Skip to main content
SpringerLink
Log in

Natural convection flows of Prabhakar-like fractional Maxwell fluids with generalized thermal transport

  • Published:
Journal of Thermal Analysis and Calorimetry Aims and scope Submit manuscript

Abstract

Natural convective flows of Prabhakar-like fractional viscoelastic fluids over an infinite vertical heated wall are studied by introducing the generalized fractional constitutive equations for the stress-shear rate and thermal flux density vector. The generalized memory effects are described by the time-fractional Prabhakar derivative. Closed-form solutions for the non-dimensional velocity and temperature fields are determined using the method of integral transform. The velocity and heat transfer of Prabhakar-like fractional Maxwell fluids with generalized thermal transport are compared with ordinary Maxwell fluids with generalized thermal transport and with the ordinary viscoelastic fluids with classical Fourier thermal flux. Solutions of the generalized model are particularized into solutions corresponding to flows and heat transfer with Caputo memory, respectively, to flows of the ordinary fluids with ordinary heat transfer. The use of Prabhakar operators shows the possibility of a convenient choice of fractional parameters such that to have a very good fitting between theoretical and experimental data.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Erickson LE, Fan LT, Fox VG. Heat and mass transfer on moving continuous flat plate with suction or injection. Ind Eng Chem Fundamen. 1966;5(1):19–25. https://doi.org/10.1021/i160017a004.

    Article  CAS  Google Scholar 

  2. Seth GS, Tripathi R, Sharma R, Chamkha AJ. MHD double diffusive natural convection flow over exponentially accelerated inclined plate. J Mech. 2017;33(1):87–99. https://doi.org/10.1017/.2016.56.

    Article  CAS  Google Scholar 

  3. Ahmad S, Nadeem S. Flow analysis by Cattaneo–Christov heat flux in presence of Thompson and Troian slip condition. Appl Nanosci. 2020. https://doi.org/10.1007/s13204-020-01267-4.

    Article  Google Scholar 

  4. Ullah N, Nadeem S, Khan AU. Finite element simulations for natural convective flow of nanofluid in a rectangular cavity having corrugated heated rods. J. Therm Anal Calorim. 2020. https://doi.org/10.1007/s10973-020-09378-4.

    Article  Google Scholar 

  5. Nadeem S, Malik MY, Abbas N. Heat transfer of three-dimensional micropolar fluid on a Riga plate. Can J Phys. 2020;98(1):32–8.

    Article  CAS  Google Scholar 

  6. Raju MC, Veeresh C, Varma SVK, Kumar R, Kumar V. Heat and mass transfer in MHD mixed convection flow on a moving inclined porous plate. J Appl Comput Math. 2015. https://doi.org/10.4172/2168-9679.1000259.

    Article  Google Scholar 

  7. Das M, Mahanta G, Shaw S, Parida SB. Unsteady MHD chemically reactive double-diffusive Casson fluid past a flat plate in porous medium with heat and mass transfer. Heat Transf Asian Res. 2019;48:1761–77. https://doi.org/10.1002/htj.21456.

    Article  Google Scholar 

  8. Ellahi R, Hassan M, Zeeshan MA. A study of heat transfer in power law nanofluid. Therm Sci. 2016;20(6):2015–26.

    Article  Google Scholar 

  9. Raza M, Ellahi R, Salt SM, Sarafraz MM, Shadloo MS, Waheed I. Enhancement of heat transfer in peristaltic flow in a permeable channel under induced magnetic field using different CNTs. J Therm Anal Calorim. 2020. https://doi.org/10.1007/s10973-019-0997-5.

    Article  Google Scholar 

  10. Wang S, Tan W. Stability analysis of Soret-driven double-diffusive convection of Maxwell fluid in a porous medium. Int J Heat Fluid Flow. 2011;32:88–94. https://doi.org/10.1016/j.ijheatfluidflow.2010.10.005.

    Article  Google Scholar 

  11. Wang X, Xu H, Qi H. Numerical analysis for rotating electro-osmotic flow of fractional Maxwell fluids. Appl Math Lett. 2020;103:106179. https://doi.org/10.1016/j.aml.2019.106179.

    Article  Google Scholar 

  12. Sabatier J, Agrawal OP, Tenreiro Machado JA. Advances in fractional calculus: theoretical developments and applications in physics and engineering. Dordrecht: Springer; 2007.

    Book  Google Scholar 

  13. Baleanu D, Diethelm K, Scalase Trujillo JJ. Fractional calculus: models and numerical methods. Singapore: World Scientific; 2011.

    Google Scholar 

  14. Ortigueira MD, Machado JT. Fractional derivatives: the perspective of system theory. Mathematics. 2019;7:150. https://doi.org/10.3390/math7020150.

    Article  Google Scholar 

  15. Hristov J. Response functions in linear viscoelastic constitutive equations and related fractional operators. Math Model Nat Phenom. 2019;14(3):305.

    Article  CAS  Google Scholar 

  16. Hristov J. Linear viscoelastic responses and constitutive equations in terms of fractional operators with non-singular kernels Pragmatic approach, memory kernel correspondence requirement and analyses. Eur Phys J Plus. 2019;134:283. https://doi.org/10.1140/epjp/i2019-12697-7.

    Article  Google Scholar 

  17. Bazhlekova E, Bazhlekov I. Subordination approach to space-time fractional diffusion. Mathematics. 2019;7:415. https://doi.org/10.3390/math7050415.

    Article  Google Scholar 

  18. Baleanu D, Mousalou A, Rezapour S. The extended fractional Caputo–Fabrizio derivative of order 0 ≤ σ<0 on CR[0, 1] and the existence of solutions for two higher-order series-type differential equations. Adv Differ Equ. 2018. https://doi.org/10.1186/s13662-018-1696-6.

    Article  PubMed  PubMed Central  Google Scholar 

  19. Garra R, Garrappa R. The Prabhakar or three parameter Mittag–Leffler function: theory and application. Commun Nonlinear Sci Numer Simul. 2018;56:314–29. https://doi.org/10.1016/j.cnsns.2017.08.018.

    Article  Google Scholar 

  20. Yang XJ. General fractional derivatives: theory, methods and applications. Boca Raton: CRC Press; 2019.

    Book  Google Scholar 

  21. Giusti A, Colombaro I. Prabhakar-like fractional viscoelasticity. Commun Nonlinear Sci Numer Simul. 2018;56:138–43. https://doi.org/10.1016/j.cnsns.2017.08.002.

    Article  Google Scholar 

  22. Srivastava HM, Fernandez A, Baleanu D. Some new fractional-calculus connections between Mittag–Leffler functions. Mathematics. 2019;7:485. https://doi.org/10.3390/math7060485.

    Article  Google Scholar 

  23. Polito F, Tomovski Z. Some properties of Prabhakar-type fractional calculus operators. Fract Diff Calculus. 2016;6(1):73–94. https://doi.org/10.7153/fdc-06-05.

    Article  Google Scholar 

  24. Gorenflo R, Kilbas AA, Mainardi F, Rogosin SV. Mittag-Leffler Functions, Related Topics and Applications. Berlin: Springer; 2014.

    Book  Google Scholar 

  25. Haubold HJ, Mathai AM, Saxena RK. Mittag-Leffler functions and their applications. J Appl Math. 2011. https://doi.org/10.1155/2011/298628.

    Article  Google Scholar 

  26. Mittag-Leffler GF. Sur la nouvelle fonction eα(x). CR Acad Sci Paris. 1903;137:554–8.

    Google Scholar 

  27. Wiman A. Über den fundamental satz in der theorie der funcktionen, Eα(x). Acta Math. 1905;29:191–201.

    Article  Google Scholar 

  28. Prabhakar TR. A singular integral equation with a generalized Mittag–Leffler function in the kernel. Yokohama Math J. 1971;19:7–15.

    Google Scholar 

  29. Kilbas A, Saigo M, Saxena R. Generalized Mittag–Leffler function and generalized fractional calculus operators. Integr Transforms Spec Funct. 2004;15:31–49. https://doi.org/10.1080/10652460310.0.0160.0717.

    Article  Google Scholar 

  30. dos Santos MAF. Fractional Prabhakar derivative in diffusion equation with non-static stochastic resetting. Physics. 2019. https://doi.org/10.3390/physics1010005.

    Article  Google Scholar 

  31. Caputo M, Fabrizio M. A new definition of fractional derivative without singular kernel. Progr Fract Differ Appl. 2015;1(2):73–85. https://doi.org/10.12785/pfda/010201.

    Article  Google Scholar 

  32. Hetnarski RB. An algorithm for generating some inverse Laplace transforms of exponential forms. J Appl Math Phys (ZAMP). 1975;26:249–53.

    Article  Google Scholar 

  33. Jacquot RG, Steadman JW, Rhodine CN. The Gaver–Stehfest algorithm for approximate inversion of Laplace transforms. Circuits Syst Mag. 1983;5(1):4–8.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Informetrics Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Nehad Ali Shah

  2. Faculty of Mathematics & Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Nehad Ali Shah

  3. Academy of Romanian Scientists, 050094, Bucharest, Romania

    Constantin Fetecau

  4. Department of Theoretical Mechanics, Technical University “Gheorghe Asachi” of Iași, Iasi, Romania

    Dumitru Vieru

Authors
  1. Nehad Ali Shah
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Constantin Fetecau
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dumitru Vieru
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dumitru Vieru.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shah, N.A., Fetecau, C. & Vieru, D. Natural convection flows of Prabhakar-like fractional Maxwell fluids with generalized thermal transport. J Therm Anal Calorim 143, 2245–2258 (2021). https://doi.org/10.1007/s10973-020-09835-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10973-020-09835-0

Keywords

Search

Navigation