Skip to main content
SpringerLink
Log in

MHD Squeezed Radiative Flow of Casson Hybrid Nanofluid Between Parallel Plates with Joule Heating

  • Original Paper
  • Published:
International Journal of Applied and Computational Mathematics Aims and scope Submit manuscript

Abstract

The physical significance of this study stems from a complex fluid dynamics situation that takes note of several real-world aspects. It is essential to figure out how hybrid nanofluids respond when exposed to various phenomena for a variety of applications, such as environmental engineering, industrial processes, and heat transfer systems. Overall, this research adds valuable insights to the field of fluid dynamics, particularly in the context of advanced nanofluid applications. The purport of the study is to dig into 2-dimensional unsteady hybrid nanofluid flow of Casson Cu–Al2O3/water between parallel plates in a channel which is placed horizontally where the lower plate is considered as stretchable. Oscillation of upper plate is there relative to plate which is placed at lower position. Here we examine the possessions of MHD during the flow of fluid when applied on lower plate along with suction/injection in the company of porous medium in conjunction with consequences of radiation, joule heating and heat source under the consequences of viscous dissipation while fluid is in motion. Further, we iron out the governing differential equation using non-dimensional parameters with the help of optimal homotopy analysis method. Moreover, results as well as conclusions have been carried out with the help of tables and graphs. We concluded that the liquid’s temperature rises with growing value of thermal radiation parameter. There is diminishing in the rate of heat transfer by 0.32% on the lower plate is observed for the varying numeric value of magnetic parameter.

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
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18

Similar content being viewed by others

Data Availability

No data associated in the manuscript.

Abbreviations

\(b\) :

Constant

B :

Time dependent magnetic field

\(B_{0}\) :

Constant

\(C_{f1}\) :

Skin friction coefficient of lower plate

\(C_{f2}\) :

Skin friction coefficient of upper plate

\(C_{p}\) :

Specific heat \(({\text{J}}\,{\text{kg}}^{ - 1} \,{\text{K}}^{ - 1} )\)

\(Ec\) :

Eckert number

\(f\left( {Subscript} \right)\) :

Base fluid

\(f\left( \eta \right)\) :

Stream function

\(hnf\left( {Subscript} \right)\) :

Hybrid nanofluid

\(h\left( t \right)\) :

Distance between the plates (m)

\(Hs\) :

Heat source parameter

\(\kappa\) :

Thermal conductivity \(({\text{W}}\,{\text{m}}^{ - 1} \,{\text{K}}^{ - 1} )\)

\(K_{P}^{*}\) :

Porous medium’s permeability (m2)

\(K_{P}^{{}}\) :

Porosity parameter

\(M\) :

Magnetic parameter

\(k^{*}\) :

Mean absorption coefficient

\(Nu_{x1}\) :

Nusselt number for lower plate

\(Nu_{x2}\) :

Nusselt number for upper plate

\(nf\left( {Subscript} \right)\) :

Nanofluid

\(\Pr\) :

Prandtl number

\(Q_{0}\) :

Heat absorption coefficient

\(q_{r}\) :

Radiative heat flux (W)

\(Rd\) :

Radiation parameter

\({\text{Re}}\) :

Reynolds number

\(Sq\) :

Unsteady squeezing parameter

\(S\) :

Suction/injection parameter

\(s_{1} \left( {Subscript} \right)\) :

\({\text{Al}}_{2} {\text{O}}_{3}\) Nanoparticle

\(s_{2} \left( {Subscript} \right)\) :

\({\text{Cu}}\) Nanoparticle

\(T\) :

Temperature of hybrid nanofluid (K)

\(T_{0}\) :

Reference temperature (K)

\(T_{1}\) :

Lower plate temperature (K)

\(T_{2}\) :

Upper plate temperature (K)

\(t\) :

Time

\(u\) :

Fluid velocity in x-direction (m s−1)

\(u_{w}\) :

Velocity of the stretching lower plate (m s−1)

\(v\) :

Fluid velocity in y-direction (m s−1)

\(v_{w}\) :

Velocity of the wall mass porous lower plate (m s−1)

\(V_{0}\) :

Constant

\(V_{h}\) :

Velocity of the upper plate moving towards or away from lower plate (m s−1)

\(x,y\) :

Cartesian coordinates

\(\alpha\) :

Constant

\(\beta^{*}\) :

Casson parameter

\(\gamma\) :

Constant

\(\delta\) :

Constant

\(\eta\) :

Similarity transformation

\(\theta\) :

Dimensionless fluid temperature

\(\lambda\) :

Stretching parameter

\(\mu\) :

Dynamic viscosity \(({\text{kg}}\,{\text{m}}^{ - 1} \,{\text{s}}^{ - 1} )\)

\(\nu\) :

Kinematic viscosity \(({\text{m}}^{2} \,{\text{s}}^{ - 1} )\)

\(\rho\) :

Density of fluid \(({\text{kg}}\,{\text{m}}^{ - 3} )\)

\(\rho C_{p}\) :

Heat capacity \(({\text{J}}\,{\text{K}}^{ - 1} \,{\text{m}}^{ - 3} )\) s

\(\sigma^{*}\) :

Stefan–Boltzmann constant

\(\sigma\) :

Electrical conductivity \(({\text{S}}\,{\text{m}}^{ - 1} )\)

\(\phi_{1} ,\phi_{2}\) :

Nanoparticles concentration

References

  1. Azam, M.: Effects of Cattaneo–Christov heat flux and nonlinear thermal radiation on MHD Maxwell nanofluid with Arrhenius activation energy. Case Stud. Therm. Eng. 34, 1–13 (2022)

    Article  Google Scholar 

  2. Bhaskar, K., Sharma, K., Gupta, S., Mehta, R.: MHD fluid flow with cross-diffusion effects through channel using optimal homotopy analysis method. Sci. Technol. Asia 25, 19–30 (2020)

    Google Scholar 

  3. Bhatti, M.M., Arain, M.B., Zeeshan, A., Ellahi, R., Doranehgard, M.H.: Swimming of gyrotactic microorganism in MHD Williamson nanofluid flow between rotating circular plates embedded in porous medium: application of thermal energy storage. J. Energy Storage 45, 1–12 (2022)

    Article  Google Scholar 

  4. Bilal, M., Arshad, H., Ramzan, M., Shah, Z., Kumam, P.: Unsteady hybrid-nanofluid flow comprising ferrousoxide and CNTs through porous horizontal channel with dilating/squeezing walls. Sci. Rep. 11, 1–16 (2021)

    Article  Google Scholar 

  5. Bouslimi, J., Alkathiri, A.A., Alharbi, A.N., Jamshed, W., Eid, M.R., Bouazizi, M.L.: Dynamics of convective slippery constraints on hybrid radiative Sutterby nanofluid flow by Galerkin finite element simulation. Nanotechnol. Rev. 11, 1219–1236 (2022)

    Article  Google Scholar 

  6. Chabani, I., Mebarek-Oudina, F., Ismail, L.A.I.: MHD flow of a hybrid nano-fluid in a triangular enclosure with zigzags and an elliptic obstacle. Micromachines 13, 1–18 (2022)

    Article  Google Scholar 

  7. Choi, S. U., Eastman, J. A.: Enhancing thermal conductivity of fluids with nanoparticles, pp. 1–8 (1995)

  8. Chu, Y.M., Bashir, S., Ramzan, M., Malik, M.Y.: Model-based comparative study of magnetohydrodynamics unsteady hybrid nanofluid flow between two infinite parallel plates with particle shape effects. Math. Methods Appl. Sci. 2022, 1–15 (2022)

    Google Scholar 

  9. Dawar, A., Wakif, A., Thumma, T., Shah, N.A.: Towards a new MHD non-homogeneous convective nanofluid flow model for simulating a rotating inclined thin layer of sodium alginate-based iron oxide exposed to incident solar energy. Int. Commun. Heat Mass Transf. 130, 1–11 (2022)

    Article  Google Scholar 

  10. Dawar, A., Shah, Z., Alshehri, H.M., Islam, S., Kumam, P.: Magnetized and non-magnetized Casson fluid flow with gyrotactic microorganisms over a stratified stretching cylinder. Sci. Rep. 11, 1–14 (2021)

    Article  Google Scholar 

  11. Gul, T., Usman, M., Khan, I., Nasir, S., Saeed, A., Khan, A., Ishaq, M.: Magneto hydrodynamic and dissipated nanofluid flow over an unsteady turning disk. Adv. Mech. Eng. 13, 1–11 (2021)

    Article  Google Scholar 

  12. Jameel, M., Shah, Z., Rooman, M., Alshehri, M.H., Vrinceanu, N.: Entropy generation analysis on Darcy–Forchheimer Maxwell nanofluid flow past a porous stretching sheet with threshold non-Fourier heat flux model and Joule heating. Case Stud. Therm. Eng. 52, 1–16 (2023)

    Article  Google Scholar 

  13. Jan, W.U., Farooq, M., Khan, A., Alharbi, A., Shah, R.A., Jan, R., Idris, S.A.: A parametric analysis of the effect of hybrid nanoparticles on the flow field and homogeneous-heterogeneous reaction between squeezing plates. Adv. Math. Phys. 2022, 1–22 (2022)

    Article  MathSciNet  Google Scholar 

  14. Jawad, M., Nisar, K.S.: Upper-convected flow of Maxwell fluid near stagnation point through porous surface using Cattaneo–Christov heat flux model. Case Stud. Therm. Eng. 48, 1–11 (2023)

    Article  Google Scholar 

  15. Jawad, M., Hameed, M.K., Majeed, A., Nisar, K.S.: Arrhenius energy and heat transport activates effect on gyrotactic microorganism flowing in Maxwell bio-nanofluid with Nield boundary conditions. Case Stud. Therm. Eng. 41, 1–13 (2023)

    Article  Google Scholar 

  16. Jawad, M., Hameed, M.K., Nisar, K.S., Majeed, A.H.: Darcy–Forchheimer flow of maxwell nanofluid flow over a porous stretching sheet with Arrhenius activation energy and Nield boundary conditions. Case Stud. Therm. Eng. 44, 1–11 (2023)

    Article  Google Scholar 

  17. Jawad, M., et al.: Numerical simulation of chemically reacting Darcy–Forchheimer flow of Buongiorno Maxwell fluid with Arrhenius energy in the appearance of nanoparticles. Case Stud. Therm. Eng. 50, 1–11 (2023)

    Article  Google Scholar 

  18. Khan, A., Shah, Z., Alzahrani, E., Islam, S.: Entropy generation and thermal analysis for rotary motion of hydromagnetic Casson nanofluid past a rotating cylinder with Joule heating effect. Int. Commun. Heat Mass Transf. 119, 1–8 (2020)

    Article  Google Scholar 

  19. Khan, M.R., Al-Johani, A.S., Elsiddieg, A.M.A., Saeed, T., Abd Allah, A.M.: The computational study of heat transfer and friction drag in an unsteady MHD radiated Casson fluid flow across a stretching/shrinking surface. Int. Commun. Heat Mass Transf. 130, 1–7 (2022)

    Article  Google Scholar 

  20. Khashi’ie, N.S., Waini, I., Arifin, N.M., Pop, I.: Unsteady squeezing flow of Cu–Al2O3/water hybrid nanofluid in a horizontal channel with magnetic field. Sci. Rep. 11, 1–11 (2021)

    Article  Google Scholar 

  21. Korei, Z., Louali, K.: Prediction of hybrid nanofluids behavior and entropy generation during the cooling of an electronic chip using the Lagrangian–Eulerian approach. Heat Transf. 2022, 1–21 (2022)

    Google Scholar 

  22. Kumar, R.N., Gowda, R.P., Madhukesh, J., Prasannakumara, B., Ramesh, G.: Impact of thermophoretic particle deposition on heat and mass transfer across the dynamics of Casson fluid flow over a moving thin needle. Phys. Scr. 96, 1–14 (2021)

    Article  Google Scholar 

  23. Lim, Y.J., Zakaria, M.N., Isa, S.M., Zin, N.M., Mohamad, A.Q., Shafie, S.: VON Kármán Casson fluid flow with Navier’s slip and Cattaneo–Christov heat flux. Case Stud. Therm. Eng. 28, 1–15 (2021)

    Article  Google Scholar 

  24. Mishra, A., Pandey, A.K., Kumar, M.: Thermal performance of Ag–water nanofluid flow over a curved surface due to chemical reaction using Buongiorno’s model. Heat Transf. 50, 257–278 (2021)

    Article  Google Scholar 

  25. Murad, N.M., Rawi, N.A., Shafie, S., Mahat, R.: Numerical solution for Falkner–Skan flow of hybrid nanofluid with porosity effect. J. Appl. Sci. Eng. 25, 457–463 (2021)

    Google Scholar 

  26. Murtaza, S., Kumam, P., Ahmad, Z., Sitthithakerngkiet, K., Ali, I.E.: Finite difference simulation of fractal-fractional model of electro-osmotic flow of Casson fluid in a micro channel. IEEE Access 10, 26681–26692 (2022)

    Article  Google Scholar 

  27. Nasir, S., Berrouk, A.S., Gul, T., Zari, I.: Chemically radioactive unsteady nonlinear convective couple stress Casson hybrid nanofluid flow over a gyrating sphere. J. Therm. Anal. Calorim. 148, 1–13 (2023)

    Article  Google Scholar 

  28. Nasir, S., Berrouk, A.S., Aamir, A., Gul, T.: Significance of chemical reactions and entropy on Darcy–Forchheimer flow of H2O and C2H6O2 convening magnetized nanoparticles. Int. J. Thermofluids 17, 1–12 (2023)

    Article  Google Scholar 

  29. Nasir, S., Berrouk, A.S., Aamir, A., Gul, T., Ali, I.: Features of flow and heat transport of MoS2+ GO hybrid nanofluid with nonlinear chemical reaction, radiation and energy source around a whirling sphere. Heliyon 9, 1–13 (2023)

    Article  Google Scholar 

  30. Nasir, S., Berrouk, A.S., Tassaddiq, A., Aamir, A., Akkurt, N., Gul, T.: Impact of entropy analysis and radiation on transportation of MHD advance nanofluid in porous surface using Darcy–Forchheimer model. Chem. Phys. Lett. 811, 1–9 (2023)

    Article  Google Scholar 

  31. Nazeer, M., Hussain, F., Khan, M.I., El-Zahar, E.R., Chu, Y.-M., Malik, M.Y.: Theoretical study of MHD electro-osmotically flow of third-grade fluid in micro channel. Appl. Math. Comput. 420, 1–13 (2022)

    MathSciNet  Google Scholar 

  32. Painuly, A., Mishra, N.K., Zainith, P.: Heat transfer enhancement in a helically corrugated tube by employing W/EG based non-Newtonian hybrid nanofluid under turbulent conditions. J. Enhanc. Heat Transf. 29, 1–25 (2022)

    Article  Google Scholar 

  33. Rashid, U., Baleanu, D., Liang, H., Abbas, M., Iqbal, A., Rahman, Ju.: Marangoni boundary layer flow and heat transfer of graphene–water nanofluid with particle shape effects. Processes 8, 1–17 (2020)

    Article  Google Scholar 

  34. Saeed, A., Algehyne, E.A., Aldhabani, M.S., Dawar, A., Kumam, P., Kumam, W.: Mixed convective flow of a magnetohydrodynamic Casson fluid through a permeable stretching sheet with first-order chemical reaction. PLoS ONE 17, 1–15 (2022)

    Article  Google Scholar 

  35. Shah, Z., Dawar, A., Islam, S., Alshehri, A., Alrabaiah, H.: A comparative analysis of MHD Casson and Maxwell flows past a stretching sheet with mixed convection and chemical reaction. Waves Random Complex Med. 1–16 (2021)

  36. Sharma, K., Bhaskar, K.: Influence of Soret and Dufour on three-dimensional MHD flow considering thermal radiation and chemical reaction. Int. J. Appl. Comput. Math. 6, 1–17 (2020)

    Article  MathSciNet  Google Scholar 

  37. Ullah, M.Z., Abuzaid, D., Asma, M., Bariq, A.: Couple stress hybrid nanofluid flow through a converging-diverging channel. J. Nanomater. 2021, 1–13 (2021)

    Article  Google Scholar 

Download references

Acknowledgements

I would like to express my sincere gratitude to Manipal University Jaipur for their generous financial support through Dr. Ramdas Pai scholarship. This scholarship has played a supportive role in enabling me to pursue my research work.

Funding

This research not received grant from any funding agency.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Manipal University Jaipur, Jaipur, Rajasthan, India

    Kajal Bhaskar & Kalpna Sharma

  2. Department of Mathematics, S. S. Jain Subodh P.G. (Autonomous) College, Jaipur, Rajasthan, India

    Khushbu Bhaskar

Authors
  1. Kajal Bhaskar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kalpna Sharma
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Khushbu Bhaskar
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Conceptualization, KB and Dr. KS; supervision, Dr. KS; writing-original draft, KB and KB; methodology, KB and KB; software, KB.

Corresponding author

Correspondence to Kalpna Sharma.

Ethics declarations

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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

Bhaskar, K., Sharma, K. & Bhaskar, K. MHD Squeezed Radiative Flow of Casson Hybrid Nanofluid Between Parallel Plates with Joule Heating. Int. J. Appl. Comput. Math 10, 80 (2024). https://doi.org/10.1007/s40819-024-01720-w

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40819-024-01720-w

Keywords

Search

Navigation