Effects of temperature dependent viscosity and thermal conductivity on natural convection flow along a curved surface in the presence of exothermic catalytic chemical reaction

Uzma Ahmad; Muhammad Ashraf; A. Al-Zubaidi; Aamir Ali; Salman Saleem

doi:10.1371/journal.pone.0252485

Peer Review History

Original SubmissionFebruary 11, 2021
10 Mar 2021 Decision Letter - Naramgari Sandeep, Editor PONE-D-21-04681 Effects of Temperature Dependent Viscosity and Thermal Conductivity on Natural Convection Flow along a Curved Surface in the Presence of Exothermic Catalytic Chemical Reaction PLOS ONE Dear Dr. Ali, Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process. Please submit your revised manuscript by Apr 10 2021 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file. Please include the following items when submitting your revised manuscript: A rebuttal letter that responds to each point raised by the academic editor and reviewer(s). You should upload this letter as a separate file labeled 'Response to Reviewers'. A marked-up copy of your manuscript that highlights changes made to the original version. You should upload this as a separate file labeled 'Revised Manuscript with Track Changes'. An unmarked version of your revised paper without tracked changes. You should upload this as a separate file labeled 'Manuscript'. If you would like to make changes to your financial disclosure, please include your updated statement in your cover letter. Guidelines for resubmitting your figure files are available below the reviewer comments at the end of this letter. If applicable, we recommend that you deposit your laboratory protocols in protocols.io to enhance the reproducibility of your results. Protocols.io assigns your protocol its own identifier (DOI) so that it can be cited independently in the future. For instructions see: http://journals.plos.org/plosone/s/submission-guidelines#loc-laboratory-protocols We look forward to receiving your revised manuscript. Kind regards, Naramgari Sandeep, Ph.D Academic Editor PLOS ONE Journal Requirements: When submitting your revision, we need you to address these additional requirements. 1) Please ensure that your manuscript meets PLOS ONE's style requirements, including those for file naming. The PLOS ONE style templates can be found at https://journals.plos.org/plosone/s/file?id=wjVg/PLOSOne_formatting_sample_main_body.pdf and https://journals.plos.org/plosone/s/file?id=ba62/PLOSOne_formatting_sample_title_authors_affiliations.pdf 2) PLOS requires an ORCID iD for the corresponding author in Editorial Manager on papers submitted after December 6th, 2016. Please ensure that you have an ORCID iD and that it is validated in Editorial Manager. To do this, go to ‘Update my Information’ (in the upper left-hand corner of the main menu), and click on the Fetch/Validate link next to the ORCID field. This will take you to the ORCID site and allow you to create a new iD or authenticate a pre-existing iD in Editorial Manager. Please see the following video for instructions on linking an ORCID iD to your Editorial Manager account: https://www.youtube.com/watch?v=_xcclfuvtxQ 3) We note you have included a table to which you do not refer in the text of your manuscript. Please ensure that you refer to Tables 1 and 2 in your text; if accepted, production will need this reference to link the reader to the Tables. 4) Thank you for stating the following financial disclosure: [The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.]. At this time, please address the following queries: Please clarify the sources of funding (financial or material support) for your study. List the grants or organizations that supported your study, including funding received from your institution. State what role the funders took in the study. If the funders had no role in your study, please state: “The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.” If any authors received a salary from any of your funders, please state which authors and which funders. If you did not receive any funding for this study, please state: “The authors received no specific funding for this work.” Please include your amended statements within your cover letter; we will change the online submission form on your behalf. Reviewers' comments: 5. Review Comments to the Author Reviewer #1: The report presents the natural convection boundary layer flow of a viscous incompressible fluid with temperature-dependent viscosity and temperature-dependent thermal conductivity in the presence of exothermic catalytic chemical reaction along a curved surface. I recommend major revision due to the following facts. 1. In Eq. (3), the authors added (u^2/2x)n to the convective acceleration term. Comment: What does the new term means? How did you derive it? 2. After Eq. (6), it was written that n = P(x) + Q(x). Whereas () is the wall temperature function and the () is the body shape function. Comment: The authors refer Readers to check "Pop, I. and Takhar, H. S. Free convection from a curved surface. Journal of Applied Mathematics and Mechanics, 73 (1993), pp. 534 -539" for details. Comment: This is not acceptable. Update the research methodology with the derivation of such momentum equation. 3. Research question is needed to harmonize the contribution of the report to the body of knowledge. Comment: Research questions are needed at the end of the introduction section. Research questions are needed. Note that the results in this report are typical answers to unknown questions. This is true because the manuscript provides some powerful answers to unknown questions. Note that the research questions must connect the title to the analysis of results, and conclusion. This would guide authors not to generate many results that are not consistent to provide insight. The author should update the manuscript with appropriate and relevant research questions at the end of the introduction section. This would guide the author to structure logical analysis of results. Logical questions are expected. This would help readers to link what is known in the literature with the novelty of this study. 4. The similarity variables presented as Eq. (7) are dimensional. The unit of the first, u, is m^0.5 while that of v is m^{-1/4}, and that of y is also dimensional. Hence, the variables used to non-dimenzionalize the governing equation are not appropriate. Reviewer #2: Recommendation: Minor revision Authors should revise the manuscript according to the following comments. • Nomenclature is must. • Some grammatical and typo mistakes are found in the manuscript. • The governing differential equations describing the flow are non-linear. Is the solution obtained unique? • Specify the applications of considered physical model or geometry. • Introduction section should be made more concise to show previous work in the field. At present a lot of related researches are stated in the introduction. However, not sufficient analysis is presented. The authors should ask themselves: what are the problems with the presented researches? Why is the recent work needed? Hope these can improve the present work by following past articles. • Effect of Joule heating on MHD non‐Newtonian fluid flow past an exponentially stretching curved surface • Magnetohydrodynamic Cattaneo-Christov flow past a cone and a wedge with variable heat source/sink • Heat and mass transfer in MHD Casson nanofluid flow past a stretching sheet with thermophoresis and Brownian motion • Effect of asymmetrical heat rise/fall on the film flow of magnetohydrodynamic hybrid ferrofluid • Simultaneous solutions for first order and second order slips on micropolar fluid flow across a convective surface in the presence of Lorentz force and variable heat source/sink • Effect of thermal radiation on MHD Casson fluid flow over an exponentially stretching curved sheet • Influence of non-uniform heat source/sink on the three-dimensional magnetohydrodynamic Carreau fluid flow past a stretching surface with modified Fourier’s law • MHD Carreau Fluid Flow Past a Melting Surface with Cattaneo-Christov Heat Flux • Physical aspects on unsteady MHD‐free convective stagnation point flow of micropolar fluid over a stretching surface • Influence of viscous dissipation on MHD flow of micropolar fluid over a slendering stretching surface with modified heat flux model • A non‐Fourier heat flux model for magnetohydrodynamic micropolar liquid flow across a coagulated sheet Note: I want to see the revised version of the manuscript. [NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files.] While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com/. PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Registration is free. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email PLOS at figures@plos.org. Please note that Supporting Information files do not need this step. https://doi.org/10.1371/journal.pone.0252485.r001
Revision 1
13 Apr 2021 Author Response Response to Reviewers Comments (PONE-D-21-04681) Effects of Temperature Dependent Viscosity and Thermal Conductivity on Natural Convection Flow along a Curved Surface in the Presence of Exothermic Catalytic Chemical Reaction Reviewer #1: The report presents the natural convection boundary layer flow of a viscous incompressible fluid with temperature-dependent viscosity and temperature-dependent thermal conductivity in the presence of exothermic catalytic chemical reaction along a curved surface. I recommend major revision due to the following facts. In Eq. (3), the authors added (u^2/2x)n to the convective acceleration term. Comment: What does the new term means? How did you derive it. Response: During dimensionalization of the momentum equation we have equation of the form u ∂u/∂x+v ∂u/∂y+u^2/2 1/g_x (dg_x)/dx=γ_μ (∂θ/∂y)(∂u/∂y) +(1+γ_μ θ)((∂^2 u)/(∂y^2 ))+θ+ϕ, Here, as 1/x ̅ (P(x ̅)+Q(x ̅))=1/g_x (dg_x)/(dx ̅ ) and P(x ̅ )+Q(x ̅ )=n and thus 1/x ̅ (n)=1/g_x (dg_x)/(dx ̅ ) u ∂u/∂x+v ∂u/∂y+u^2/2x n=γ_μ (∂θ/∂y)(∂u/∂y) +(1+γ_μ θ)((∂^2 u)/(∂y^2 ))+θ+ϕ, 2. After Eq. (6), it was written that n = P(x) + Q(x). Whereas () is the wall temperature function and the () is the body shape function. Comment: The authors refer Readers to check "Pop, I. and Takhar, H. S. Free convection from a curved surface. Journal of Applied Mathematics and Mechanics, 73 (1993), pp. 534 -539" for details. Comment: This is not acceptable. Update the research methodology with the derivation of such momentum equation. Response: The detail derivation of n=P(x)+Q(x) is given in below steps: Since we know that d/dx (lnx)=1/x and d(lnx)=1/x dx which implies that 1/(d(lnx))=x/dx. (1) Also consider d/dx (ln g_x )=1/g_x (dg_x)/dx which implies d(ln g_x )=1/g_x dg_x. (2) Now combining the above two equations (1) and (2), we have (d lng_x)/(d lnx)=x/g_x (dg_x)/dx. Now using the non-dimensionless variables, we may write the above equations as (d lng_x)/(d lnx ̅l)=(x ̅l)/g_x (dg_x)/(dx ̅l) which implies that (d lng_x)/(d lnx ̅l)=x ̅/g_x (dg_x)/(dx ̅ ) and thus we have 1/x ̅ (d lng_x)/(d lnx ̅l)=1/g_x (dg_x)/(dx ̅ ) (3) Also, we consider d/dx (lnT_w )=1/T_w (dT_w)/dx=0 which implies that d lnT_w=0 and thus, we can write (d lnT_w)/(d lnx ̅l)=0 (4) Now from equation (3) and (4), we have 1/x ̅ ((d lnT_w)/(d lnx ̅ l)+(d lng_x)/(d lnx ̅ l))=1/g_x (dg_x)/(dx ̅ ) which can also be written as 1/x ̅ (P(x ̅)+Q(x ̅))=1/g_x (dg_x)/(dx ̅ ) (5) Here, P(x ̅), the wall temperature function and Q(x ̅) the body shape function, are defined as P(x ̅ )=(d lnT_w)/(d lnx ̅ l), Q(x ̅ )=(d lng_x)/(d lnx ̅ l) . (6) 3. Research question is needed to harmonize the contribution of the report to the body of knowledge. Comment: Research questions are needed at the end of the introduction section. Research questions are needed. Note that the results in this report are typical answers to unknown questions. This is true because the manuscript provides some powerful answers to unknown questions. Note that the research questions must connect the title to the analysis of results, and conclusion. This would guide authors not to generate many results that are not consistent to provide insight. The author should update the manuscript with appropriate and relevant research questions at the end of the introduction section. This would guide the author to structure logical analysis of results. Logical questions are expected. This would help readers to link what is known in the literature with the novelty of this study. Response: The overall objectives of this research are to develop the mathematical model to study and compare the different modes of coupling the exothermic and catalytic chemical reaction via momentum, energy and mass concentration equation. Model predictions are used to assess the effects of different parameters on conversion of exothermic catalytic chemical reaction at heated curve generated by tangential component of acceleration due to gravity. The proposed model is used as the guidelines for the selection of a suitable coupling to achieve the desired applications. A numerical technique Finite Difference Method in conjunction with primitive variable formulation is used to investigate the coupling of exothermic reaction with catalyst particles. Furthermore, parametric effects of heated wall and mass concentration along the curved surface are studied and highlighted graphically and as well as in tabular form. 4. The similarity variables presented as Eq. (7) are dimensional. The unit of the first, u, is m^0.5 while that of v is m^{-1/4}, and that of y is also dimensional. Hence, the variables used to non-dimenzionalize the governing equation are not appropriate. Response: In the section of numerical analysis, we have used primitive variable formulation given as below: u=x^(1⁄2) U(X,Y),v=x^((-1)⁄4) V(X,Y), x=X,y=x^(1⁄4) Y, θ=Θ(X.Y), C=Φ(X,Y). It is pertinent to mention that primitive variable are dimensionless, we use this transformation to get primitive form of partial differential equations, please see red terms in each equation After using the aforementioned transformations, we have the following form of system of equations: U/2+X ∂U/∂X-Y/4 ∂U/∂Y+∂V/∂Y=0, [1/2+n/2] U^2+XU ∂U/∂X+(V-YU/4) ∂U/∂Y=γ_μ ∂Θ/∂Y ∂U/∂Y+(1+γ_μ Θ) (∂^2 U)/(∂Y^2 )+Θ+Φ, XU ∂Θ/∂X+(V-YU/4) ∂Θ/∂Y=1/Pr [ξ(∂Θ/∂Y)^2+(1+ξΘ) (∂^2 Θ)/(∂Y^2 )]+βλ^2 X^(1⁄2) (1+nγΘ)exp((-E)/(1+γθ))Θ, XU ∂Φ/∂X+(V-YU/4) ∂Φ/∂Y=1/Sc (∂^2 Φ)/(∂Y^2 )+λ^2 X^(1⁄2) (1+nγΘ)exp((-E)/(1+γθ))Φ. Reviewer #2: Recommendation: Minor revision Authors should revise the manuscript according to the following comments. • Nomenclature is must. Response: The nomenclature has been included in the revised manuscript. • Some grammatical and typo mistakes are found in the manuscript. Response: In the revised manuscript we have correct the grammatical issues. • The governing differential equations describing the flow are non-linear. Is the solution obtained unique? Response: Please if you focus on the graphs and boundary conditions U_(i,j)=0, V_(i,j)=0,Θ_(i,j) =1, Φ_(i,j)=1 at Y_j=0 U_(i,j)→0, 〖 θ〗_(i,j)→0, Φ_(i.j)→0 as Y_j→∞ In these graphs and all other graphs given in this study, the solutions are satisfying by the boundary conditions, it is evident of the correctness of the obtained results. • Specify the applications of considered physical model or geometry. Response: The proposed model is used as the guidelines for the selection of a suitable coupling to achieve the desired applications. The curved shaped geometry is used to design many problems of civil engineering as pressure barrier. • Introduction section should be made more concise to show previous work in the field. At present a lot of related researches are stated in the introduction. However, not sufficient analysis is presented. The authors should ask themselves: what are the problems with the presented researches? Why is the recent work needed? Hope these can improve the present work by following past articles. • Effect of Joule heating on MHD non‐Newtonian fluid flow past an exponentially stretching curved surface • Magneto hydrodynamic Cattaneo-Christov flow past a cone and a wedge with variable heat source/sink • Heat and mass transfer in MHD Casson nanofluid flow past a stretching sheet with thermophoresis and Brownian motion. • Effect of asymmetrical heat rise/fall on the film flow of magneto hydrodynamic hybrid Ferro fluid • Simultaneous solutions for first order and second order slips on micro polar fluid flow across a convective surface in the presence of Lorentz force and variable heat source/sink. • Effect of thermal radiation on MHD Casson fluid flow over an exponentially stretching curved sheet • Influence of non-uniform heat source/sink on the three-dimensional magneto hydrodynamic Carreau fluid flow past a stretching surface with modified Fourier’s law • MHD Carreau Fluid Flow Past a Melting Surface with Cattaneo-Christov Heat Flux • Physical aspects on unsteady MHD‐free convective stagnation point flow of micro polar fluid over a stretching surface • Influence of viscous dissipation on MHD flow of micro polar fluid over a slandering stretching surface with modified heat flux model • A non‐Fourier heat flux model for magneto hydrodynamics micro polar liquid flow across a coagulated sheet Note: I want to see the revised version of the manuscript. Response: Most relevant papers are added to enrich introduction (see in introduction ref [23] - [30]) in the revised manuscript. Attachments Attachment Submitted filename: Response to Reviewers Comments.docx https://doi.org/10.1371/journal.pone.0252485.r002
28 Apr 2021 Decision Letter - Naramgari Sandeep, Editor PONE-D-21-04681R1 Effects of Temperature Dependent Viscosity and Thermal Conductivity on Natural Convection Flow along a Curved Surface in the Presence of Exothermic Catalytic Chemical Reaction PLOS ONE Dear Dr. Ali, Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process. Please submit your revised manuscript by Jun 12 2021 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file. Please include the following items when submitting your revised manuscript: A rebuttal letter that responds to each point raised by the academic editor and reviewer(s). You should upload this letter as a separate file labeled 'Response to Reviewers'. A marked-up copy of your manuscript that highlights changes made to the original version. You should upload this as a separate file labeled 'Revised Manuscript with Track Changes'. An unmarked version of your revised paper without tracked changes. You should upload this as a separate file labeled 'Manuscript'. If you would like to make changes to your financial disclosure, please include your updated statement in your cover letter. Guidelines for resubmitting your figure files are available below the reviewer comments at the end of this letter. If applicable, we recommend that you deposit your laboratory protocols in protocols.io to enhance the reproducibility of your results. Protocols.io assigns your protocol its own identifier (DOI) so that it can be cited independently in the future. For instructions see: http://journals.plos.org/plosone/s/submission-guidelines#loc-laboratory-protocols. Additionally, PLOS ONE offers an option for publishing peer-reviewed Lab Protocol articles, which describe protocols hosted on protocols.io. Read more information on sharing protocols at https://plos.org/protocols?utm_medium=editorial-email&utm_source=authorletters&utm_campaign=protocols. We look forward to receiving your revised manuscript. Kind regards, Naramgari Sandeep, Ph.D Academic Editor PLOS ONE Journal Requirements: Please review your reference list to ensure that it is complete and correct. If you have cited papers that have been retracted, please include the rationale for doing so in the manuscript text, or remove these references and replace them with relevant current references. Any changes to the reference list should be mentioned in the rebuttal letter that accompanies your revised manuscript. If you need to cite a retracted article, indicate the article’s retracted status in the References list and also include a citation and full reference for the retraction notice. Reviewers' comments: Reviewer's Responses to Questions 6. Review Comments to the Author Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters) Reviewer #1: The report presents the natural convection boundary layer flow of a viscous incompressible fluid with temperature-dependent viscosity and temperature-dependent thermal conductivity in the presence of exothermic catalytic chemical reaction along a curved surface. I recommend minor revision due to the following facts. Q1. In Eq. (3), the authors added (u^2/2x)n to the convective acceleration term. Comment: What does the new term means? How did you derive it. In the report, the derivation presented by the Authors is in isolation. You are expected to derive it as a convective acceleration of momentum equation. Q2. After Eq. (6), it was written that n = P(x) + Q(x). Whereas () is the wall temperature function and the () is the body shape function. This is the exact contribution of the report to the body of knowledge and it should be introduced under the introduction section. In fact, a paragraph is needed to announce this novelty. Q3. How do you study a fluid flow along a vertical surface without buoyancy forces? This is true because there is nothing like the associated dimensionless parameter called buoyancy parameter or Grashof number. Q4. Before Eq. 7, was defined as the tangential component of acceleration due to gravity. What is r in the definition? Reviewer #2: The revisions are Good but some references are not arranged properly. Hence I recommend for possible publication. [NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files.] While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com/. PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Registration is free. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email PLOS at figures@plos.org. Please note that Supporting Information files do not need this step. https://doi.org/10.1371/journal.pone.0252485.r003
Revision 2
14 May 2021 Author Response Reply to Review PONE-D-21-04681R1 Effects of Temperature Dependent Viscosity and Thermal Conductivity on Natural Convection Flow along a Curved Surface in the Presence of Exothermic Catalytic Chemical Reaction PLOS ONE Reviewer #1: The report presents the natural convection boundary layer flow of a viscous incompressible fluid with temperature-dependent viscosity and temperature-dependent thermal conductivity in the presence of exothermic catalytic chemical reaction along a curved surface. I recommend minor revision due to the following facts. Q1. In Eq. (3), the authors added (u^2/2x)n to the convective acceleration term. Comment: What does the new term means? How did you derive it. In the report, the derivation presented by the Authors is in isolation. You are expected to derive it as a convective acceleration of momentum equation. Response: The detail derivation of momentum equation is given as below: The dimensioned form of momentum equation in article is given as under u ∂u/∂x+v ∂u/∂y=1/ρ ∂/∂y (μ ∂u/∂y)+g_x β_T (T-T_∞ )+g_x β_C (C-C_∞) (1) Dimensionless variables x ̅=x/l,y ̅=y/l Gr^(1/4),u ̅=u/U_s ,v ̅=v/V_s Gr^(1/4),θ=(T-T_∞)/(T_w-T_∞ ) ,ϕ =(C-C_∞)/(C_w-C_∞ ) (2) To convert the equation (1) into dimensionless form we use the dimensionless variables defined in equation (2), for this we find the term s one by one appeared in the equation (1) u ∂u/∂x=U_s u ̅ (∂U_s u ̅)/(∂x ̅l)=(U_s^2)/l u ̅ (∂u ̅)/(∂x ̅ )+〖U_s u ̅〗^2/l (∂U_s)/(∂x ̅ ) u ∂u/∂x=(U_s^2)/l u ̅ (∂u ̅)/(∂x ̅ )+〖U_s u ̅〗^2/l ∂/(∂x ̅ ) (g_x β_T ΔT)^(1/2) u ∂u/∂x=(U_s^2)/l u ̅ (∂u ̅)/(∂x ̅ )+〖U_s u ̅〗^2/l (g_x β_T ΔT)^(1/2) 1/(2√(g_x )) (dg_x)/(dx ̅ ) u ∂u/∂x=(U_s^2)/l [u ̅ (∂u ̅)/(∂x ̅ )+u ̅^2/2 1/g_x (dg_x)/(dx ̅ )] (3) Now v ∂u/∂y=U_s v ̅Gr^(-1/4) (∂U_s u ̅)/(∂ly ̅Gr^(-1/4) )=(U_s^2)/l v ̅ (∂u ̅)/(∂y ̅ ) (4) Now 1/ρ ∂/∂y (μ ∂u/∂y)=1/ρ [∂μ/∂y ∂u/∂y+μ (∂^2 u)/(∂y^2 )] (5) Considering ∂μ/∂y=∂/∂y μ_o (1+γ^* (T-T_∞)) ∂μ/∂y=∂/(∂y ̅lGr^(-1/4) ) (μ_o (1+γ^* ΔTθ)) ∂μ/∂y=(μ_o γ^* ΔT)/(lGr^(-1/4) ) ∂θ/(∂y ̅ ) ∂μ/∂y=(μ_o γ_T)/(lGr^(-1/4) ) ∂θ/(∂y ̅ ) (6) ∂u/∂y=(∂U_s u ̅)/(∂y ̅lGr^(-1/4) )=U_s/(lGr^(-1/4) ) (∂u ̅)/(∂y ̅ ) (7) (∂^2 u)/(∂y^2 )=(∂U_s u ̅)/(∂y ̅lGr^(-1/4) )=U_s/(lGr^(-1/2) ) (∂^2 u ̅)/(∂y ̅^2 ) (8) Put eqs. (6)-(8) in eq. (5) we have 1/ρ ∂/∂y (μ ∂u/∂y)=1/ρ [(μ_o γ_μ)/(lGr^(-1/4) ) ∂θ/(∂y ̅ ) U_s/(lGr^(-1/4) ) (∂u ̅)/(∂y ̅ )+(μU_s)/(lGr^(-1/2) ) (∂^2 u ̅)/(∂y ̅^2 )] 1/ρ ∂/∂y (μ ∂u/∂y)=1/ρ [(μ_o γ_μ U_s)/(l^2 Gr^(-1/2) ) ∂θ/(∂y ̅ ) (∂u ̅)/(∂y ̅ )+〖μ_o (1+γ_μ θ)U〗_s/(l^2 Gr^(-1/2) ) (∂^2 u ̅)/(∂y ̅^2 )] 1/ρ ∂/∂y (μ ∂u/∂y)=(μ_o U_s)/(ρl^2 Gr^(-1/2) ) [γ_μ ∂θ/(∂y ̅ ) (∂u ̅)/(∂y ̅ )+(1+γ_μ θ)(∂^2 u ̅)/(∂y ̅^2 )] Where γ_(μ=) γ^* ΔT 1/ρ ∂/∂y (μ ∂u/∂y)=(μ_o U_s)/(ρl^2 Gr^(-1/2) ) [γ_μ ∂θ/(∂y ̅ ) (∂u ̅)/(∂y ̅ )+(1+γ_μ θ)(∂^2 u ̅)/(∂y ̅^2 )] (9) Put the eqs. (3-4) and Eq. (9) in eq. (1) (U_s^2)/l [u ̅ (∂u ̅)/(∂x ̅ )+u ̅^2/2 1/g_x (dg_x)/(dx ̅ )+v ̅ (∂u ̅)/(∂y ̅ )]=(μ_o U_s Gr^(1/2))/(ρl^2 ) [γ_μ ∂θ/(∂y ̅ ) (∂u ̅)/(∂y ̅ )+(1+γ_μ θ)(∂^2 u ̅)/(∂y ̅^2 )]+g_x β_T ΔTθ+g_x β_C ΔCϕ [u ̅ (∂u ̅)/(∂x ̅ )+u ̅^2/2 1/g_x (dg_x)/(dx ̅ )+v ̅ (∂u ̅)/(∂y ̅ )]=(μ_o Gr^(1/2))/(ρlU_s ) [γ_μ ∂θ/(∂y ̅ ) (∂u ̅)/(∂y ̅ )+(1+γ_μ θ)(∂^2 u ̅)/(∂y ̅^2 )]+(〖lg〗_x β_T ΔTθ)/(U_s^2 )+(〖lg〗_x β_C ΔTϕ)/(U_s^2 ) u ̅ (∂u ̅)/(∂x ̅ )+u ̅^2/2 1/g_x (dg_x)/(dx ̅ )+v ̅ (∂u ̅)/(∂y ̅ )=(νGr^(1/2))/(lU_s ) [γ_μ ∂θ/(∂y ̅ ) (∂u ̅)/(∂y ̅ )+(1+γ_μ θ)(∂^2 u ̅)/(∂y ̅^2 )]+(〖lg〗_x β_T ΔTθ)/(U_s^2 )+(〖lg〗_x β_C ΔCϕ)/(U_s^2 ) u ̅ (∂u ̅)/(∂x ̅ )+u ̅^2/2 1/g_x (dg_x)/(dx ̅ )+v ̅ (∂u ̅)/(∂y ̅ )=(νGr^(1/2))/(lU_s ) [γ_μ ∂θ/(∂y ̅ ) (∂u ̅)/(∂y ̅ )+(1+γ_μ θ)(∂^2 u ̅)/(∂y ̅^2 )]+(〖lg〗_x β_T ΔTθ)/(U_s^2 )+(〖lg〗_x β_C ΔCϕ)/(U_s^2 ) Where U_s=(g_x β_T ΔTl)^(1/2),Gr=(g_x β_T ΔTl^3)/ν^2 and U_sc=(g_x β_T ΔCl)^(1/2),Gr^=(g_x β_T ΔCl^3)/ν^2 So u ̅ (∂u ̅)/(∂x ̅ )+u ̅^2/2 1/g_x (dg_x)/(dx ̅ )+v ̅ (∂u ̅)/(∂y ̅ )=(νGr^(1/2))/(lU_s ) [γ_μ ∂θ/(∂y ̅ ) (∂u ̅)/(∂y ̅ )+(1+γ_μ θ)(∂^2 u ̅)/(∂y ̅^2 )]+(U_s^2 θ)/(U_s^2 )+(U_s^2 ϕ)/(U_s^2 ) u ̅ (∂u ̅)/(∂x ̅ )+u ̅^2/2 1/g_x (dg_x)/(dx ̅ )+v ̅ (∂u ̅)/(∂y ̅ )=γ_μ ∂θ/(∂y ̅ ) (∂u ̅)/(∂y ̅ )+(1+γ_μ θ)(∂^2 u ̅)/(∂y ̅^2 )+θ+ϕ (10) Where the detail derivation of n=P(x)+Q(x) is given in below steps: Since we know that d/dx (lnx)=1/x and d(lnx)=1/x dx which implies that1/(d(lnx))=x/dx. Also considerd/dx (ln g_x )=1/g_x (dg_x)/dx which impliesd(ln g_x )=1/g_x dg_x. (d lng_x)/(d lnx)=x/g_x (dg_x)/dx. Now using the non-dimensionless variables, we may write the above equations as (d lng_x)/(d lnx ̅l)=(x ̅l)/g_x (dg_x)/(dx ̅l) which implies that (d lng_x)/(d lnx ̅l)=x ̅/g_x (dg_x)/(dx ̅ ) and thus we have 1/x ̅ (d lng_x)/(d lnx ̅l)=1/g_x (dg_x)/(dx ̅ ) Also, we consider d/dx (lnT_w )=1/T_w (dT_w)/dx=0 which implies that d lnT_w=0 and thus, we can write (d lnT_w)/(d lnx ̅l)=0 Now from equation (c) and (d), we have 1/x ̅ ((d lnT_w)/(d lnx ̅ l)+(d lng_x)/(d lnx ̅ l))=1/g_x (dg_x)/(dx ̅ ) which can also be written as 1/x ̅ (P(x ̅)+Q(x ̅))=1/g_x (dg_x)/(dx ̅ ) Here, P(x ̅), the wall temperature function and Q(x ̅) the body shape function, are defined as P(x ̅ )=(d lnT_w)/(d lnx ̅ l), Q(x ̅ )=(d lng_x)/(d lnx ̅ l) . Now (10) becomes u ̅ (∂u ̅)/(∂x ̅ )+u ̅^2/2(P(x ̅ )+Q(x ̅))+v ̅ (∂u ̅)/(∂y ̅ )=γ_μ ∂θ/(∂y ̅ ) (∂u ̅)/(∂y ̅ )+(1+γ_μ θ)(∂^2 u ̅)/(∂y ̅^2 )+θ+ϕ (11) Now P(x ̅ )+Q(x ̅ )=n Thus (11) becomes u ̅ (∂u ̅)/(∂x ̅ )+u ̅^2/2 1/g_x (dg_x)/(dx ̅ )+v ̅ (∂u ̅)/(∂y ̅ )=γ_μ ∂θ/(∂y ̅ ) (∂u ̅)/(∂y ̅ )+(1+γ_μ θ)(∂^2 u ̅)/(∂y ̅^2 )+θ+ϕ Which is dimensionless form of momentum equation along curved surface, and n is the body shape parameter. Q2. After Eq. (6), it was written that n = P(x) + Q(x). Whereas () is the wall temperature function and the () is the body shape function. This is the exact contribution of the report to the body of knowledge and it should be introduced under the introduction section. In fact, a paragraph is needed to announce this novelty. Response: In current paper we are investigated the natural convection flow over a two dimensional body of arbitrary geometric configuration in the presence of exothermic catalytic chemical reaction. The momentum, energy, and mass concentration equations are a general form suitable for laminar natural convection flows along curved surface in the inclusion of exothermic catalytic reaction. For body of arbitrary shape, the special case in which P(x ̅ )+Q(x ̅ )=n, thus the body shape parameter (index parameter) n has chosen 0<n≤1/2 which is main novelty of this work. Note: This paragraph has inserted in the introduction section. Q3. How do you study a fluid flow along a vertical surface without buoyancy forces? This is true because there is nothing like the associated dimensionless parameter called buoyancy parameter or Grashof number. Response: Please if you see the dimensionless variables used to dimensionalize the flow model Dimensionless variables x ̅=x/l,y ̅=y/l Gr^(1/4),u ̅=u/U_s ,v ̅=v/V_s Gr^(1/4),θ=(T-T_∞)/(T_w-T_∞ ) ,ϕ =(C-C_∞)/(C_w-C_∞ ) Here Grashof number is used as y ̅=y/l Gr^(1/4), v ̅=v/V_s Gr^(1/4) where Gr=(gβ∆Tx^3)/ν^2 , which is infact responsible for buoyancy force and it is used in dimensionalization (frequently available in literature (you can see Convective Heat Transfer by Ion Pop and D. B. Ingham for the cases of free or natural convection heat transfer). Dimensionless temperature variable θ=(T-T_∞)/(T_w-T_∞ ), it is the ratio of temperature difference. The problem under consideration is based on natural convection heat transfer, natural convection occurred because of the difference in density, and difference in density is due to the difference in temperature. The contribution of θ in momentum is obvious and sufficient answer to question. Q4. Before Eq. 7, was defined as the tangential component of acceleration due to gravity. What is r in the definition? Response: Here, x is the boundary layer coordinate along the curved surface while r and z are the horizontal and vertical coordinates of the points of the shape curve z=z(r) respectively. It is pertinent to mention that here flow is along the curved surface and dr/dz=0 means there is no flow inner side of the curve. The coordinate z is the vertical distance measure from the lower stagnation point (Please see fig. 1 flow configuration and coordinate system). Reviewer #2: The revisions are Good but some references are not arranged properly. Hence I recommend for possible publication. Response: We have focused on references, all the references are now in good order and well targeted to the problem under consideration. Attachments Attachment Submitted filename: Response to Reviewers Comments.docx* https://doi.org/10.1371/journal.pone.0252485.r004
17 May 2021 Decision Letter - Naramgari Sandeep, Editor Effects of Temperature Dependent Viscosity and Thermal Conductivity on Natural Convection Flow along a Curved Surface in the Presence of Exothermic Catalytic Chemical Reaction PONE-D-21-04681R2 Dear Dr. Ali, We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements. Within one week, you’ll receive an e-mail detailing the required amendments. When these have been addressed, you’ll receive a formal acceptance letter and your manuscript will be scheduled for publication. An invoice for payment will follow shortly after the formal acceptance. To ensure an efficient process, please log into Editorial Manager at http://www.editorialmanager.com/pone/, click the 'Update My Information' link at the top of the page, and double check that your user information is up-to-date. If you have any billing related questions, please contact our Author Billing department directly at authorbilling@plos.org. If your institution or institutions have a press office, please notify them about your upcoming paper to help maximize its impact. If they’ll be preparing press materials, please inform our press team as soon as possible -- no later than 48 hours after receiving the formal acceptance. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information, please contact onepress@plos.org. Kind regards, Naramgari Sandeep, Ph.D Academic Editor PLOS ONE Additional Editor Comments (optional): The revised version is acceptable for publication. Reviewers' comments: https://doi.org/10.1371/journal.pone.0252485.r005
Formally Accepted
19 Jul 2021 Acceptance Letter - Naramgari Sandeep, Editor PONE-D-21-04681R2 Effects of Temperature Dependent Viscosity and Thermal Conductivity on Natural Convection Flow along a Curved Surface in the Presence of Exothermic Catalytic Chemical Reaction Dear Dr. Ali: I'm pleased to inform you that your manuscript has been deemed suitable for publication in PLOS ONE. Congratulations! Your manuscript is now with our production department. If your institution or institutions have a press office, please let them know about your upcoming paper now to help maximize its impact. If they'll be preparing press materials, please inform our press team within the next 48 hours. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information please contact onepress@plos.org. If we can help with anything else, please email us at plosone@plos.org. Thank you for submitting your work to PLOS ONE and supporting open access. Kind regards, PLOS ONE Editorial Office Staff on behalf of Dr. Naramgari Sandeep Academic Editor PLOS ONE https://doi.org/10.1371/journal.pone.0252485.r006

Open letter on the publication of peer review reports

PLOS recognizes the benefits of transparency in the peer review process. Therefore, we enable the publication of all of the content of peer review and author responses alongside final, published articles. Reviewers remain anonymous, unless they choose to reveal their names.

We encourage other journals to join us in this initiative. We hope that our action inspires the community, including researchers, research funders, and research institutions, to recognize the benefits of published peer review reports for all parts of the research system.

Learn more at ASAPbio .