Analysis of a Thin Layer Formation of Third-Grade Fluid

Round 1

Reviewer 1 Report

Tareq Manzoor et al study the lubrication theory-based calculation of third-grade fluid in blade coating. Although this work demonstrated the complex lubrication theory and the solution by means of OHAM, some inconsistent and incorrect throughout exist.

Overall, this work require major revision and authors should greatly clarify this work. Please consider the following comments.

Abbreviation of OHAM is not clarified. I guess it is Optimal Homotopy Asymptotic Method. In line60, tan CE should be tan fai. Abbreviation of LAT is not clarified. in the final sentence in page 2, the definition of y-direction is not defined. Grammar should be carefully corrected. Authors do not define the s=0. Is it a boundary between coating substrate and NNF? As seen in Figure 2, it is not boundary, but there is no clear definition. Schematics in Figure 1 is not understandable. Please point which the blade is & NNF is. It is unclear which coating situations authors have considered. Does author consider the situation that the NNF thickness is too thick to consider the substrate-NNF interfacial parameter? Does this model required to consider the influence of interfacial energy? In Figure 2 & Figure 3, the perpendicular axis is incorrect. It should be u(s) not mu. Figure 3,5,7&9, authors show 6 graphs but all of them are plotted from same equations. Author should clarify these figures (e.g. summarize at most in two captions & increase the size of graphs). Readers may get confused. s,r, u(s,r) v(s,r) is not normalized but there is no unit in each variable characters. The definition of beta is unclear. In real situation, what is key factor for determining beta? In a legend in Figure 4, the variables should be italic. In Figure 3&5, Authors plot u(s) over s ranging 0 to 10-12; While that in Figure 1, 2, that over s ranging 0 to 1. Is there specific reason of this inconsistent?

Author Response

We wrote all answers to each questions.

On behalf of all authors, as one of corresponding authors, I am very thankful for reviewing our manuscript.

Author Response File: Author Response.docx

Reviewer 2 Report

In the present study, the authors investigated the surface protection layer progression of a third-grade fluid, where fluid transport within the micro passage made by the firm blade. Generally, the manuscript was very well written and organized. The figures are of high quality. In addition, the findings in this works would provide very useful information for coating engineering. Therefore, I recommend this work to be accepted.

Author Response

On behalf of all authors, as one of corresponding authors, I am very thankful for reviewing our manuscript.

Reviewer 3 Report

This paper applies OHAM to the nonlinear problem arising in coating. It is mathematically correct and suitable for publication after minor modification:   

1. English requires some improvement.   

2. OHAM is also used as optimal homotopy analysis method, this method is a direct plagiarized one from the optimal homotopy asymptotic method. A historical remark is necessary.   

3. Though this paper shows OHAM works well, but the Homotopy perturbation method with an auxiliary parameter might be more suitable for the present study.   

4. In the discussion section, the author should point out that a variational principle for a thin film equation can be established, and a fractal modification with fractal derivatives can effective describe the effect of unsmooth surface or porous surface on the coating ability.   

Author Response

1. English Requires some Improvement.

We tried our best to rectify all the grammatical mistakes and improved the text.

2. OHAM is also used as optimal homotopy analysis method, this method is a direct plagiarized one from the optimal homotopy asymptotic method. A historical remark is necessary.

Optimal Homotopy Asymptotic Method (OHAM) is actually different than Optimal Homotopy Analysis Method. This technique is developed by V. Marinca and N. Herisanu, and S. Iqbal et. Al. updated the said method for PDEs and fractional PDEs etc., one can see in ref [1-6]. This technique is rapidly convergent as compared to other techniques.

Optimal homotopy analysis method is a powerful tool for nonlinear differential equations. In this method, the convergence of the series solutions is controlled by one or more parameters which can be determined by minimizing a certain function. [1]. The optimal homotopy asymptotic method (OHAM) is capable to handle a wide variety of linear and nonlinear problems effectively.

3. Though this paper shows OHAM works well, but the Homotopy perturbation method with an auxiliary parameter might be more suitable for the present study.

The numerical results given by OHAM in are compared with the exact solutions and the solutions obtained by Adomian decomposition (ADM), variational iteration (VIM), homotopy perturbation method (HPM), and variational iteration decomposition method (VIDM). The results show that the proposed method is more effective and reliable. In , the optimal homotopy asymptotic method (OHAM) is implemented  for obtaining the approximate solution of modified Kawahara equations. The OHAM results are compared with Variational Iteration Method (VIM), Homotopy Perturbation Method (HPM) and Exact solutions. The comparison of OHAM with these methods reveals that OHAM is very effective, reliable and efficient.

4. In the discussion section the author should point out that a variational principle for a thin film equation can be established, and a fractional modification with fractal derivatives can effective describe the effect of unsmooth surface or porous surface on the coating ability.

For the model of thin film equation in equation (12), an effective tool to finding a needed variational principle for a practical problem (12), the derivation process is explained step by step in the equations (1)-(11), so that it can be easily described [7].

Fractional calculus is generalized of classical calculus. Fractional models of different real life problems, provides the best result as compared to classical models. Due to this reason, Fractional derivative technique is used to investigate the behavior of coatings. The model equation (12), first applied to investigate Brownian motion and the diffusion mode of chemical reactions, is now largely employed, in various generalized forms, in physics, chemistry, engineering and biology.

 References:

[1]- Fan, T. and You, X., 2013. Optimal homotopy analysis method for nonlinear differential equations in the boundary layer. Numerical Algorithms, 62(2), pp.337-354.

[2]  Ali, J., Islam, S., Khan, H., Shah, A. and Inayat, S., 2012. The optimal homotopy asymptotic method for the solution of higher-order boundary value problems in finite domains.In Abstract and Applied Analysis (Vol. 2012).Hindawi.

[3] Ullah, H., Nawaz, R., Islam, S., Idrees, M. and Fiza, M., 2015. The optimal homotopy asymptotic method with application to modified Kawahara equation. Journal of the Association of Arab Universities for Basic and Applied Sciences, 18, pp.82-88.

[4]  Younas, H.M., Mustahsan, M., Manzoor, T., Salamat, N. and Iqbal, S., 2019. Dynamical Study of Fokker-Planck Equations by Using Optimal Homotopy Asymptotic Method. Mathematics, 7(3), p.264.

[5] V. Marinca, N. Herisanu, Determination of periodic solutions for the motion of a particle on a rotating parabola by means of the optimal homotopy asymptotic method, J.SoundVib.329(2010)1450–1459.

[6] S. Iqbal, M. Idrees, A.M. Siddiqui, A.R. Ansari, Some solutions of the linear and nonlinear Klein–Gordon equations using the optimal homotopy asymptotic method, Appl. Math. Comput. 216 (2010) 2898–2909.

[7] He, J.H. and Sun, C., 2019. A variational principle for a thin film equation. Journal of Mathematical Chemistry, 57(9), pp.2075-2081.

Round 2

Reviewer 1 Report

Authors corrected and clarified the paper.

Author Response

Thank you for your consideration.