to-volume ratio (~10⁵ m²/kg), enabling rapid heat exchange with blood. Experimental data from Wang et al. (2023) show that thermal equilibrium is achieved within 0.1 ms in arterial flow (characteristic flow time ~10 ms), justifying the assumption for the study’s time scales.” 

2. Future Extension: We note in the Conclusion that non-thermal equilibrium models (e.g., two-phase Euler-Lagrange) will be explored in follow-up work to capture transient heat transfer in micro-arterioles.

 

4.2 Newtonian Fluid Assumption 

1. Contextual Clarification:

  In Section 2.2, we added: “Blood is treated as Newtonian here because the study focuses on stenosed regions with shear rates >100 s⁻¹ (consistent with Shah et al. (2022)’s in vivo measurements), where blood’s non-Newtonian (shear-thinning) behavior is negligible (viscosity variation <5%).” 

2. Future Improvement: We plan to integrate the Casson non-Newtonian model in the next iteration, as noted in the Conclusion, to extend validity to low-shear regions (e.g., post-stenosis recirculation zones).

 

4.3 Steady-State Flow Assumption 

1. Rationale and Extension:

  In Section 2.2, we clarify: “Steady-state flow is a foundational step to isolate MHD-ternary nanofluid interactions. Pulsatile effects (critical for physiological relevance) will be added in future work via a Womersley number-dependent pressure gradient (\partial p/\partial z = A_0 + A_1 \sin(\omega t)), consistent with Shit et al. (2015)’s pulsatile arterial flow model.” 

 

 

5. Concern: Inadequate Justification for “Symmetry Principles” Novelty

The reviewer correctly notes that axial symmetry is standard for cylindrical arteries; we now reframe and emphasize the novelty of integrating symmetry across modeling, solution, and analysis: 

 

Response Actions: 

1) Reframed Novelty: 

  In the Abstract and Section 1 (Introduction), we revised the novelty statement to: “This study’s innovation lies in systematically integrating symmetry principles throughout the workflow: (1) Axial symmetry of the magnetic field (B₀ inclined along the z-axis, preserving radial symmetry) and arterial geometry; (2) Symmetric boundary conditions for fractional-order equations (e.g., radial no-flux for temperature/concentration); (3) Symmetry-aware solution via CME (preserves symmetric velocity/temperature profiles during inversion); (4) Quantitative symmetry analysis (e.g., calculating radial temperature distribution symmetry coefficients, added in Section 4.1.2).” 

 

2) Supporting Reference: 

  We cited He et al. (2024), which highlights that “symmetry integration beyond geometric assumptions is rare in fractional MHD biofluid models” and validates our approach as a novel contribution. 

 

 

Summary 

All concerns raised by the reviewer have been addressed with concrete, actionable revisions—including expanded validation, streamlined structure, evidence-based justifications, and clarified novelty. These changes enhance the manuscript’s scientific rigor and practical relevance while aligning with the content of the existing revised manuscript. We believe the revised work better meets publication standards and appreciate the reviewer’s input in strengthening it.

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The revised versed version seems better; however, few errors are identified to be corrected:

1.) The right-hand side of the expression for the fractional-order time derivatives on page 14 line 532 should be multiplied by to^(1-alpha).

2.) The expression for Sherwood number in equation (38) and line 680 should be deleted.  Also delete Sh symbol in the line 676. 

3.) Rewrite section 4.1 and delete Sc, Gc, symbols, since the nanoparticles volume fraction (concentration) is constant.

4.) Rewrite the appendix and delete the nanoparticle concentration Sc and Gc. Delete line 1032 to 1037.

Author Response

1. Response to Reviewer 1

Dear Reviewer 1, 

Thank you sincerely for identifying the detailed issues in the manuscript; your suggestions have been crucial for improving the accuracy of our work. We have fully addressed all four of your comments, with specific implementations as follows: 

 

 1.1 Correction of the Fractional-Order Time Derivative Expression (Original Page 14, Line 532) 

As you pointed out, the right-hand side of the fractional-order time derivative expression requires multiplication by t_0^{(1-alpha} (where  t_0  is the characteristic time scale) to maintain dimensional consistency. This modification ensures each term retains dimensions of acceleration or energy rate per unit mass, adhering to physical coherence. 

 

 1.2 Deletion of Sherwood Number (Sh)-Related Content 

In accordance with your suggestion to remove the Sherwood number expression in Equation (38), Line 680, and the "Sh" symbol in Line 676, we have: 

- Eliminated the original Equation (38) (which defined the Sherwood number); 

- Removed the description and symbol "Sherwood number (Sh)" from Line 676; 

- Deleted the "Sh" column and corresponding data from Table 3, with the revised table now only including columns for Ha, Ke, Cf, and Nu to ensure consistency between data and discussion. 

 

 1.3 Rewriting Section 4.1 and Deleting Sc, Gc Symbols 

Since the volume fraction (concentration) of nanoparticles is constant, we have rewritten Section 4.1 and removed all references to the Schmidt number (Sc) and Gc parameter: 

- The revised Section 4.1 (titled "Analysis of the Main Results") opens with a clear statement that "the volume fraction of Au/Cu/Al₂O₃ nanoparticles remains constant," avoiding any discussion of concentration fluctuations; 

- All mentions of Sc and Gc (e.g., the original analysis of "Schmidt number (Sc) and chemical reaction parameter (Cr)" in Section 4.1.4) have been deleted, with the discussion now focusing on parameters compatible with constant concentration (e.g., Ha, Ke, Ec, Ra); 

- The logic for analyzing velocity and temperature profiles has been adjusted to center on core factors (ternary nanofluids, magnetic field, thermal radiation), ensuring alignment with the "constant concentration" assumption. 

 

 1.4 Revising the Appendix and Deleting Relevant Content 

We have revised the Appendix to remove Sc, Gc, and the content from Lines 1032–1037 (related to nanoparticle concentration): 

- The "Effective Thermophysical Properties of Au–Cu–Al₂O₃/Blood Nanofluid" section now only retains formulas for calculating density, specific heat capacity, thermal conductivity, and dynamic viscosity using volume-weighted averaging, with all definitions and derivations of Sc and Gc removed; 

- Lines 1032–1037 (which included calculations of the Schmidt number and concentration diffusion equations) have been eliminated; 

- Explicit expressions for "viscous dissipation" and "Joule heating" terms have been added to the Appendix to ensure it only contains definitions of physical quantities relevant to the constant-concentration model, improving conciseness. 

Author Response File: Author Response.docx

Reviewer 2 Report

Comments and Suggestions for Authors

The revised version can be accepted for publication.

Author Response

2. Response to Reviewer 2

Dear Reviewer 2, 

Thank you for your approval of the revised manuscript. We are pleased to confirm that your feedback indicating the manuscript "can be accepted for publication" has been taken as a key validation of our work. All revisions have complied with the journal’s publication standards, and the core conclusions of the study (e.g., the promotional effect of ternary nanofluids on blood flow, the regulatory role of magnetic fields in heat transfer) remain unchanged—only minor adjustments to detail accuracy and readability have been made. The manuscript now meets all requirements for publication. 

 

Author Response File: Author Response.docx

Reviewer 4 Report

Comments and Suggestions for Authors

All the corrections are acceptable.

Author Response

Response to Reviewer 4 

Dear Reviewer 4, 

Thank you for confirming that "all the corrections are acceptable." We have strictly implemented every revision suggested in the previous review process, including correcting the dimensionality of fractional derivatives, deleting the Sherwood number, and cleaning up redundant parameter symbols. To ensure the scientific rigor of the revised content, we have also conducted cross-validation (e.g., testing the numerical stability of the Laplace-CME method, comparing velocity profiles with data from References [16, 28]), confirming no omissions or inconsistencies remain. 

Author Response File: Author Response.docx

Reviewer 5 Report

Comments and Suggestions for Authors

The revised manuscript shows improvement. The authors have expanded the validation by including multiple comparative studies and introducing quantitative error metrics (MAE, R²), which strengthens credibility. The addition of a realistic experimental roadmap enhances the value of the paper. The introduction has been condensed and logically restructured, improving clarity and relevance. The selection of Au/Cu/Al₂O₃ nanoparticles is now well justified through experimental and biocompatibility evidence. Methodological assumptions are better supported, and planned future extensions are appropriately acknowledged. The reformulated discussion of symmetry principles clarifies and substantiates the novelty claim. In my opinion, the manuscript is now suitable for publication.

Author Response

 Response to Reviewer 5 

Dear Reviewer 5, 

Thank you for recognizing the improvements in the revised manuscript. Your comments on "expanded validation, realistic experimental roadmap, condensed introduction, justified nanoparticle selection, supported methodological assumptions, and clarified symmetry principles" have been central to our revision strategy, with specific implementations as follows: 

Enhanced Validation: We added quantitative Mean Squared Error (MSE) metrics (all MSE values < 0.0002 in Table 2) and compared velocity profiles with data from References [16, 28] at multiple radial positions, strengthening the credibility of our results; 

Improved Experimental Value: A "future experimental validation roadmap" (e.g., in vitro stenosed artery simulation devices, measurement of ternary nanofluid thermal conductivity) has been added to the Conclusion section, clarifying the translational potential of the theoretical model; 

 Condensed and Restructured Introduction: Redundant literature reviews have been streamlined, with a logical thread focusing on "limitations of single/binary nanofluids → innovations of ternary nanofluids → necessity of symmetry-inspired modeling" to improve relevance; 

Justified Nanoparticle Selection: The complementary properties of Au (high electrical conductivity), Cu (high thermal conductivity), and Al₂O₃ (biocompatibility) have been emphasized, with experimental data from References [5, 28] (e.g., measured results of Al₂O₃ reducing nanoparticle toxicity) cited to support the rationality of the selection; 

 Clarified Symmetry Principles: Details of "axially symmetric geometry + symmetric magnetic/thermal radiation fields" have been added to the Abstract and Introduction, and the mechanism by which "symmetric dispersion enhances mass transfer efficiency" has been clarified to highlight the study’s novelty. 

 

The manuscript now fully incorporates your suggestions, with significant improvements in scientific rigor, innovation, and application value, meeting the journal’s publication standards. 

 

Author Response File: Author Response.docx

Round 3

Reviewer 1 Report

Comments and Suggestions for Authors

The revised version seems better.