Skip to main content
SpringerLink
Log in

Fractional analysis of unsteady radiative brinkman-type nanofluid flow comprised of CoFe2O3 nanoparticles across a vertical plate

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

Abstract

An incompressible magneto hydrodynamic (MHD) nanofluid flow across a vertical sheet has been examined. The first-order chemical reaction, thermal radiation, Newtonian heating, slip conditions and heat generation/absorption effect has been also analyzed on the Brinkman-type nanofluid model. The nanofluid is prepared by the dispersion of CoFe2O3 (Cobalt ferrite) nanoparticles (NPs) in the water. The exceptional mechanical and chemical equilibrium stabilities of CoFe2O3-NPs at atmospheric temperatures, as well as their significant coercive power and magnetism, are the main reasons for their increased interest of researchers. CoFe2O3-NPs are used in magnetic cards, magnetic drug delivery, recording devices, biotechnology and medicine. Due to these remarkable uses of Brinkman-type nanofluid consist of CoFe2O3-NPs, we have modeled the fluid flow in terms of coupled PDEs. The system of PDEs is further generalized by employing the CPCFD (Constant proportional Caputo fractional derivative). The non-integer case Laplace transform is utilized to derive the exact solutions for the proposed model. To check the accuracy and validity, the results are estimated with the existing study. It can be noticed that the energy transference rate rises up to 13.92% by the addition of CoFe2O3-NPs, while the mass communication rate intensifies up to 17.102%.

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

Similar content being viewed by others

Data availability

All data used in this manuscript have been presented within the article.

Abbreviations

Chemical reaction:

\({\text{Kr}}\left( {{\text{M}}^{ - 1} {\text{s}}^{ - 1} } \right)\)

Gravitational force:

\(g\left( {{\text{LT}}^{ - 2} } \right)\)

Mass concentration:

\(C\left( {{\text{kg}}\;{\text{m}}^{-3} } \right)\)

Ambient concentration:

\(C_{\text{a}} \left( {{\text{kg}}\;{\text{m}}^{-3} } \right)\)

Dynamic viscosity:

\(\mu_{\text{nf}} \left( {{\text{kg}}\;{{\text{m}}^{-1}}\;{\text{s}}^{-1}} \right)\)

Thermal expansion:

\(\left( {\beta_{{\text{T}}} } \right)_{\text{nf}} \left( {{\text{K}}^{ - 1} } \right)\)

Mass diffusion rate:

\(D_{\text{nf}} \left( {{\text{m}}^{2} \;{\text{s}}^{-1}} \right)\)

Prandtl number:

Pr

Mass Grashof number:

\({\text{Gm}}\)

Thermal Grashof number:

\({\text{Gr}}\)

Schmidt number:

\({\text{Sc}}\)

Brinkman factor:

\(\beta^{*}\)

Heat absorption/generation:

\(Q_{0} \left( {{\text{W}}\;{\text{m}}^{-3} } \right)\)

Velocity on \(x - {\text{axis}}\) :

\(u\left( {{\text{LT}}^{ - 1} } \right)\)

Permeability of the medium:

\(K\left( {{\text{H}}/{\text{m}}} \right)\)

Temperature:

\(T\left( {\text{K}} \right)\)

Specific heat capacity:

\(\left( {\rho C_{\text{p}} } \right)_{\text{nf}} \left( {{\text{J}}{\text{kg}}^{-1}\;{\text{K}}^{-1}} \right)\)

Density of nanofluid:

\(\rho_{\text{nf}} \left( {{\text{kg}}\,{\text{m}}^{-3} } \right)\)

Thermal conductivity:

\(k_{\text{nf}} \left( {{\text{w}}{\text{m}}^{-1}\;{\text{K}}^{-1}} \right)\)

Electrical conductivity:

\(\sigma_{\text{nf}} \left( {{\text{S}}\,{\text{m}}^{-1}} \right)\)

Magnetic factor:

\(M\)

Porosity parameter:

\(K\)

Heat generation:

\(S\)

Chemical reaction factor:

\(\lambda_{0}\)

References

  1. Ghalambaz M, Groşan T, Pop I. Mixed convection boundary layer flow and heat transfer over a vertical plate embedded in a porous medium filled with a suspension of nano-encapsulated phase change materials. J Mol Liq. 2019;293:111432.

    CAS  Google Scholar 

  2. Sulochana C, Poornima M. Unsteady MHD Casson fluid flow through vertical plate in the presence of Hall current. SN Appl Sci. 2019;1:1–14.

    Google Scholar 

  3. Ali Shah N, Ahmed N, Elnaqeeb T, Rashidi MM. Magnetohydrodynamic free convection flows with thermal memory over a moving vertical plate in porous medium. J Appl Comput Mech. 2019;5(1):150–61.

    Google Scholar 

  4. Khan MI, Qayyum S, Kadry S, Khan WA, Abbas SZ. Irreversibility analysis and heat transport in squeezing nanoliquid flow of non-Newtonian (second-grade) fluid between infinite plates with activation energy. Arab J Sci Eng. 2020;45:4939–47.

    CAS  Google Scholar 

  5. Awati VB, Goravar A, Wakif A. Analysis of unsteady boundary layer flow of nanofluids with heat transfer over a permeable stretching/shrinking sheet via a shifted Chebyshev collocation method. In: Sebbah P, editor. Waves in random and complex media. Springer: Berlin; 2022. p. 1–27.

    Google Scholar 

  6. El-Zahar ER, Algelany AM, Rashad AM. Sinusoidal natural convective flow of non-newtonian nanoliquid over a radiative vertical plate in a saturated porous medium. IEEE Access. 2020;8:136131–40.

    Google Scholar 

  7. Kumar MA, Reddy YD, Rao VS, Goud BS. Thermal radiation impact on MHD heat transfer natural convective nano fluid flow over an impulsively started vertical plate. Case Stud Therm Eng. 2021;24:100826.

    Google Scholar 

  8. Mahanthesh B, Mackolil J, Radhika M, Al-Kouz W. Significance of quadratic thermal radiation and quadratic convection on boundary layer two-phase flow of a dusty nanoliquid past a vertical plate. Int Commun Heat Mass Transfer. 2021;120:105029.

    CAS  Google Scholar 

  9. Raza A, Khan SU, Farid S, Khan MI, Sun TC, Abbasi A, Malik MY. Thermal activity of conventional Casson nanoparticles with ramped temperature due to an infinite vertical plate via fractional derivative approach. Case Stud Therm Eng. 2021;27:101191.

    Google Scholar 

  10. Asogwa KK, Mebarek-Oudina F, Animasaun IL. Comparative investigation of water-based Al2O3 nanoparticles through water-based CuO nanoparticles over an exponentially accelerated radiative Riga plate surface via heat transport. Arab J Sci Eng. 2022;47(7):8721–38.

    CAS  Google Scholar 

  11. Bakar FNA, Soid SK. MHD stagnation-point flow and heat transfer over an exponentially stretching/shrinking vertical sheet in a micropolar fluid with a Buoyancy effect. J Adv Res Numer Heat Transf. 2022;8(1):50–5.

    Google Scholar 

  12. Wakif A. Numerical inspection of two-dimensional MHD mixed bioconvective flows of radiating Maxwell nanofluids nearby a convectively heated vertical surface. In: Sebbah P, editor. Waves in random and complex media. Springer: Berlin; 2023. p. 1–22.

    Google Scholar 

  13. Bejawada SG, Nandeppanavar MM. Effect of thermal radiation on magnetohydrodynamics heat transfer micropolar fluid flow over a vertical moving porous plate. Exp Comput Multiph Flow. 2023;5(2):149–58.

    Google Scholar 

  14. Abbas A, Wakif A, Shafique M, Ahmad H, Ul Ain Q, Muhammad T. Thermal and mass aspects of Maxwell fluid flows over a moving inclined surface via generalized Fourier’s and Fick’s laws. In: Sebbah P, editor. Waves in random and complex media. Springer: Berlin; 2023. p. 1–27.

    Google Scholar 

  15. Mahabaleshwar US, Sneha KN, Wakif A. Significance of thermo-diffusion and chemical reaction on MHD Casson fluid flows conveying CNTs over a porous stretching sheet. In: Sebbah P, editor. Waves in random and complex media. Springer: Berlin; 2023. p. 1–19.

    Google Scholar 

  16. Tanda G, Ahmed EN, Bottaro A. Natural convection heat transfer from a ribbed vertical plate: effect of rib size, pitch, and truncation. Exp Thermal Fluid Sci. 2023;145:110898.

    Google Scholar 

  17. Zukri NZM, Ilias MR, Ishak SS, Osman R, Makhatar NAM, Abd Rahman MN. Magnetohydrodynamic effect in mixed convection casson hybrid nanofluids flow and heat transfer over a moving vertical plate. CFD Lett. 2023;15(7):92–111.

    Google Scholar 

  18. Rajesh V, Öztop HF, Abu-Hamdeh NH. Impact of moving/exponentially accelerated vertical plate on unsteady flow and heat transfer in hybrid nanofluids. J Nanofluids. 2023;12(5):1374–82.

    Google Scholar 

  19. Kitagawa A, Kobayashi R, Denissenko P, Murai Y. Natural convection heat transfer enhancement using bubble injection between vertical parallel plates. Int J Heat Mass Transf. 2023;202:123658.

    Google Scholar 

  20. Ramzan M, Gul N, Chung JD, Kadry S, Chu YM. Numerical treatment of radiative Nickel-Zinc ferrite-Ethylene glycol nanofluid flow past a curved surface with thermal stratification and slip conditions. Sci Rep. 2020;10(1):16832.

    CAS  PubMed  PubMed Central  Google Scholar 

  21. Alrashed AA, Karimipour A, Bagherzadeh SA, Safaei MR, Afrand M. Electro-and thermophysical properties of water-based nanofluids containing copper ferrite nanoparticles coated with silica: experimental data, modeling through enhanced ANN and curve fitting. Int J Heat Mass Transf. 2018;127:925–35.

    CAS  Google Scholar 

  22. Kurian J, Lahiri BB, Mathew MJ, Philip J. High magnetic fluid hyperthermia efficiency in copper ferrite nanoparticles prepared by solvothermal and hydrothermal methods. J Magn Magn Mater. 2021;538:168233.

    CAS  Google Scholar 

  23. Murtaza S, Kumam P, Bilal M, Sutthibutpong T, Rujisamphan N, Ahmad Z. Parametric simulation of hybrid nanofluid flow consisting of cobalt ferrite nanoparticles with second-order slip and variable viscosity over an extending surface. Nanotechnol Rev. 2023;12(1):20220533.

    CAS  Google Scholar 

  24. Salawu SO, Obalalu AM, Shamshuddin MD. Nonlinear solar thermal radiation efficiency and energy optimization for magnetized hybrid Prandtl–Eyring nanoliquid in aircraft. Arab J Sci Eng. 2023;48(3):3061–72.

    CAS  Google Scholar 

  25. Basit MA, Farooq U, Imran M, Fatima N, Alhushaybari A, Noreen S, Akgül A. Comprehensive investigations of (Au-Ag/Blood and Cu-Fe3O4/Blood) hybrid nanofluid over two rotating disks: numerical and computational approach. Alex Eng J. 2023;72:19–36.

    Google Scholar 

  26. Sharma J, Ahammad NA, Wakif A, Shah NA, Chung JD, Weera W. Solutal effects on thermal sensitivity of casson nanofluids with comparative investigations on Newtonian (water) and non-Newtonian (blood) base liquids. Alex Eng J. 2023;71:387–400.

    Google Scholar 

  27. Elboughdiri N, Ghernaout D, Muhammad T, Alshehri A, Sadat R, Ali MR, Wakif A. Towards a novel EMHD dissipative stagnation point flow model for radiating copper-based ethylene glycol nanofluids: An unsteady two-dimensional homogeneous second-grade flow case study. Case Stud Therm Eng. 2023;45:102914.

    Google Scholar 

  28. Murtaza S, Kumam P, Ahmad Z, Ramzan M, Ali I, Saeed A. Computational simulation of unsteady squeezing hybrid nanofluid flow through a horizontal channel comprised of metallic nanoparticles. J Nanofluids. 2023;12(5):1327–34.

    Google Scholar 

  29. Mourad A, Aissa A, Abed AM, Toghraie D, Akbari OA, Guedri K, Marzouki R. MHD natural convection of Fe3O4-MWCNT/Water hybrid nanofluid filled in a porous annulus between a circular cylinder and Koch snowflake. Alex Eng J. 2023;65:367–82.

    Google Scholar 

  30. Goedbloed H, Goedbloed JP, Keppens R, Poedts S. Magnetohydrodynamics: of laboratory and astrophysical plasmas. Cambridge: Cambridge University Press; 2019.

    Google Scholar 

  31. Khashi’ie NS, Arifin NM, Nazar R, Hafidzuddin EH, Wahi N, Pop I. Magnetohydrodynamics (MHD) axisymmetric flow and heat transfer of a hybrid nanofluid past a radially permeable stretching/shrinking sheet with Joule heating. Chin J Phys. 2020;64:251–63.

    CAS  Google Scholar 

  32. Ali B, Rasool G, Hussain S, Baleanu D, Bano S. Finite element study of magnetohydrodynamics (MHD) and activation energy in Darcy-Forchheimer rotating flow of Casson Carreau nanofluid. Processes. 2020;8(9):1185.

    CAS  Google Scholar 

  33. Ur Rasheed H, AL-Zubaidi A, Islam S, Saleem S, Khan Z, Khan W. Effects of Joule heating and viscous dissipation on magnetohydrodynamic boundary layer flow of Jeffrey nanofluid over a vertically stretching cylinder. Coatings. 2021;11(3):353.

    CAS  Google Scholar 

  34. Mogharrebi ARD, Ganji AR, Hosseinzadeh K, Roghani S, Asadi A, Fazlollahtabar A. Investigation of magnetohydrodynamic nanofluid flow contain motile oxytactic microorganisms over rotating cone. Int J Numer Meth Heat Fluid Flow. 2021;31(11):3394–412.

    Google Scholar 

  35. Waini I, Khashi’ie NS, Kasim ARM, Zainal NA, Hamzah KB, Arifin N, Pop I. Unsteady magnetohydrodynamics (MHD) flow of hybrid ferrofluid due to a rotating disk. Mathematics. 2022;10(10):1658.

    Google Scholar 

  36. Waqas M, Khan U, Zaib A, Ishak A, Albaqami MD, Waini I, Pop I. Radiation effect on MHD three-dimensional stagnation-point flow comprising water-based graphene oxide nanofluid induced by a nonuniform heat source/sink over a horizontal plane surface. Int J Mod Phys B. 2023;37(15):2350146.

    CAS  Google Scholar 

  37. Biswas N, Mandal DK, Manna NK, Gorla RSR, Chamkha AJ. Magnetohydrodynamic thermal characteristics of water-based hybrid nanofluid-filled non-Darcian porous wavy enclosure: effect of undulation. Int J Numer Meth Heat Fluid Flow. 2022;32(5):1742–77.

    Google Scholar 

  38. Vijay N, Sharma K. Magnetohydrodynamic hybrid nanofluid flow over a decelerating rotating disk with Soret and Dufour effects. Multidiscip Model Mater Struct. 2023;19(2):253–76.

    CAS  Google Scholar 

  39. Khan U, Zaib A, Ishak A, Waini I, Wakif A, Galal AM. Agrawal nanofluid flow towards a stagnation point past a moving disk with smoluchowski temperature and Maxwell velocity slip boundary conditions: The case of Buongiorno’s model. ZAMM J Appl Math Mech Zeitschrift für Angewandte Mathematik und Mechanik. 2023;103(3):e202200051.

    Google Scholar 

  40. Shankar Goud B, Yanala DR, Wakif A. Numerical analysis on the heat and mass transfer MHD flow characteristics of nanofluid on an inclined spinning disk with heat absorption and chemical reaction. Heat Transf. 2023;52(5):3615–39.

    Google Scholar 

  41. Mahabaleshwar US, Maranna T, Perez LM, Nayakar SR. An effect of magnetohydrodynamic and radiation on axisymmetric flow of non-Newtonian fluid past a porous shrinking/stretching surface. J Magn Magn Mater. 2023;571:170538.

    CAS  Google Scholar 

  42. Zhu S, Wang T, Jiang C, Wu Z, Yu G, Hu J, Luo E. Experimental and numerical study of a liquid metal magnetohydrodynamic generator for thermoacoustic power generation. Appl Energy. 2023;348:121453.

    Google Scholar 

  43. Rauf A, Shah NA, Mushtaq A, Botmart T. Heat transport and magnetohydrodynamic hybrid micropolar ferrofluid flow over a non-linearly stretching sheet. AIMS Math. 2023;8(1):164–93.

    Google Scholar 

  44. Zhao TH, Khan MI, Chu YM. Artificial neural networking (ANN) analysis for heat and entropy generation in flow of non-Newtonian fluid between two rotating disks. Math Methods Appl Sci. 2023;46(3):3012–30.

    Google Scholar 

  45. Oldham K, Spanier J. The fractional calculus theory and applications of differentiation and integration to arbitrary order. Amsterdam: Elsevier; 1974.

    Google Scholar 

  46. Samko SG. Fractional integrals and derivatives. Theory and applications. 1993.

  47. Machado JT, Kiryakova V, Mainardi F. Recent history of fractional calculus. Commun Nonlinear Sci Numer Simul. 2011;16(3):1140–53.

    Google Scholar 

  48. Kumar S, Chauhan RP, Momani S, Hadid S. Numerical investigations on COVID-19 model through singular and non-singular fractional operators. Numer Methods Partial Differ Equ. 2020. https://doi.org/10.1002/num.22707.

    Article  Google Scholar 

  49. Kumar S, Kumar A, Samet B, Dutta H. A study on fractional host–parasitoid population dynamical model to describe insect species. Numer Methods Partial Differ Equ. 2021;37(2):1673–92.

    Google Scholar 

  50. Khan MA, Ullah S, Kumar S. A robust study on 2019-nCOV outbreaks through non-singular derivative. Euro Phys J Plus. 2021;136:1–20.

    Google Scholar 

  51. Kumar S, Kumar R, Osman MS, Samet B. A wavelet based numerical scheme for fractional order SEIR epidemic of measles by using Genocchi polynomials. Numer Methods Partial Differ Equ. 2021;37(2):1250–68.

    Google Scholar 

  52. Mohammadi H, Kumar S, Rezapour S, Etemad S. A theoretical study of the Caputo-Fabrizio fractional modeling for hearing loss due to Mumps virus with optimal control. Chaos Solitons Fractals. 2021;144:110668.

    Google Scholar 

  53. Kumar S, Kumar R, Cattani C, Samet B. Chaotic behaviour of fractional predator-prey dynamical system. Chaos Solitons Fractals. 2020;135:109811.

    Google Scholar 

  54. Murtaza S, Kumam P, Ahmad Z, Sitthithakerngkiet K, Ali IE. Finite difference simulation of fractal-fractional model of electro-osmotic flow of casson fluid in a micro channel. IEEE Access. 2022;10:26681–92.

    Google Scholar 

  55. Murtaza S, Ahmad Z, Ali IE, Akhtar Z, Tchier F, Ahmad H, Yao SW. Analysis and numerical simulation of fractal-fractional order non-linear couple stress nanofluid with cadmium telluride nanoparticles. J King Saud Univ Sci. 2023;35(4):102618.

    Google Scholar 

  56. Ahmad Z, El-Kafrawy SA, Alandijany TA, Giannino F, Mirza AA, El-Daly MM, Azhar EI. A global report on the dynamics of COVID-19 with quarantine and hospitalization: a fractional order model with non-local kernel. Comput Biol Chem. 2022;98:107645.

    CAS  PubMed  PubMed Central  Google Scholar 

  57. Ahmad Z, Bonanomi G, di Serafino D, Giannino F. Transmission dynamics and sensitivity analysis of pine wilt disease with asymptomatic carriers via fractal-fractional differential operator of Mittag-Leffler kernel. Appl Numer Math. 2023;185:446–65.

    Google Scholar 

  58. Ahmad M, Imran MA, Nazar M. Mathematical modeling of (Cu−Al2 O3) water-based Maxwell hybrid nanofluids with Caputo-Fabrizio fractional derivative. Adv Mech Eng. 2020;12(9):1687814020958841.

    CAS  Google Scholar 

  59. Khan I, Hussanan A, Saqib M, Shafie S. Convective heat transfer in drilling nanofluid with clay nanoparticles: applications in water cleaning process. BioNanoScience. 2019;9(2):453–60.

    Google Scholar 

  60. Baleanu D, Fernandez A, Akgül A. On a fractional operator combining proportional and classical differintegrals. Mathematics. 2020;8(3):360.

    Google Scholar 

  61. Acharya N, Maity S, Kundu PK. Framing the hydrothermal features of magnetized TiO2–CoFe2O4 water-based steady hybrid nanofluid flow over a radiative revolving disk. Multidiscip Model Mater Struct. 2019;16(4):765–90.

    Google Scholar 

  62. Kumam P, Tassaddiq A, Watthayu W, Shah Z, Anwar T. Modeling and simulation-based investigation of unsteady MHD radiative flow of rate type fluid; a comparative fractional analysis. Math Comput Simul. 2021;201:486–507.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sheikh Taimur Academic Block-II, Department of Mathematics, University of Peshawar, Peshawar, Khyber Pakhtunkhwa, 25120, Pakistan

    Muhammad Bilal

  2. School of Mathematical Sciences, Jiangsu University, Zhenjiang, 212013, Jiangsu, China

    Aatif Ali

  3. GRC Department, Applied College, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    Samy Refahy Mahmoud

  4. Faculty of Engineering and Technology, Future University in Egypt, New Cairo, 11835, Egypt

    Elsayed Tag-Eldin

  5. Department of Industrial Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    Mohammed Balubaid

Authors
  1. Muhammad Bilal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Aatif Ali
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Samy Refahy Mahmoud
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Elsayed Tag-Eldin
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Mohammed Balubaid
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Muhammad Bilal.

Ethics declarations

Conflicts of interest

The authors declare no conflict of interest.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bilal, M., Ali, A., Mahmoud, S.R. et al. Fractional analysis of unsteady radiative brinkman-type nanofluid flow comprised of CoFe2O3 nanoparticles across a vertical plate. J Therm Anal Calorim 148, 13869–13882 (2023). https://doi.org/10.1007/s10973-023-12705-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10973-023-12705-0

Keywords

Search

Navigation