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New Direction of Atangana–Baleanu Fractional Derivative with Mittag-Leffler Kernel for Non-Newtonian Channel Flow

  • Muhammad Saqib
  • Ilyas KhanEmail author
  • Sharidan Shafie
Chapter
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 194)

Abstract

This book chapter higlights a new direction of Atangana–Baleanu fractional derivative to channel flow of non-Newtonian fluids. Because the idea to apply fractional derivatives with Mittag-Leffler kernel is a quite new direction for non-Newtonian fluids when flow is in a parallel plate channel. This new and inreresting fractional derivative launched by Atangana and Baleanu with a new fractional operator namely, Atangana–Baleanu fractional operator with Mittag-Leffler function as the kernel of integration has attracted the interest of the researchers. Because this new operator is an efficient tool to model complex and real-world problems. Therefore, this chapter deals with modeling and solution of generalized magnetohydrodynamic (MHD) flow of Casson fluid in a microchannel. The microchannel is taken of infinite length in the vertical direction and of finite width in the horizontal direction. The flow is modeled in terms of a set of partial differential equations involving Atangana–Baleanu time fractional operator with physical initial and boundary conditions. The partial differential equations are transformed to ordinary differential equations via fractional Laplace transformation and solved for exact solutions. To explore the physical significance of various pertinent parameters, the solutions are numerically computed and plotted in different graphs with a physical explanation. The results obtained here may have useful industrial and engineering applications.

Keywords

Fractional calculus Atangana–Baleanu fractional derivative Non-Newtonian fluid 

Notes

Acknowledgements

The authors would like to acknowledge Ministry of Higher Education (MOHE) and Research Management Centre-UTM, Universiti Teknologi Malaysia UTM for the financial support through vote numbers 15H80 and 13H74 for this research.

References

  1. 1.
    Ali, F., Saqib, M., Khan, I., Sheikh, N.A.: Application of Caputo-Fabrizio derivatives to MHD free convection flow of generalized Walters’-B fluid model. Eur. Phys. J. Plus 131(10), 1–10 (2016)Google Scholar
  2. 2.
    Ali, F., Sheikh, N.A., Khan, I., Saqib, M.: Solutions with Wright function for time fractional free convection flow of Casson fluid. Arab. J. Sci. Eng. 42(6), 2565–2572 (2017)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Gómez-Aguilar, J.F., Torres, L., Yépez-Martínez, H., Baleanu, D., Reyes, J.M., Sosa, I.O.: Fractional Liénard type model of a pipeline within the fractional derivative without singular kernel. Adv. Differ. Equ. 2016(1), 1–17 (2016)zbMATHGoogle Scholar
  4. 4.
    Gómez-Aguilar, J.F., Dumitru, B.: Fractional transmission line with losses. Z. für Naturforschung A 69(10–11), 539–546 (2014)Google Scholar
  5. 5.
    Gómez-Aguilar, J.F., Yépez-Martínez, H., Escobar-Jiménez, R.F., Astorga-Zaragoza, C.M., Reyes-Reyes, J.: Analytical and numerical solutions of electrical circuits described by fractional derivatives. Appl. Math. Model. 40(21–22), 9079–9094 (2016)MathSciNetGoogle Scholar
  6. 6.
    Morales-Delgado, V.F., Taneco-Hernández, M.A., Gómez-Aguilar, J.F.: On the solutions of fractional order of evolution equations. Eur. Phys. J. Plus 132(1), 1–17 (2017)Google Scholar
  7. 7.
    Alegría-Zamudio, M., Escobar-Jiménez, R.F., Gómez-Aguilar, J.F.: Fault tolerant system based on non-integers order observers: application in a heat exchanger. ISA Trans. 80, 286–296 (2018)Google Scholar
  8. 8.
    Ali, F., Sheikh, N.A., Khan, I., Saqib, M.: Magnetic field effect on blood flow of Casson fluid in axisymmetric cylindrical tube: a fractional model. J. Magn. Magn. Mater. 423, 327–336 (2017)Google Scholar
  9. 9.
    Cuahutenango-Barro, B., Taneco-Hernández, M.A., Gómez-Aguilar, J.F.: On the solutions of fractional-time wave equation with memory effect involving operators with regular kernel. Chaos Solitons Fractals 115, 283–299 (2018)MathSciNetGoogle Scholar
  10. 10.
    Coronel-Escamilla, A., Gómez-Aguilar, J.F., Baleanu, D., Córdova-Fraga, T., Escobar-Jiménez, R.F., Olivares-Peregrino, V.H., Qurashi, M.M.A.: Bateman-Feshbach tikochinsky and Caldirola-Kanai oscillators with new fractional differentiation. Entropy 19(2), 1–21 (2017)Google Scholar
  11. 11.
    Dalir, M., Bashour, M.: Applications of fractional calculus. Appl. Math. Sci. 4(21), 1021–1032 (2010)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Coronel-Escamilla, A., Gómez-Aguilar, J.F., Baleanu, D., Escobar-Jiménez, R.F., Olivares-Peregrino, V.H., Abundez-Pliego, A.: Formulation of Euler-Lagrange and Hamilton equations involving fractional operators with regular kernel. Adv. Differ. Equ. 2016(1), 1–21 (2016)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Gómez-Aguilar, J.F., Yépez-Martínez, H., Escobar-Jiménez, R.F., Astorga-Zaragoza, C.M., Morales-Mendoza, L.J., González-Lee, M.: Universal character of the fractional space-time electromagnetic waves in dielectric media. J. Electromagn. Waves Appl. 29(6), 727–740 (2015)Google Scholar
  14. 14.
    Singh, J., Kumar, D., Hammouch, Z., Atangana, A.: A fractional epidemiological model for computer viruses pertaining to a new fractional derivative. Appl. Math. Comput. 316, 504–515 (2018)MathSciNetGoogle Scholar
  15. 15.
    Atangana, A., Owolabi, K.M.: New numerical approach for fractional differential equations. Math. Model. Nat. Phenom. 13(1), 1–13 (2018)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Gómez-Aguilar, J.F., Atangana, A.: New insight in fractional differentiation: power, exponential decay and Mittag-Leffler laws and applications. Eur. Phys. J. Plus 132(1), 1–19 (2017)Google Scholar
  17. 17.
    Atangana, A., Gómez-Aguilar, J.F.: Fractional derivatives with no-index law property: application to chaos and statistics. Chaos Solitons Fractals 114, 516–535 (2018)MathSciNetGoogle Scholar
  18. 18.
    Coronel-Escamilla, A., Gómez-Aguilar, J.F., López-López, M.G., Alvarado-Martínez, V.M., Guerrero-Ramírez, G.V.: Triple pendulum model involving fractional derivatives with different kernels. Chaos Solitons Fractals 91, 248–261 (2016)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Caputo, M.: Linear models of dissipation whose Q is almost frequency independent. Ann. Geophys. 19(4), 383–393 (1966)MathSciNetGoogle Scholar
  20. 20.
    Caputo, M.: Linear models of dissipation whose Q is almost frequency independent–II. Geophys. J. Int. 13(5), 529–539 (1967)Google Scholar
  21. 21.
    Atangana, A., Gómez-Aguilar, J.F.: Numerical approximation of Riemann-Liouville definition of fractional derivative: from Riemann-Liouville to Atangana-Baleanu. Numer. Methods Part. Differ. Equ. 34(5), 1502–1523 (2018)MathSciNetGoogle Scholar
  22. 22.
    Bakkyaraj, T., Sahadevan, R.: Invariant analysis of nonlinear fractional ordinary differential equations with Riemann-Liouville fractional derivative. Nonlinear Dyn. 80(1–2), 447–455 (2015)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Sousa, E., Li, C.: A weighted finite difference method for the fractional diffusion equation based on the Riemann-Liouville derivative. Appl. Numer. Math. 90, 22–37 (2015)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Saqib, M., Ali, F., Khan, I., Sheikh, N.A., Jan, S.A.A.: Exact solutions for free convection flow of generalized Jeffrey fluid: a Caputo-Fabrizio fractional model. Alex. Eng. J. 1, 1–10 (2017)Google Scholar
  25. 25.
    Sheikh, N.A., Ali, F., Khan, I., Saqib, M.: A modern approach of Caputo-Fabrizio time-fractional derivative to MHD free convection flow of generalized second-grade fluid in a porous medium. Neural Comput. Appl. 30(6), 1865–1875 (2018)Google Scholar
  26. 26.
    Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Progr. Fract. Differ. Appl 1(2), 1–13 (2015)Google Scholar
  27. 27.
    Atangana, A., Gómez-Aguilar, J.F.: Hyperchaotic behaviour obtained via a nonlocal operator with exponential decay and Mittag-Leffler laws. Chaos Solitons Fractals 102, 285–294 (2017)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Atangana, A., Gómez-Aguilar, J.F.: A new derivative with normal distribution kernel: theory, methods and applications. Phys. A Stat. Mech. Appl. 476, 1–14 (2017)MathSciNetGoogle Scholar
  29. 29.
    Atangana, A., Baleanu, D.: New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Therm. Sci. 20(2), 763–769 (2016)Google Scholar
  30. 30.
    Al-Salti, FAMN, Karimov E., Initial and Boundary Value Problems for Fractional differential equations involving Atangana-Baleanu Derivative. arXiv:1706.00740, 2017
  31. 31.
    Atangana, A., Alqahtani, R.T.: Modelling the spread of river blindness disease via the caputo fractional derivative and the beta-derivative. Entropy 18(2), 1–18 (2016)Google Scholar
  32. 32.
    Atangana, A., Baleanu, D.: Caputo-Fabrizio derivative applied to groundwater flow within confined aquifer. J. Eng. Mech. 143(5), 1–18 (2017)Google Scholar
  33. 33.
    Sheikh, N.A., Ali, F., Saqib, M., Khan, I., Jan, S.A.A.: A comparative study of Atangana-Baleanu and Caputo-Fabrizio fractional derivatives to the convective flow of a generalized Casson fluid. Eur. Phys. J. Plus 132(1), 1–15 (2017)Google Scholar
  34. 34.
    Sheikh, N.A., Ali, F., Saqib, M., Khan, I., Jan, S.A.A., Alshomrani, A.S., Alghamdi, M.S.: Comparison and analysis of the Atangana-Baleanu and Caputo-Fabrizio fractional derivatives for generalized Casson fluid model with heat generation and chemical reaction. Results Phys. 7, 789–800 (2017)Google Scholar
  35. 35.
    Jan, S.A.A., Ali, F., Sheikh, N.A., Khan, I., Saqib, M., Gohar, M.: Engine oil based generalized brinkman-type nano-liquid with molybdenum disulphide nanoparticles of spherical shape: Atangana-Baleanu fractional model. Numer. Methods Part. Differ. Equ. 34(5), 1472–1488 (2018)MathSciNetGoogle Scholar
  36. 36.
    Saqib, M., Khan, I., Shafie, S.: Application of Atangana-Baleanu fractional derivative to MHD channel flow of CMC-based-CNT’s nanofluid through a porous medium. Chaos Solitons Fractals 116, 79–85 (2018)MathSciNetGoogle Scholar
  37. 37.
    Narahari, M., Pendyala, R.: Exact solution of the unsteady natural convective radiating gas flow in a vertical channel. AIP Conf. Proc. 1557(1), 121–124 (2013)Google Scholar
  38. 38.
    Seth, G.S., Sharma, R., Kumbhakar, B.: Effects of Hall current on unsteady MHD convective Couette flow of heat absorbing fluid due to accelerated movement of one of the plates of the channel in a porous medium. J. Porous Media 19(1), 13–30 (2016)Google Scholar
  39. 39.
    Singh, A.K., Gholami, H.R., Soundalgekar, V.M.: Transient free convection flow between two vertical parallel plates. Heat Mass Transf. 31(5), 329–331 (1996)Google Scholar
  40. 40.
    Saqib, M., Ali, F., Khan, I., Sheikh, N.A.: Heat and mass transfer phenomena in the flow of Casson fluid over an infinite oscillating plate in the presence of first-order chemical reaction and slip effect. Neural Comput. Appl. 30(7), 2159–2172 (2018)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematical Sciences, Faculty of ScienceUniversiti Teknologi MalaysiaSkudai, Johor BahruMalaysia
  2. 2.Faculty of Mathematics and StatisticsTon Duc Thang UniversityHo Chi Minh CityVietnam

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