Some applications of extended calculus to non-Newtonian flow in pipes


Fractional and non-Newtonian calculus are an extension of classical calculus, usually known for providing new mathematical tools useful in science, developed from alternative approaches. Among fractional calculus, Riemann-Liouville and Caputo fractional derivatives have been the most popular operators employed in spite of their complexity. In this work, two novel and compact methods are presented as an alternative to the fractional calculation options. To test the feasibility of proposed methods, three classical fluid mechanic problems are studied: the flow through circular pipe, parallel plates and annulus, by modifying the constitutive equations into their fractional equivalent. On the other hand, a new weighted non-Newtonian derivative is proposed to extend the possibilities to model fluid viscosity based on the influence of nonadjacent layers, using the pipe flow as an example. Results show that proposed fractional models can describe shear-thinning and shear-thickening behaviors depending on the fractional order of the derivative, while the weighted derivative allows to expand the way viscosity is modeled, demonstrating the suitability of these approaches to describe physical problems.

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  1. 1.

    Águila B, Vasco DA, Galvez P, Zapata PA (2018) Effect of temperature and CuO-nanoparticle concentration on the thermal conductivity and viscosity of an organic phase-change material. Int J Heat Mass Transf 120:1009–1019

    Article  Google Scholar 

  2. 2.

    Barrera C, Letelier M, Siginer D, Stockle J (2016) The Graetz problem in tubes of arbitrary cross section. Acta Mech 227:3239–3246

    MathSciNet  Article  Google Scholar 

  3. 3.

    Letelier MF, Barrera C, Siginer DA, González A (2018) Elastoviscoplastic fluid flow in non-circular tubes: transversal field and interplay of elasticity and plasticity. Appl Math Model 54:768–781

    MathSciNet  Article  Google Scholar 

  4. 4.

    Sonder I, Zimanowski B, Büttner R (2006) Non-Newtonian viscosity of basaltic magma. Geophys Res Lett 33(2):L02303

    Article  Google Scholar 

  5. 5.

    Pastor M, Blanc T, Pastor M (2009) A depth-integrated viscoplastic model for dilatant saturated cohesive-frictional fluidized mixtures: application to fast catastrophic landslides. J Non-Newton Fluid Mech 158(1):142–153 (Visco-plastic fluids: from theory to application)

    Article  Google Scholar 

  6. 6.

    Leibniz G, Gerhardt K, Pertz G (1858) Leibnizens mathematische Schriften: Mathematik. [Leibnizens gesammelte Werke aus den Handschriften der Königlichen Bibliothek zu Hannover herausgewgeben von Georg Heinrich Pertz. Dritte Folge]. A. Asher & Company

  7. 7.

    Ross B (1975) A brief history and exposition of the fundamental theory of fractional calculus. In: Fractional calculus and its applications. Springer, pp 1–36

  8. 8.

    Oldham K, Spanier J (1974) The fractional calculus theory and applications of differentiation and integration to arbitrary order, vol 111. Elsevier, Amsterdam

    Google Scholar 

  9. 9.

    Liouville J (1832) Mémoire sur quelques questions de géométrie et de mécanique, et sur un nouveau genre de calcul pour résoudre ces questions

  10. 10.

    Ortigueira MD, Machado JT (2015) What is a fractional derivative? J Comput Phys 293:4–13

    MathSciNet  Article  Google Scholar 

  11. 11.

    Khalil R, Al Horani M, Yousef A, Sababheh M (2014) A new definition of fractional derivative. J Comput Appl Math 264:65–70

    MathSciNet  Article  Google Scholar 

  12. 12.

    Atangana A, Baleanu D (2016) New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. arXiv preprint arXiv:1602.03408

  13. 13.

    Delgado VM, Aguilar JG, Taneco-Hernandez M (2018) Analytical solutions of electrical circuits described by fractional conformable derivatives in Liouville-Caputo sense. AEU Int J Electron Commun 85:108–117

    Article  Google Scholar 

  14. 14.

    Atanackovic TM, Pilipovic S (2018) On a constitutive equation of heat conduction with fractional derivatives of complex order. Acta Mech 229(3):1111–1121

    MathSciNet  Article  Google Scholar 

  15. 15.

    Saadia A, Rashdi A (2018) Incorporating fractional calculus in echo-cardiographic image denoising. Comput Electr Eng 67:134–144

    Article  Google Scholar 

  16. 16.

    Magin R (2006) Fractional calculus in bioengineering. Begell House Inc. Publishers, Redding

    Google Scholar 

  17. 17.

    Sun H, Zhang Y, Baleanu D, Chen W, Chen Y (2018) A new collection of real world applications of fractional calculus in science and engineering. Commun Nonlinear Sci Numer Simul 64:213–231

    Article  Google Scholar 

  18. 18.

    Gemant A (1936) A method of analyzing experimental results obtained from elasto-viscous bodies. Physics 7(8):311–317

    Article  Google Scholar 

  19. 19.

    Grossman M, Katz R (1972) Non-Newtonian calculus: a self-contained. Elementary exposition of the authors’ investigations, non-Newtonian calculus

  20. 20.

    Filip D, Piatecki C (2014) A non-Newtonian examination of the theory of exogenous economic growth

  21. 21.

    Jasiulewicz H, Kordecki W (2016) Multiplicative parameters and estimators: applications in economics and finance. Ann Oper Res 238(1–2):299–313

    MathSciNet  Article  Google Scholar 

  22. 22.

    Aniszewska D, Rybaczuk M (2009) Fractal characteristics of defects evolution in parallel fibre reinforced composite in quasi-static process of fracture. Theor Appl Fract Mech 52(2):91–95

    Article  Google Scholar 

  23. 23.

    Rybaczuk M, Stoppel P (2000) The fractal growth of fatigue defects in materials. Int J Fract 103(1):71–94

    Article  Google Scholar 

  24. 24.

    Bashirov AE, Kurpınar EM, Özyapıcı A (2008) Multiplicative calculus and its applications. J Math Anal Appl 337(1):36–48

    MathSciNet  Article  Google Scholar 

  25. 25.

    Ortigueira M, Tenreiro Machado J (2014) What is a fractional derivative? J Comput Phys 293:4–13

    MathSciNet  Article  Google Scholar 

  26. 26.

    Wei Y, Chen Y, Cheng S, Wang Y (2017) Discussion on fractional order derivatives. IFAC-PapersOnLine 50(1):7002–7006

    Article  Google Scholar 

  27. 27.

    Hatcher C (2013) Fractional derivatives, fractional differential equations, and their numerical approximation. Ph.D. thesis, Tennessee Technological University

  28. 28.

    Cruz-Duarte JM, Rosales-Garcia J, Correa-Cely CR, Garcia-Perez A, Avina-Cervantes JG (2018) A closed form expression for the Gaussian-based Caputo-Fabrizio fractional derivative for signal processing applications. Commun Nonlinear Sci Numer Simul 61:138–148

    MathSciNet  Article  Google Scholar 

  29. 29.

    Liu Z, Cheng A, Li X (2017) A second order Crank–Nicolson scheme for fractional Cattaneo equation based on new fractional derivative. Appl Math Comput 311:361–374

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Sazmand A, Behroozifar M (2018) Application Jacobi spectral method for solving the time-fractional differential equation. J Comput Appl Math 339:49–68

    MathSciNet  Article  Google Scholar 

  31. 31.

    Atangana A (2018) Blind in a commutative world: simple illustrations with functions and chaotic attractors. Chaos Solitons Fractals 114:347–363

    MathSciNet  Article  Google Scholar 

  32. 32.

    Atangana A, Gómez-Aguilar J (2018) Decolonisation of fractional calculus rules: breaking commutativity and associativity to capture more natural phenomena. Eur Phys J Plus 133:1–22

    Article  Google Scholar 

  33. 33.

    Atangana A (2018) Non validity of index law in fractional calculus: a fractional differential operator with Markovian and non-Markovian properties. Physica A 505:688–706

    MathSciNet  Article  Google Scholar 

  34. 34.

    Aniszewska D, Rybaczuk M (2016) Multiplicative hénon map. In: AIP Conference Proceedings, vol 1738. AIP Publishing, p 480060

  35. 35.

    Sun H, Zhang Y, Wei S, Zhu J, Chen W (2018) A space fractional constitutive equation model for non-Newtonian fluid flow. Commun Nonlinear Sci Numer Simul 62:409–417

    MathSciNet  Article  Google Scholar 

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Letelier, M., Stockle, J. Some applications of extended calculus to non-Newtonian flow in pipes. J Braz. Soc. Mech. Sci. Eng. 43, 62 (2021).

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  • Fractional calculus
  • Non-Newtonian calculus
  • Non-Newtonian fluids
  • Pipe flow