Crank Nicholson scheme to examine the fractional-order unsteady nanofluid flow of free convection of viscous fluids.

Fractional fluid models are usually difficult to solve analytically due to complicated mathematical calculations. This difficulty in considering fractional model further increases when one considers nth order chemical reaction. Therefore, in this work an incompressible nanofluid flow as well as the benefits of free convection across an isothermal vertical sheet is examined numerically. An nth order chemical reaction is considered in the chemical species model. The specified velocity (wall's) is time-based, and its motion is translational into mathematical form. The fractional differential equations are used to express the governing flow equations (FDEs). The non-dimensional controlling system is given appropriate transformations. A Crank Nicholson method is used to find solutions for temperature, solute concentration, and velocity. Variation in concentration, velocity, and temperature profiles is produced as a result of changes in discussed parameters for both Ag-based and Cu-based nanofluid values. Water is taken as base fluid. The fractional-order time evaluation has opened the new gateways to study the problem into a new direction and it also increased the choices due to the extended version. It records the hidden figures of the problem between the defined domain of the time evaluation. The suggested technique has good accuracy, dependability, effectiveness and it also cover the better physics of the problem specially with concepts of fractional calculus.

1. Introduction

Natural convection process is that type of flow situation in which a liquid such as water as an example of Newtonian fluid, in which the fluid motion is produced by an external source instead by some parts of the fluid being heavier than other parts. More exactly, it is a specific kind of self-persistent flow with a high-temperature gradient, as a natural convection flow. This factor is then endorsed in order to get the non-uniform density. Because of changes in density and gravitational field, buoyancy effects promote current movement. The aforementioned occurrences occur often in nature and have been documented in a variety of technical and engineering settings [1]. The most common model in convective flow models is natural convection, which involves the movement of heat and mass near a moving sheet. The aforementioned concept is often used in solar energy collectors, nuclear reactor architecture, and electronic devices. Various writers discussed the wonders of natural convection with the transfer of heat and mass, as well as the Newtonian/Non-Newtonian character of the fluid, due to its wide range of possible uses. The effect of convection (or free convection) on the accelerating plate in a perpendicular position was examined in [2], where they utilized the Laplace transformation technique (LTM) to examine the solution for two distinct circumstances, namely, the constant heat flux and isothermal plate. Refer to [3] for a free-convection flow issue in which a vertical plate is constructed in such a manner that it increases exponentially. Some of the prospective information may be examined methodically in references [4-11].

Following the contribution of Choi [12], the discipline of fluid mechanics received valuable concertation. Because he focused on thermal conductivity enrichment ideas related to fluids, he created the nano-fluids discipline. He demonstrated that nanoparticles, which are microscopic and small particles, may be put into fluids to convert them into nanofluids. The extensive research and experimental results revealed that it improves the thermal characteristics of the conventional fluid. As a result of this dramatic modernization, this area acquired considerable significance, and a large body of work is accessible in the literature. Sheikholelsami and Ganji [13] investigated the convective heat transport of nanofluids. [14-32] provide a comprehensive examination of nano-fluids and applications from different perspectives.

Previously, the fractional calculus theory sparked widespread interest due to its wide range of applications in physics and engineering [10]. This kind of research has made use of multidimensional dynamics such as wave, viscoelastic, and relaxation activities. Because of the operators, we developed a straightforward method for introducing fractional ordered derivatives into linear viscous models, which drew much attention to this area.

This research looked at the physical elements of the issue of fractional-order derivatives between certain domains. The fractional calculus makes visible contributions to various technical and scientific circumstances, including neurology, capacitor theory, viscoelasticity, electro-analytical chemistry, and electrical circuits [33,34]. Several authors [35-39] suggested several techniques for dealing with the nonlinearity of fractional differential equations. Despite the fact that there is extensive research literature on fluid flows, numerous mathematical and fluid models are developed and effectively solved using the fractional calculus method; see, for example, some useful investigations in this direction [7,40-42].

The above literature shows that several investigations are done on convection heat transfer using classical/fractional models [43-47] and [48-53]. However, in all these models, particularly those they are involved with fractional derivatives, no attention is given to nth order chemical reaction in species concentration and the free convection flow of viscous nanofluid using fractional derivatives. Therefore, the main objective of this work is to fill this gap. More exactly, in this article, water-based nanofluid is considered with Ag and Cu nanoparticles. The fractional differential equations are used to express the governing flow, heat, and species concentration equations. The Crank Nicholson technique is used to generate numerical results [54]. Variation in numerical implications of concentration, velocity, and temperature profiles is shown as a result of variations in various parameters for both Ag-based and Cu-based nanofluid values. The collected findings demonstrate the suggested technique’s accuracy, dependability, and effectiveness.

2. Mathematical and geometrical analysis

Consider the mass and energy (heat) transmission performance of a nanofluid that is unsteady free convection, incompressible, one dimensional, viscous, and radiative, and is limited among specified plates that are parallel, filled with a porous material, and have distance d. Initially (t = 0), the fluid and plates are assumed to be stationary, and T∞ and C∞ are the constant temperature and constant concentration, respectively. For t > 0, the heat transfer process and surface temperature are proportionate. The nth-order chemical reaction is taken into account. Flow may be described using the following partial differential equations in light of the Boussinesq approximation. Where u(y, t), T(y, t), C(y, t), g, ν, ρ, β, σ, (C), k are the velocity, temperature, concentration gravitational acceleration, kinematics viscosity, density, heat transfer constant, electrical conductivity, heat capacity, and thermal conductivity. , D, γ, n and ϕ are the parameters for current density, mass diffusion, rate of chemical reaction, order of chemical reaction, and porosity, respectively. Where in Eqs (1)–(4), ρ, ρ, β, β, μ, μ, σ, σ, σ, k, k and ϕ are density, the density of solid particle, heat transfer constant, heat transfer constant for solid particle, viscosity, viscosity, viscosity of solid particle, electrical conductivity, the electrical conductivity of solid, thermal conductivity, thermal conductivity for solid, where the subscripts nf and f are for nanofluid and fluid, respectively. The current density value is where E is the electric field. Cogley et al. shows that [30]: Where k, e, w are the absorption coefficient, plank function and value at the wall y = d. Substituting the values from Eqs (4–6) into Eqs (1–3), once obtained where

The associated initial and boundary conditions for Eqs (7)–(9) are: for t > 0,

To nondimensionalize the above system of PDEs let us consider the following transformations:

Using the above transformation into (7–9), once obtained

Moreover A, B, C1, D1, E, F are just constants which introduced for simplicity and is given by:

Transformed form of IC and BC’s are given as:

The Caputo time fractional form of Eqs (8) and (9) are explained as follow also replacing and by t, y, u, T and C: where .

3. Finite difference scheme

Crank Nicolson method (CNM) is projected to construct the numerical solution of problem (13–15) in this section. Consider the problem (13–15) for n = 1:

The boundary condition associated with above system given in above. In above 0 ≤ α ≤ 1 is Caputo derivative of fractional order. Consider that the above fractional-order system (16)–(18) has sufficiently smooth and has unique. Assume that x = jh, 0 ≤ j ≤ M with Mh = 1 and t = n, 0 ≤ n ≤ N. Here h and τ indicate the space and time step length, M and N are represents the number of grids point. Fractional order derivate can discretize as [34]: and the second-order derivative using Crank-Nicholson idea can be discretized as under:

Using the above-discretized formulas, system (16)–(18) takes the following form: where for n ≥ 1,

In above and b are represents the block matrices which are defined as follow: where the matrices and present in [33,34] and u and T are given as:

4. Discussion about numerical outcomes

The parametric study is provided to investigate the physics of the problem described in the preceding section. The fractional finite difference technique is used to find a numerical solution. Figs 1–11 show dimensionless velocity, temperature, and concentration plotted against the change of various parameters listed in Table 1.

Fig 1

Variation in u(y, t) against M for Pr = 6.2, Gr = 0.9, N = 0.9, ϕ = 0.2, R = 0.9.

Fig 11

Behavior of C(y, t) against t for δ = 0.9 Le = 2.

Table 1

The expressions of parameters used in Eqs (10)–(12) are given in tabular form.

Parameter	Expression	Parameter	Expression
Grashof number	Gr=gβ_fT_w-T_∞_d^3νf2	Prandtl number	Pr=μ_fc_P_fk_f
Hartmann number	M^2=σB02d^2μ_f	Radiation	R=4Id^2k_f
Porosity	N=d^2ϕ_mk_f	Lewis number	Le=ν_fD
Reaction rate	δ=γd^2C_w-C_∞^n-1ν_f	Fractional	α

Figs 1–5 depict the behaviour of velocity for water based nanofluid (with Prandtl number Pr = 6.2) containing copper (Cu) and silver (Ag) nanoparticles) for various values of fractional parameter α, as well as Hartmann number (M), porosity parameter (N), Grashof number (Gr), time (t) and solid volume fraction (ϕ) [43]. At time t = 0.3, the decreasing behaviour of velocity for Hartmann number (M2) and fractional parameter α is shown in Fig 1.

Fig 5

Variation in u(y, t) against t for Gr = 0.5 R = 0.5, M = 2, N = 0.9, ϕ = 0.2, Pr = 6.2.

M arises in the problem as a significance of substantial magnetic field effects, as is well known. As a consequence, magnetic forces working against the flow process become stronger, resulting in a decrease in velocity. Normally, this parameter causes the temperature to rise and the collision process to accelerate, which has a noticeable influence on the velocity, as seen in Fig 1. The fractional parameter stores the time evaluation values of velocity fluids, indicating that the velocity is steadily decreasing and nearing the fractional parameter’s integer value. As a result, the fractional parameter traces the location of the fluid particles.

Fig 2 illustrates the effect of changing numerical values N and on velocity at t = 0.3. As the value of the porosity parameter is increased, the velocity impact diminishes (N). Increased porosity implies an increase in the degree of resistance. That is why the velocity of nanofluid decreases as the porosity parameter increases (N).

Fig 2

Variation in u(y, t) against N for Pr = 6.2, Gr = 0.9, M = 0.9, ϕ = 0.2, R = 0.9.

The effect of Gr on flow velocity is seen in Fig 3, which demonstrates that velocity increases as the size of Gr increases. As seen in the preceding section, the Grashof number is inversely related to the viscosity μ. Increased values of the Grashof number indicate that the viscosity is reducing, which explains why the velocity function is growing. Generally, an increase in any buoyancy-related parameter such as the Grashof number in the present case, causes an increase in the wall temperature, which weakens the bond(s) between the fluids, reduces internal friction pressure, and makes gravity stronger (i.e. makes the specific weight appreciably different between the immediate fluid layers adjacent to the wall). For detailed analysis of Grashof number and its effect on the fluid motion, three different scenarios (transport phenomenon) are discussed, namely: when (a) Grashof number is greater than 1, (b) Grashof number is less than 1, and (c) Grashof number is small (Gr = 0.01). In this first case, it is observed that velocity increases with increasing value of temperature-dependent viscosity parameter when Grashof number is greater than 1. In the second case, the observation showed that velocity decreases with increasing values of temperature-dependent viscosity parameter when Grashof number is less than 1. The third case examines the flow situation when the when Grashof number is moderately small i.e. Gr = 0.01. The above observation shows that the Grashof number in convection flow plays and important role (Fig 3) [44,45].

Fig 3

Variation in u(y, t) against Gr for Pr = 6.2 R = 0.9, M = 2, N = 0.2, ϕ = 0.2.

Volume fraction is often used in solid materials science and engineering to refer to the concentration of a given phase, that is, the ratio of the volume of the particular phase to the total volume of the sample. Here in our study, ϕ shows the solid volume fraction of nanofluid. Additionally, Fig 4 has been displayed for various numerical values. As seen in Fig 4, velocity decreases as the numerical value of for nanofluid increases. In Fig 5, the velocity behaviour for changing t has been illustrated, demonstrating that flow velocity steadily increases with time.

Fig 4

Variation in u(y, t) against R for Pr = 6.2 R = 0.5, Gr = 0.5, M = 2, N = 0.9.

The temperature performance of nanofluids (based on copper (Cu) and silver (Ag)) has been presented in Figs 6–9 for various numerical parameter values and fractional parameter values. Fig 6 demonstrates that the temperature of the nanofluid decreases as the radiation effects N increase, indicating that the nanofluid is radiative in nature and radiates energy. As a result of this, radiation (in the form of electromagnetic waves) wastes energy, reducing the thickness of the thermal boundary layer and therefore the temperature. The temperature increases as the value of the solid volume fraction (ϕ) disclosed in Fig 7 which is increased. In Fig 8, a similar response of the temperature of the nanofluid was seen when the numerical values t and α were changed. Moreover, as seen in Fig 8, nano fluids based on silver have a greater temperature than nano fluids based on copper.

Fig 6

Behavior of T(y, t) against R for Pr = 6.2 ϕ = 0.1.

Fig 9

Behavior of C(y, t) against Le for δ = 0.9.

Fig 7

Behavior of T(y, t) against ϕ for Pr = 6.2.

Fig 8

Variation in T(y, t) against t for R = 0.1 ϕ = 0.1, Pr = 6.2.

Enactment of concentration of solute is presented on Figs 9–11 for diverse values of parameters. Figs 9 and 10 show the performance of concentration of solute for upsurging numerical values of Lewis number (Le) and δ. Fig 9 shows that concentration and concentration boundary layer thickness is decreeing as we are increasing the magnitude of Le. The motive behind is that when the Le increases, the diffusion process decreases because of the inversely proportional relationship between Le and diffusion. As the diffusion process decreases, the concentration of solute is also decreased. On the other hand, the response rate parameter exhibits the inverse relationship. In this situation, we can readily assert that the rate of chemical reaction rises as the reaction rate parameter increases. This becomes the cause for the growing behaviour of the solute’s concentration. In Fig 10, the concentration of solute increases with increasing δ and decreases with increasing fractional parameter. In the last Fig 11, the concentration of solute increases with the passage of time and with the decreasing values of the fractional parameter (α).

Fig 10

Variation in C(y, t) against δ for Le = 1.

5. Conclusion

The viscous, incompressible, and convection-free fluidic flow near an isothermal vertical plate is theoretically investigated in this article. The plate’s velocity varies with time, and its motion is translational. The fractional differential equations are used to express the governing flow equations (FDEs). A mixture of finite difference and Crank Nicolson techniques is used to find numerical solutions for solute concentration, velocity, and temperature. As a result, the following is the main summary of our research:

The outlines of temperature, velocity, and concentration reduced, while the total numerical values of parameter α decreased.

Values for velocity and temperature for the case of Ag-based nanofluid is more than Cu-based nanofluid

Ordinary fluid flow is more leisurely than fractional fluid flow.

The velocity, temperature, and concentration profiles all showed declining behaviour as time passed.

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5 Oct 2021

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18 Oct 2021

Response to reviewer’s comments

Manuscript ID: PONE-D-21-31025

Journal Name: PLOS ONE

Dear Editor-in-Chief,

Thank you for your useful comments on our manuscript. We have considered your additional editorial comments and made the following changes in our paper entitled “Crank Nicholson Scheme to Examine the Fractional-order Unsteady Nanofluid Flow of Free Convection of Viscous Fluids". We have modified the manuscript accordingly, and detailed corrections are listed below point by point. All the changes of are highlighted in red colour.

Additional Editor Comments & Response:

1. Strengthen the mathematical modelling of the problem.

Response: Geometry of the problem has been added to fix the query.

2. Update the introduction with relevant references only. Also arrange the references in proper format.

Response: Introduction section has been updated according to the instructions. The format of the references has been improved.

3. Validate the results.

Response: Graphical results and further analysis on the base of presented figures has been added.

With Regards,

Tamour Zubair

Submitted filename: Response Letter.docx

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25 Oct 2021

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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 manuscript presents the outcome of a research study on the fractional-order of unsteady

nanofluid flow due to free convection using Crank Nicholson integration scheme. The contribution of the report to the body of knowledge is significant and novel. Also, the aim and objectives of the study are within the scope of PLOS ONE. However, the present form of the report needs revision. The author should consider the following points:

Q1. The title is wrongly written. It was written, "Crank Nicholson Scheme to Examine the Fractional-order Unsteady

Nanofluid Flow of Free Convection of Viscous Fluids". Meanwhile, the report actually presents, "fractional-order of unsteady

nanofluid flow due to free convection using Crank Nicholson integration scheme"

Comment: Revise the title or consider, "Analysis of fractional-order of unsteady nanofluid flow due to free convection using Crank Nicholson integration scheme."

Q2. An effective Abstract is a global paragraph that prepares the reader for the rest of the document. The present form of the abstract is not acceptable because it is not appropriate.

Comment 1: Use

(1) one sentence to present the significance of the study,

(2) one sentence to present the aim of the study,

(3) one sentence to present the research methodology, and

(4) two sentences to present the conclusion drawn from the study.

Comment 2: Do not use the pronoun, "we" in a scientific report like this. Also, the first sentence of the abstract should focus on the significance of the study on, "Analysis of fractional-order of unsteady nanofluid flow due to free convection using Crank Nicholson integration scheme"

Q3. The present form of the introduction lacks structure and strongly rejected. It is important to release two comments which are

Comment 1: Writing is constructed by putting sentences in sequence, one after another. Meaning should flow from one sentence to the next, carrying the argument or point of view forward clearly and concisely.

Comment 2: Use the style mentioned above to revise not only the introduction section but also the analysis and discussion of results. This is necessary because some paragraphs are somehow scanty and not structure.

Comment 3:

After reading the introduction section, it is worth noticing that the present form of the theoretical and empirical reviews seems okay but not connecting related facts to open a gap which this report intends to fill.

Comment 4: Revise. The authors may check the following videos for insight: https://youtu.be/ugFJnflnsF0

https://youtu.be/1A27y6eUsqA

https://youtu.be/5GiT-N9Y_Ig

Q4. Revise the introduction. Update with scientific facts like:

Q4a) The manuscript lacks important facts on the choice of the value for Prandtl number.

Comment: Update the manuscript with the fact that "…the appropriate Prandtl number for (a) Methanol based nanofluid is 7.3786, (b) water-based nanofluid is 6.1723, (c) blood-based nanofluid is 22.9540, and Ethylene Glycol based nanofluid is 150.46."

Comment: Cite the Source - Physica Scripta 95(9), 095205. https://doi.org/10.1088/1402-4896/aba8c6

Q5. Update the discussion of the results illustrated as Fig. 13 and Fig. 14 with the fact that

a) An increase in the Grashof number or some other buoyancy-related parameter means an increase in the wall temperature, which weakens the bond(s) between the fluids, reduces internal friction pressure, and makes gravity stronger (i.e. makes the specific weight appreciably different between the immediate fluid layers adjacent to the wall).

You are expected to revise the fact above using your group of words.

Cite the source: Journal of Molecular Liquids 249, 980 – 990, 2018. https://doi.org/10.1016/j.molliq.2017.11.042

Q6. The analysis of the effect of Grashof number is not logical and insightful. Permit me to remind you of heating the surface and cooling the surface.

Comment: Do you know that there are three different implications of increasing Grashof number proportional to the buoyancy force? Could you try to extend your simulation to check what will happen to the transport phenomenon when (a) Grashof number is greater than 1, (b) Grashof number is less than 1, and (c) Grashof number is small like 0.01. Relate your observation(s) with the first conclusion (i.e. Velocity increases with increasing value of temperature-dependent viscosity parameter when Grashof number is greater than 1; velocity decreases with increasing values of temperature-dependent viscosity parameter when Grashof number is less than 1. The two effects exist when Grashof number is moderately small) reached in page 117 of Open Journal of Fluid Dynamics, 2015, 5, 106-120. http://dx.doi.org/10.4236/ojfd.2015.52013.

Q7. Section three is loaded with new results that are not clear. Nothing is known on the direction of thought. Research Questions are needed at the end of the introduction to harmonize the findings of the study.

Comment 1: The video below may be useful to guide the authors

https://youtu.be/cn4iRvtwf8M

Comment 2: 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 a 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. Samples of Research Questions can be found in the following reports

A. https://doi.org/10.1038/s41598-021-81417-y

B. https://doi.org/10.1007/s10973-021-10550-7

Reviewer #2:

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9 Dec 2021

Response to Reviewer’s Comments

Manuscript ID: PONE-D-21-31025R1

Journal Name: Plos One

Dear Editor-in-Chief,

Thank you for your useful comments on our manuscript. We have considered your editorial or reviewer's comments and made the following changes in our paper entitled “Analysis of Fractional-order of Unsteady Nanofluid Flow Due to Free Convection Using Crank Nicholson Integration Scheme ”. We have modified the manuscript accordingly, and detailed corrections are listed below point by point. All the changes of referee-I and referee-II is highlighted in red and purple colors respectively.

Comments from Reviewers-I & Response:

1. The title is wrongly written. It was written, "Crank Nicholson Scheme to Examine the Fractional-order Unsteady Nanofluid Flow of Free Convection of Viscous Fluids". Meanwhile, the report actually presents, "fractional-order of unsteady nanofluid flow due to free convection using Crank Nicholson integration scheme"

Comment: Revise the title or consider, "Analysis of fractional-order of unsteady nanofluid flow due to free convection using Crank Nicholson integration scheme."

Response: Revised as suggested.

2. An effective Abstract is a global paragraph that prepares the reader for the rest of the document. The present form of the abstract is not acceptable because it is not appropriate.

Comment 1: Use (1) one sentence to present the significance of the study, (2) one sentence to present the aim of the study, (3) one sentence to present the research methodology, and (4) two sentences to present the conclusion drawn from the study.

Comment 2: Do not use the pronoun, "we" in a scientific report like this. Also, the first sentence of the abstract should focus on the significance of the study on, "Analysis of fractional-order of unsteady nanofluid flow due to free convection using Crank Nicholson integration scheme"

Response: Revised as suggested.

3. The present form of the introduction lacks structure and strongly rejected. It is important to release two comments which are

Comment 1: Writing is constructed by putting sentences in sequence, one after another. Meaning should flow from one sentence to the next, carrying the argument or point of view forward clearly and concisely.

Comment 2: Use the style mentioned above to revise not only the introduction section but also the analysis and discussion of results. This is necessary because some paragraphs are somehow scanty and not structure.

Comment 3: After reading the introduction section, it is worth noticing that the present form of the theoretical and empirical reviews seems okay but not connecting related facts to open a gap which this report intends to fill.

Comment 4: Revise. The authors may check the following videos for insight: https://youtu.be/ugFJnflnsF0 https://youtu.be/1A27y6eUsqA https://youtu.be/5GiT-N9Y_Ig

Response: Response: Revised as suggested. All the four comments are included.

4. Revise the introduction. Update with scientific facts like:

a) The manuscript lacks important facts on the choice of the value for Prandtl number.

Comment: Update the manuscript with the fact that "…the appropriate Prandtl number for (a) Methanol based nanofluid is 7.3786, (b) water-based nanofluid is 6.1723, (c) blood-based nanofluid is 22.9540, and Ethylene Glycol based nanofluid is 150.46."

Comment: Cite the Source - Physica Scripta 95(9), 095205. https://doi.org/10.1088/1402-4896/aba8c6

Response: As we have considered viscous fluid model, therefore, the base fluid is limited to water only with Prandtl number as 6.2. This correction is done in all figures, and the figure for variation of Prandtl number is deleted. All calculations are now done for water-based nanofluid with Pr=6.2, with its fixed value. The suggested reference is also included in the revised manuscript, see Ref. [44].

5. Update the discussion of the results illustrated as Fig. 13 and Fig. 14 with the fact that a) An increase in the Grashof number or some other buoyancy-related parameter means an increase in the wall temperature, which weakens the bond(s) between the fluids, reduces internal friction pressure, and makes gravity stronger (i.e. makes the specific weight appreciably different between the immediate fluid layers adjacent to the wall). You are expected to revise the fact above using your group of words. Cite the source: Journal of Molecular Liquids 249, 980 – 990, 2018. https://doi.org/10.1016/j.molliq.2017.11.042

Response: Dear referee, in this manuscript we don’t have Fig. 13 and Fig. 14, however, the suggested correction is done in Fig. 3, as this include the Grashof number related information. The suggested reference is also included in the revised manuscript, see Ref. [45].

6. The analysis of the effect of Grashof number is not logical and insightful. Permit me to remind you of heating the surface and cooling the surface.

Comment: Do you know that there are three different implications of increasing Grashof number proportional to the buoyancy force? Could you try to extend your simulation to check what will happen to the transport phenomenon when (a) Grashof number is greater than 1, (b) Grashof number is less than 1, and (c) Grashof number is small like 0.01. Relate your observation(s) with the first conclusion (i.e. Velocity increases with increasing value of temperature-dependent viscosity parameter when Grashof number is greater than 1; velocity decreases with increasing values of temperature-dependent viscosity parameter when Grashof number is less than 1. The two effects exist when Grashof number is moderately small) reached in page 117 of Open Journal of Fluid Dynamics, 2015, 5, 106-120. http://dx.doi.org/10.4236/ojfd.2015.52013.

Response: Dear referee, the above suggestions are included, and the authors are thankful for such a useful suggestion. The manuscript is revised accordingly, please refer to the discussion of Fig, 3. The suggested reference is also included in the revised manuscript, see Ref. [46].

7. Section three is loaded with new results that are not clear. Nothing is known on the direction of thought. Research Questions are needed at the end of the introduction to harmonize the findings of the study.

Comment 1: The video below may be useful to guide the authors https://youtu.be/cn4iRvtwf8M Comment 2: 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 a 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. Samples of Research Questions can be found in the following reports A. https://doi.org/10.1038/s41598-021-81417-y B. https://doi.org/10.1007/s10973-021-10550-7

Response: The manuscript is revised accordingly. The two suggested report are also included please refer to Res. [47] and [48].

Comments from Reviewers-II & Response:

1. Write at least two sentences that highlights the practical application of this research.

Response: Our study is very significant and is applicable in different area of science as follow.

• It is applicable in antibacterial study.

• Effects of shape control is applicable in inorganic materials.

• The study of nano-particles in term are applicable in biomedical.

All the relevant applications (along with references) of our study are updated in the paper and suitable references are also added.

2. Paper should be carefully revised for punctuation, grammar, space, and spelling mistakes. There are number of such kind of mistakes that must be tackled.

Response: The manuscript revised carefully and all mistakes are removed.

3. Nomenclature can help the readers to understand the number of variables used in the article. So add a nomenclature with the units.

Response: Nomenclature is added.

4. Physical reasoning is missing in most of figures. Write in detail the true physical reason for increasing or decreasing.

Response: Result and discussion section has been improved with physical reasons.

5. The introduction section needs to be carefully revised. It is suggested that most related articles should be cited. The following articles can help you to improve your introduction section. https://doi.org/10.1016/j.icheatmasstransfer.2021.105563 https://doi.org/10.1177/09544089211043605 https://doi.org/10.1002/htj.22280 https://doi.org/10.1002/htj.21947 https://doi.org/10.1007/s10973-020-09943-x https://doi.org/10.1080/01430750.2020.1831593 https://doi.org/10.1038/s41598-020-61215-8

Response: The references have been cited at suitable positions.

6. The abstract should be updated with fruitful outcomes of the problem.

Response: The abstract has been improved.

7. All governing equations need to be referenced properly.

Response: All governing equations have been cited accordingly.

8. Some unclear words are used in the paper rectify it.

Response: Done as suggested.

With Regards,

Tamour Zubair

Submitted filename: Response Letter.docx

Click here for additional data file.

13 Dec 2021

Crank Nicholson Scheme to Examine the Fractional-order Unsteady Nanofluid Flow of Free Convection of Viscous Fluids

PONE-D-21-31025R2

Dear Dr. Zubair,

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.

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Reviewers' comments:

Reviewer's Responses to Questions

6. Review Comments to the Author

Reviewer #1: Checking through the revised version, it is worth mentioning that

a. the manuscript contains an interesting and novel aim,

b. the title is informative and relevant,

c. the introduction, literature review, methodology, results, discussion of results, conclusion and references are of high standard,

d. Author(s) have rigorously revised the manuscript. The present form of the whole report is also of high standard, and

e. the contribution of the report to the body of knowledge is significant.

Based on these aforementioned facts, it is worth concluding that the article is error free and suitable for publication. I hereby recommend "Acceptance".

7 Jan 2022

PONE-D-21-31025R2

Crank Nicholson Scheme to Examine the Fractional-order  Unsteady Nanofluid Flow of Free Convection of Viscous Fluids

Dear Dr. Zubair:

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.

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on behalf of

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Academic Editor

PLOS ONE