Review Reports - Development and Optimization of Cattaneo–Christov Carreau–Yasuda Tri-Hybrid Nanofluid Using Artificial Neural Networks

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In this paper, the authors develop a computational framework combining ANN-LM to study CCHMF-THNF model. However, the novelty and physical insights of the work remain limited, and many aspects of the methodology and validation require further clarification.

Overall, I recommend major revision.

Comment #1: The authors should clarify the novelty of coupling ANN and LM framework with the CCHMF-THNF model by emphasizing the specific physical insights of computational advantages, other than simple curve-fitting based on the existing tools (e.g. ANN and LM).

Comment #2: The mesh independence study should be analyzed for the numerical solver settings as the reference data is generated using MATLAB. The authors should ensure the labels used for ANN training are accurate enough. Besides, there are no experimental results to verify the equations and numerical results, which limit the confidence in the physical model.

Comment #3: There are numerous grammatical errors and unclear sentences in the manuscript. Readability should be improved before publication. (e.g., Page 9: repeation; line 193: cohearant…). Please check the whole manuscript carefully.

Comment #4: The description for figures and data is not clear. For example, Figures 4 shows the We are 0.6, 1.8, 2.5 and Ha are 0.6, 1.3, 1.7, how about other parameters? Are they fixed? What are the initial conditions?

Author Response

Reviewer 1: Comment 1

The authors should clarify the novelty of coupling ANN and LM framework with the CCHMF-THNF model by emphasizing the specific physical insights of computational advantages, other than simple curve-fitting based on the existing tools (e.g. ANN and LM).

Author Response: The authors thank the reviewer for this insightful comment. We agree that the novelty should be clearly distinguished from the standalone use of ANN or the Levenberg–Marquardt (LM) algorithm as a curve-fitting tool.

In the revised manuscript, we have clarified that the contribution of this work lies in integrating the ANN–LM framework with the nonlinear Cattaneo–Christov Carreau–Yasuda tri-hybrid nanofluid (CCHMF–THNF) model to improve computational efficiency and enable rapid evaluation of complex transport behavior.

Reviewer 1: Comment 2

The mesh independence study should be analyzed for the numerical solver settings as the reference data is generated using MATLAB. The authors should ensure the labels used for ANN training are accurate enough. Besides, there are no experimental results to verify the equations and numerical results, which limit the confidence in the physical model.

Author Response: We appreciate the reviewer’s concern regarding numerical reliability and lack of experimental validation. In response, we have added a mesh independence and tolerance sensitivity analysis for the bvp4c solver to ensure the accuracy of the generated dataset. Furthermore, we clarified that the ANN is trained on numerically converged solutions rather than raw approximations.

Due to the complexity of the governing non-Newtonian tri-hybrid nanofluid model, experimental validation is currently unavailable in literature. However, the bvp4c results for this model was presented in literature and we have compared them in Figure 4 (the reference data is basically bvp4c results). These additions improve confidence in the model predictions.

Reviewer 1: Comment 3

There are numerous grammatical errors and unclear sentences in the manuscript. Readability should be improved before publication. (e.g., Page 9: repeation; line 193: cohearant…). Please check the whole manuscript carefully.

Author Response: The authors thank the reviewer for highlighting this issue. The manuscript has been thoroughly revised to correct grammatical errors, improve sentence structure, and enhance overall readability. All typographical errors such as “repeation” and “cohearant” have been corrected.

Reviewer 1: Comment 4

The description for figures and data is not clear. For example, Figures 4 shows the We are 0.6, 1.8, 2.5 and Ha are 0.6, 1.3, 1.7, how about other parameters? Are they fixed? What are the initial conditions?

Author Response: The authors thank the reviewer for this valuable observation. In the revised manuscript, the description of all figures has been clarified to improve readability and understanding. Specifically, it has now been explicitly stated that during each parametric study, only the parameter under consideration (e.g., Weissenberg number or Hartmann number) is varied, while all other governing parameters are kept fixed at their baseline values.

Furthermore, the initial and boundary conditions used in the analysis have been clearly indicated in the manuscript and are defined through Equations (7)–(9). These additions ensure that the computational setup and parameter dependencies are fully transparent to the reader.

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have used an artificial neural network (ANN) prediction model based on the Levenberg–Marquardt (LM) algorithm to predict the nonlinear heat and mass transfer characteristics of (CCHMF–THNF) Cattaneo Christov Carreau Yasuda Tri-hybrid Nanofluid flow over a porous stretching sheet.
The followings are my concerns:

Check the definition of eta between lines 229-230 for typo.
The dimensionless forms appaered in the definitions of T and C is incorrect. Also they are duplications of those a few lines above them.
The presentation of Algorithm 1 is uncommon and incomplete.
What exact value of eta was used to represent infinity?
The expression of p in the similarity variable is missinng. I presume p=1/3 as clearly defined in ref. 20.
Include a complete list of "Author Contributions"

Author Response

Comment 1:
Check the definition of eta between lines 229-230 for typo.
Author response: The authors thank the reviewer for carefully identifying this issue. The typographical error in the definition of the similarity variable η has been corrected in the revised manuscript.

Comment 2:
The dimensionless forms appaered in the definitions of T and C is incorrect. Also they are duplications of those a few lines above them.
Author response: The authors thank the reviewer for carefully identifying this issue. The incorrect and duplicated expressions for the dimensionless temperature and concentration variables have been corrected in the revised manuscript. The redundant repetitions have also been removed to improve clarity and consistency.

Comment 3:
The presentation of Algorithm 1 is uncommon and incomplete.
Author response: The authors thank the reviewer for this valuable suggestion. Algorithm 1 has been completely revised in the updated manuscript to improve clarity, completeness, and standard scientific presentation. The revised version now clearly describes the sequence of dataset generation, ANN training, validation, and testing procedures.

Comment 4:
What exact value of eta was used to represent infinity?
Author response: The authors thank the reviewer for this important question. In the present work, the semi-infinite computational domain was truncated by taking: η_∞=5 which was found sufficient to satisfy the far-field asymptotic boundary conditions and ensure convergence of the numerical solution.

Comment 5:
The expression of p in the similarity variable is missing. I presume p=1/3 as clearly defined in ref. 20.
Author response: Thank you for your valuable comment. Authors have added the value of p in revised version.

Comment 6:
Include a complete list of "Author Contributions"
Author response: Thank you for your suggestion. The Author contribution section is added in revised manuscript.

Reviewer 3 Report

Comments and Suggestions for Authors

The novelty claim is not convincing; the ANN–LM is only wrapped around a standard Cattaneo–Christov Carreau–Yasuda MHD tri-nanofluid model without introducing any new transport mechanism, closure relation, or scaling framework.
The ANN is entirely trained on bvp4c-generated data and then “validated” against the same source, making the framework self-referential with no independent verification or experimental grounding.
The extremely low MSE (~10⁻¹⁰) reflects curve-fitting of smooth numerical outputs rather than true generalization; no noise injection, uncertainty bounds, or robustness tests are included.
The governing equations include multiple high-order nonlinear and coupling terms (especially in energy/concentration equations) without clear derivation, nondimensional justification, or physical interpretation.
The bvp4c solution is treated as exact data, yet no mesh convergence, tolerance sensitivity, or numerical error analysis is reported, which undermines the reliability of the training dataset.
The ANN structure (single hidden layer, 10 neurons, log-sigmoid) is overly basic; there is no benchmarking against deeper networks, PINNs, or alternative ML frameworks to justify the choice.
The parametric/sensitivity analysis simply reproduces known behaviors (e.g., We ↑ → velocity ↑, Ha ↑ → velocity ↓) without extracting new physical insight or regime transitions.
The results lack analytical depth; no scaling analysis, limiting-case validation, or asymptotic behavior is discussed to support the numerical trends.

Author Response

Reviewer 2: Comment 1

The novelty claim is not convincing; the ANN–LM is only wrapped around a standard Cattaneo–Christov Carreau–Yasuda MHD tri-nanofluid model without introducing any new transport mechanism, closure relation, or scaling framework.

Reviewer 2: Comment 2

The ANN is entirely trained on bvp4c-generated data and then “validated” against the same source, making the framework self-referential with no independent verification or experimental grounding.

Author Response: The authors thank the reviewer for this valuable observation. We agree that the ANN–LM framework in the present study is trained and evaluated using datasets generated from the numerical solver (bvp4c), and therefore it should not be interpreted as an independently experimentally validated predictive model.

The primary objective of this work is to investigate how accurately the ANN–LM framework can learn and reproduce the solutions generated by the nonlinear bvp4c solver for the CCHMF–THNF system. In this sense, the ANN model is intended as a computational approximation tool designed to emulate the numerical solution behavior with reduced computational effort.

To ensure reliability, the generated dataset was divided into independent training, validation, and testing subsets, and the prediction accuracy was assessed using multiple statistical performance metrics including MSE, RMSE, MAE, and regression analysis.

Furthermore, this limitation and the absence of experimental verification have now been explicitly acknowledged in the revised manuscript, and future work involving experimental datasets and independent validation has been suggested.

Reviewer 2: Comment 3

The extremely low MSE (~10⁻¹⁰) reflects curve-fitting of smooth numerical outputs rather than true generalization; no noise injection, uncertainty bounds, or robustness tests are included.

Author Response: The authors thank the reviewer for this insightful observation. We agree that the very low MSE values are primarily associated with the smooth and noise-free nature of the numerically generated bvp4c dataset. In the present work, the objective of the ANN–LM framework is not to establish experimental generalization capability, but rather to investigate how accurately the neural network can reproduce the nonlinear numerical solutions generated by bvp4c.

Accordingly, the ANN–LM model was trained using independent training, validation, and testing subsets to avoid direct memorization of the complete dataset. However, we acknowledge that the present framework does not include noise injection, uncertainty quantification, or robustness testing against experimental variability.

To address this concern, additional discussion regarding the limitations of the current dataset and future incorporation of noisy or experimental data has now been included in the revised manuscript.

Reviewer 2: Comment 4

The governing equations include multiple high-order nonlinear and coupling terms (especially in energy/concentration equations) without clear derivation, nondimensional justification, or physical interpretation.

Author Response: The authors thank the reviewer for this important observation. We would like to clarify that the governing mathematical model used in the present study was not newly developed by the authors. The nonlinear governing equations, similarity transformations, and nondimensional formulation employed in this work are adopted from the established model reported in Reference [20], which specifically addresses the Cattaneo–Christov Carreau–Yasuda tri-nanofluid system.

The primary focus of the current study is not the derivation of a new physical model, but rather the computational analysis and ANN–LM based prediction of the existing nonlinear system. To avoid ambiguity, the revised manuscript now explicitly states that the governing equations and associated transformations are taken from Reference [20].

Reviewer 2: Comment 5

The bvp4c solution is treated as exact data, yet no mesh convergence, tolerance sensitivity, or numerical error analysis is reported, which undermines the reliability of the training dataset.

Reviewer 2: Comment 6

The ANN structure (single hidden layer, 10 neurons, log-sigmoid) is overly basic; there is no benchmarking against deeper networks, PINNs, or alternative ML frameworks to justify the choice.

Author Response: The authors thank the reviewer for this important observation. While advanced architectures such as deep neural networks and PINNs are indeed powerful, the focus of this study is to rigorously assess the capability of a single hidden layer ANN–LM framework for modeling the nonlinear CCHMF–THNF system.

The chosen architecture (10 neurons, log-sigmoid activation) is supported by a comprehensive sensitivity analysis, highlighted (with Turquoise color) Tables 1–4 in the revised manuscript. These results confirm that increasing model complexity beyond this configuration does not yield significant improvement in accuracy but may introduce overfitting and increased computational cost. Therefore, the selection of a single hidden layer is both deliberate and justified. The exploration of deeper architectures and physics-informed models is a valuable extension and has been explicitly stated as a future research direction.

Reviewer 2: Comment 7

The parametric/sensitivity analysis simply reproduces known behaviors (e.g., We ↑ → velocity ↑, Ha ↑ → velocity ↓) without extracting new physical insight or regime transitions.

Author Response: The authors thank the reviewer for this constructive observation. We agree that isolated monotonic trends such as and are generally established behaviors in non-Newtonian MHD flows.

In the revised manuscript, additional discussion has been incorporated to emphasize the coupled interaction between viscoelastic and magnetic effects rather than only reporting independent parameter variations. Specifically, we now highlight the competing behavior between elastic enhancement and Lorentz-force damping, which influences momentum transport differently across parameter ranges.

Furthermore, the revised discussion clarifies that at higher values of the Hartmann number, magnetic damping becomes dominant over elastic acceleration, indicating a transition from elasticity-dominated flow behavior to magnetically controlled flow behavior. These additions provide improved physical interpretation of the numerical observations.

Reviewer 2: Comment 8

The results lack analytical depth; no scaling analysis, limiting-case validation, or asymptotic behavior is discussed to support the numerical trends.

Author Response: The authors thank the reviewer for this valuable comment. In response, additional discussion regarding limiting-case behavior and physical consistency has been included in the revised manuscript.

Specifically, the revised version now discusses the limiting cases of the governing parameters, including and , which recover physically consistent hydrodynamic and near-Newtonian behavior, respectively. These observations support the validity of the numerical trends obtained in the present study.

Additional interpretation regarding the dominance of magnetic damping and relaxation effects at higher parameter values has also been incorporated to improve the analytical depth of the discussion.

Reviewer 4 Report

Comments and Suggestions for Authors

The manuscript requires major revision due to significant issues as follows:

Abstract: (a) Several grammatical and wording issues. Ex: “A mathematical description of the phenomenon is created…” can be improved to “A mathematical model of the phenomenon was developed…”. (b) Some abbreviations are over complicated. Ex: CCHMF–THNF. (c) The novelty should be stated more explicitly. (d) Important quantitative outcomes are missing or incomplete. (e) The practical significance of the study should be highlighted.
Introduction: (a) Several paragraphs contain grammatical and language-related issues. Need proofreading. Ex: “LM is a technique that is crucial for modelling Carreau-Yasuda tri-nanofluids” (b) The literature review contains excessive descriptive details for ANN fundamentals. May be shortened and more focused on recent related studies. (c) Move the novelty and research gap discussion earlier for better logical flow. (d) Some references are discussed without critically identifying their limitations. (e) Objective 2 should be improvised. “To use an ANN–LM optimization approach to solve…” → “To implement an ANN–LM optimization approach for solving…”. (f) “well known algorithm used for regression work” may be rewritten as “a widely used optimization algorithm for regression problems.” (g) “The paper has structured it such that…” may be replaced with “The remainder of the paper is organized as follows…”. (h) (j) Recent high impact articles on this topic are missing: https://doi.org/10.1016/j.cis.2025.103668.
Section 2: (a) Variables and symbols are not clearly defined when first introduced. Should be explicitly defined with physical meaning and units. (b) Some physical terms included in the equations are not adequately explained. Ex: The role of Joule heating, thermal relaxation parameter, and solutal relaxation parameter. (c) The physical interpretation of important dimensionless parameters should be briefly discussed. Ex: Ha, We, Ec, and Sc should be explained in relation to the flow physics.
Section 3: (a) “The three components of the approach are included in this section” can be improved to “This section describes the three major components of the proposed approach.” (b) The authors should specify the total number of generated samples, parameter ranges, and mesh independence criteria used in bvp4c simulations. (c) Earlier, the manuscript mentions a hidden layer with 10 neurons, while later it states “concealed layers (twenty neurones).” This inconsistency should be corrected. (d) Some terms are scientifically inappropriate or unclear.
EX: such as “language-based model,” “terrain,” “concealed layers,” and “brilliance of the proposed stochastic computing” are not suitable for a scientific manuscript and should be replaced with precise technical terminology. (e) The explanation of activation functions is acceptable, but the statement regarding log-sigmoid performance should be supported with additional reasoning or references. (f) The authors should compare the computational cost of ANN–LM and bvp4c more rigorously, including CPU time or convergence efficiency analysis.
Section 4 and 5: (a) “with an inclined in the Weissenberg number” should be corrected to “with an increase in the Weissenberg number.” (b) The temperature enhancement due to Hartmann number is scientifically appropriate. However, the authors should support the explanation with references or quantitative comparison. (c) “the time required by fluid particles to diffuse across the medium is more” should be revised as “higher values of δ reduce mass diffusivity, thereby decreasing the concentration boundary layer thickness.” (d) Words such as “terrain,” “brilliance,” “academic career,” and “ingeniousness” are inappropriate in a scientific numerical analysis section. (e) The “Real-World Application” section is interesting and improves the practical relevance of the study. However, it appears disconnected from the earlier purely numerical framework and requires stronger justification. (f) The claim of “310% improvement in heat transfer coefficient” should be carefully justified and supported with experimental uncertainty analysis and literature comparison.
Conclusion: (a) Some conclusions are descriptive rather than analytical. (b) The statement regarding prediction accuracy “within 2–4% of benchmark results extracted from the literature” should be clarified carefully. (c) The limitation statement “without experimental validation” appears inconsistent with the earlier “Experimental Validation (Prototype)” section presented in the manuscript. This inconsistency should be clarified carefully. (d) “to potentially improve generalization, ideally include experimental validation” can be revised to “to improve model generalization, incorporate experimental validation, and extend the framework to three-dimensional transient flow problems.”

Author Response

Reviewer 3: Comment 1

Abstract: (a) Several grammatical and wording issues. Ex: “A mathematical description of the phenomenon is created…” can be improved to “A mathematical model of the phenomenon was developed…”. (b) Some abbreviations are over complicated. Ex: CCHMF–THNF. (c) The novelty should be stated more explicitly. (d) Important quantitative outcomes are missing or incomplete. (e) The practical significance of the study should be highlighted.

Author Response: The authors thank the reviewer for the constructive suggestions regarding the abstract. In the revised manuscript, the abstract has been thoroughly improved to correct grammatical issues, simplify overly complex abbreviations, and clearly highlight the novelty, quantitative findings, and practical significance of the study.

Specifically, the wording has been refined for improved readability, the novelty statement has been strengthened, important quantitative outcomes including prediction accuracy and computational efficiency have been added, and the engineering relevance of the proposed ANN–LM framework has been emphasized.

Reviewer 3: Comment 2

Introduction: (a) Several paragraphs contain grammatical and language-related issues. Need proofreading. Ex: “LM is a technique that is crucial for modelling Carreau-Yasuda tri-nanofluids” (b) The literature review contains excessive descriptive details for ANN fundamentals. May be shortened and more focused on recent related studies. (c) Move the novelty and research gap discussion earlier for better logical flow. (d) Some references are discussed without critically identifying their limitations. (e) Objective 2 should be improvised. “To use an ANN–LM optimization approach to solve…” → “To implement an ANN–LM optimization approach for solving…”. (f) “well known algorithm used for regression work” may be rewritten as “a widely used optimization algorithm for regression problems.” (g) “The paper has structured it such that…” may be replaced with “The remainder of the paper is organized as follows…”. (h) (j) Recent high impact articles on this topic are missing: https://doi.org/10.1016/j.cis.2025.103668.

Author Response: The authors thank the reviewer for the detailed suggestions. The Introduction section has been thoroughly revised to improve grammar, readability, and scientific clarity. The literature review has been shortened and refocused toward recent studies directly related to ANN-assisted modeling of non-Newtonian nanofluid systems.

In addition, the novelty and research gap discussion has been moved earlier in the Introduction to improve logical flow. Several sentences identified by the reviewer have been revised using more appropriate scientific terminology. Furthermore, recent high-impact literature suggested by the reviewer has been incorporated and discussed in the revised manuscript.

Reviewer 3: Comment 3

Section 2: (a) Variables and symbols are not clearly defined when first introduced. Should be explicitly defined with physical meaning and units. (b) Some physical terms included in the equations are not adequately explained. Ex: The role of Joule heating, thermal relaxation parameter, and solutal relaxation parameter. (c) The physical interpretation of important dimensionless parameters should be briefly discussed. Ex: Ha, We, Ec, and Sc should be explained in relation to the flow physics.

Author Response: The authors thank the reviewer for these valuable comments. In the revised manuscript, all important variables and dimensionless parameters have been explicitly defined upon first appearance, along with their physical interpretation. Additional explanations regarding Joule heating, thermal relaxation, and solutal relaxation effects have also been incorporated to improve the physical understanding of the governing equations.

Reviewer 3: Comment 4

Section 3: (a) “The three components of the approach are included in this section” can be improved to “This section describes the three major components of the proposed approach.” (b) The authors should specify the total number of generated samples, parameter ranges, and mesh independence criteria used in bvp4c simulations. (c) Earlier, the manuscript mentions a hidden layer with 10 neurons, while later it states “concealed layers (twenty neurones).” This inconsistency should be corrected. (d) Some terms are scientifically inappropriate or unclear.

EX: such as “language-based model,” “terrain,” “concealed layers,” and “brilliance of the proposed stochastic computing” are not suitable for a scientific manuscript and should be replaced with precise technical terminology. (e) The explanation of activation functions is acceptable, but the statement regarding log-sigmoid performance should be supported with additional reasoning or references. (f) The authors should compare the computational cost of ANN–LM and bvp4c more rigorously, including CPU time or convergence efficiency analysis.

Author Response: The authors thank the reviewer for the constructive comments. Section 3 has been revised to improve technical terminology, remove ambiguous expressions, and ensure consistency in the ANN architecture description. The total number of samples, parameter ranges, and mesh independence criteria used in the bvp4c simulations have now been explicitly included.

Furthermore, inappropriate terms such as “terrain,” “concealed layers,” and “brilliance” have been replaced with standard scientific terminology throughout the manuscript. In the revised manuscript, a more detailed comparison between the computational cost of the ANN–LM framework and the MATLAB bvp4c solver has been incorporated, including CPU execution time and convergence efficiency analysis.

We would like to clarify that, in the present study, the data generated using the bvp4c solver is used as the target/reference dataset for ANN–LM training and testing. Therefore, the ANN–LM framework is not intended to replace the bvp4c numerical solver or the underlying physical model. Instead, the primary objective is to investigate how accurately the ANN–LM model can learn and reproduce the nonlinear numerical solutions generated by bvp4c with reduced computational effort.

The revised manuscript now explicitly states this distinction and includes computational efficiency analysis to demonstrate the prediction capability of the ANN–LM framework relative to the target numerical data.

Reviewer 3: Comment 5

Section 4 and 5: (a) “with an inclined in the Weissenberg number” should be corrected to “with an increase in the Weissenberg number.” (b) The temperature enhancement due to Hartmann number is scientifically appropriate. However, the authors should support the explanation with references or quantitative comparison. (c) “the time required by fluid particles to diffuse across the medium is more” should be revised as “higher values of δ reduce mass diffusivity, thereby decreasing the concentration boundary layer thickness.” (d) Words such as “terrain,” “brilliance,” “academic career,” and “ingeniousness” are inappropriate in a scientific numerical analysis section. (e) The “Real-World Application” section is interesting and improves the practical relevance of the study. However, it appears disconnected from the earlier purely numerical framework and requires stronger justification. (f) The claim of “310% improvement in heat transfer coefficient” should be carefully justified and supported with experimental uncertainty analysis and literature comparison.

Author Response: The authors thank the reviewer for the careful evaluation of Sections 4 and 5. The identified grammatical issues and scientifically inappropriate terms have been corrected throughout the revised manuscript. Additional physical interpretation and literature support have also been included for the Hartmann number effects and concentration transport behavior. Furthermore, the “Real-World Application” subsection has been revised to establish a clearer connection between the computational framework and practical engineering systems. The previously stated “310% improvement” has been carefully revised and clarified to avoid overstatement in the absence of experimental uncertainty analysis.

Reviewer 3: Comment 6

Conclusion: (a) Some conclusions are descriptive rather than analytical. (b) The statement regarding prediction accuracy “within 2–4% of benchmark results extracted from the literature” should be clarified carefully. (c) The limitation statement “without experimental validation” appears inconsistent with the earlier “Experimental Validation (Prototype)” section presented in the manuscript. This inconsistency should be clarified carefully. (d) “to potentially improve generalization, ideally include experimental validation” can be revised to “to improve model generalization, incorporate experimental validation, and extend the framework to three-dimensional transient flow problems.”

Author Response: The authors thank the reviewer for these insightful comments. The Conclusion section has been revised to improve analytical depth, clarify the interpretation of prediction accuracy, and resolve the inconsistency regarding experimental validation. The revised manuscript now clearly distinguishes between computational verification and experimental validation, and the future work statement has been updated accordingly.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Accept in its current form.

Reviewer 3 Report

Comments and Suggestions for Authors

The article can be accepted

Reviewer 4 Report

Comments and Suggestions for Authors

The revision is satisfactory, and the manuscript can be accepted for publication.