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Heat transfer analysis of fractional second-grade fluid subject to Newtonian heating with Caputo and Caputo-Fabrizio fractional derivatives: A comparison

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Abstract.

The present study is a comparative analysis of unsteady flows of a second-grade fluid with Newtonian heating and time-fractional derivatives, namely, the Caputo fractional derivative (singular kernel) and the Caputo-Fabrizio fractional derivative (non-singular kernel). A physical model for second-grade fluids is developed with fractional derivatives. The expressions for temperature and velocity fields in dimensionless form as well as rates of heat transfer are determined by means of the Laplace transform technique. Solutions for ordinary cases corresponding to integer order derivatives are also obtained. Numerical computations for a comparison between the solutions of the problem with the Caputo time-fractional derivative, problem with Caputo-Fabrizio time-fractional derivative and of the ordinary fluid problem were made. The influence of some flow parameters and fractional parameter \(\alpha\) on temperature field as well as velocity field was presented graphically and in tabular forms.

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References

  1. B.D. Coleman, W. Noll, Arch. Ration. Mech. Anal. 6, 355 (1960)

    Article  Google Scholar 

  2. M. Emin Erdogan, C. Erdem Imrak, Appl. Math. Model. 31, 170 (2007)

    Article  Google Scholar 

  3. T. Hayat, S. Asghar, A.M. Siddiqui, Int. J. Eng. Sci. 38, 337 (2000)

    Article  Google Scholar 

  4. C. Fetecau, J. Zierep, Acta Mech. 150, 135 (2001)

    Article  Google Scholar 

  5. M. Massoudi, T.X. Phouc, Acta Mech. 150, 23 (2001)

    Article  Google Scholar 

  6. A.M. Benharbit, A.M. Siddiqui, Acta Mech. 94, 85 (1992)

    Article  MathSciNet  Google Scholar 

  7. J.E. Dunn, K.R. Rajagopal, Int. J. Eng. Sci. 33, 689 (1995)

    Article  Google Scholar 

  8. D. Vieru, M.A. Imran, A. Rauf, Heat Transf. Res. 46, 713 (2015)

    Article  Google Scholar 

  9. W.C. Tan, F. Xian, L. Wei, China Sci. Bull. 47, 1226 (2002)

    Article  Google Scholar 

  10. M. Khan, S. Nadeem, T. Hayat, A.M. Siddiqui, Math. Comput. Model. 41, 629 (2005)

    Article  Google Scholar 

  11. T. Hayat, Z. Abbas, Chaos, Solitons Fractals 38, 556 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  12. T. Hayat, I. Khan, R. Ellahi, C. Fetecau, J. Porous Medium 11, 389 (2008)

    Article  Google Scholar 

  13. N. Mustafa, S. Asghar, M.A. Hossain, Chem. Eng. Commun. 195, 209 (2007)

    Article  Google Scholar 

  14. D. Vieru, Corina Fetecau, C. Fetecau, Niat Nigar, Z. Naturforsch. 69a, 714 (2014)

    ADS  Google Scholar 

  15. M. Khan, S. Hyder Ali, Q. Haitao, Math. Comput. Model. 49, 1519 (2009)

    Article  Google Scholar 

  16. D. Vieru, C. Fetecau, Corina Fetecau, Thermal Sci. 19, S85 (2015)

    Article  Google Scholar 

  17. M.A. Imran, I. Khan, M. Ahmad, N.A. Shah, M. Nazar, J. Mol. Liq. 229, 67 (2017)

    Article  Google Scholar 

  18. M. Caputo, M. Fabrizio, Progr. Fract. Differ. Appl. 1, 73 (2015)

    Google Scholar 

  19. A.A Zafar, C. Fetecau, Alex. Eng. J. 55, 2789 (2016)

    Article  Google Scholar 

  20. N.A. Shah, I. Khan, Eur. Phys. J. C 76, 111 (2016)

    Article  Google Scholar 

  21. N.A. Sheikh, F. Ali, I. Khan, M. Saqib, Neural Comput. Appl. (2016) DOI:10.1007/s00521-016-2815-5

  22. N.A. Sheikh, F. Ali, M. Saqib, I. Khan, S.A.A. Jan, A.S. Alshomrani, M.S. Alghamdi, Results Phys. 7, 789 (2017)

    Article  ADS  Google Scholar 

  23. J. Losada, J.J. Nieto, Progr. Fract. Differ. Appl. 1, 87 (2015)

    Google Scholar 

  24. A. Atangana, Appl. Math. Comput. 273, 948 (2016)

    MathSciNet  Google Scholar 

  25. A. Atangana, J.J. Nieto, Adv. Mech. Eng. (2015) DOI:10.1177/1687814015613758

  26. M. Caputo, M. Fabrizio, Progr. Fract. Differ. Appl. 2, 1 (2016)

    Article  Google Scholar 

  27. F. Ali, M. Saqib, I. Khan, N.A. Sheikh, Eur. Phys. J. Plus 131, 377 (2016)

    Article  Google Scholar 

  28. A. Atanganaa, I. Kocab, J. Nonlinear Sci. Appl. 9, 2467 (2016)

    MathSciNet  Google Scholar 

  29. M. Narahari, B.K. Dutta, Chem. Eng. Commun. 199, 628 (2012)

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Mathematics, University of Management and Technology Lahore, Lahore, Pakistan

    Muhammad Imran Asjad & Maryam Aleem

  2. Abdus Salam School of Mathematical Sciences, GC University Lahore, Lahore, Pakistan

    Nehad Ali Shah

  3. Basic Sciences Department, College of Engineering Majmaah University, P.O. Box 66, 11952, Majmaah, Saudi Arabia

    Ilyas Khan

Authors
  1. Muhammad Imran Asjad
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  2. Nehad Ali Shah
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  3. Maryam Aleem
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  4. Ilyas Khan
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Correspondence to Muhammad Imran Asjad.

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Asjad, M.I., Shah, N.A., Aleem, M. et al. Heat transfer analysis of fractional second-grade fluid subject to Newtonian heating with Caputo and Caputo-Fabrizio fractional derivatives: A comparison. Eur. Phys. J. Plus 132, 340 (2017). https://doi.org/10.1140/epjp/i2017-11606-6

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  • DOI: https://doi.org/10.1140/epjp/i2017-11606-6

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