Skip to main content
SpringerLink
Log in

Second law thermodynamic analysis of thermo-magnetic Jeffery–Hamel dissipative radiative hybrid nanofluid slip flow: existence of multiple solutions

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract

In this article, a detailed theoretical examination is conducted for the steady, incompressible, MHD hybrid nanofluid (Al2O3/Cu–water) dissipative slip flow between non-parallel (i.e., diverging or converging) shrinking/stretching walls in the presence of appreciable thermal radiation. Appropriate similarity transformations have been applied to render the hybrid transport model conservation equations dimensionless. The diffusion flux approximation is employed for radiative heat transfer. The well-posed boundary value problem has been analyzed for solution bifurcations with a predictor homotopy analysis method (PHAM) along with stability analysis. The eigenvalues obtained predict the upper branch (Ist solution) to be physically acceptable. The critical values (\(\chi_{c} \le \chi < 0\)) of stretching/shrinking parameter are evaluated by varying slip and magnetic body force parameters. The impact of significant parameters on skin friction coefficient, Nusselt number and entropy generation number is also visualized and elaborated in detail. With stronger magnetic field parameter (M), i.e., enhancement in magnetohydrodynamic Lorentz drag force, the existence domain of the dual solutions is shown to be expanded and the critical points of the stretching parameter (\(\chi\)) for \(M = 0.1,0.5,1\) are identified, respectively, as \(\chi_{c} = - 3.2455, - 3.2680,\)\(- 3.2987\). The upper branch of skin friction, Cfr decreases as the volume fraction \(\varphi\) increases whereas the lower branch increases with an increase in the volume fraction of copper nanoparticles. Entropy generation is observed to be elevated with stronger radiation, inertial effect (Reynolds number) and Prandtl number (i.e., lower thermal conductivity) whereas it is suppressed with increasing hybrid nanoparticle volume fraction and wall hydrodynamic slip effect. Magnetic field is found to induce a weak modification in entropy generation. Increasing angular coordinate \(\left( {\xi = \frac{\theta }{\beta }} \right)\) is observed to elevate the entropy generation. The computations find applications in thermo-magnetic nozzle design, electromagnetic propulsion systems and electroconductive materials processing.

Graphic abstract

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.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. S.U.S. Choi, J.A., Eastman, enhancing thermal conductivity of fluids with nanoparticles, Technical Report, Argonne National Lab., Illinois (United States, 1995), https://www.osti.gov/scitech/biblio/196525/. Accessed 28 March 2017

  2. O. Yıldız, Ö. Açıkgöz, G. Yıldız, M. Bayrak, A.S. Dalkılıç, S. Wongwises, Single phase flow of nanofluid including graphite and water in a microchannel. Heat Mass Transf. 56, 1–24 (2020). https://doi.org/10.1007/s00231-019-02663-5

    Article  ADS  Google Scholar 

  3. P. Rana, N. Shukla, Y. Gupta, I. Pop, Homotopy analysis method for predicting multiple solutions in the channel flow with stability analysis. Commun. Nonlinear Sci. Numer. Simul. 66, 183–193 (2019). https://doi.org/10.1016/j.cnsns.2018.06.012

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. M. Sheikholeslami Kandelousi, KKL correlation for simulation of nanofluid flow and heat transfer in a permeable channel. Phys. Lett. A 378, 3331–3339 (2014). https://doi.org/10.1016/j.physleta.2014.09.046

    Article  ADS  MATH  Google Scholar 

  5. J. Buongiorno, Convective transport in nanofluids. ASME J. Heat Transf. 128, 240–250 (2006). https://doi.org/10.1115/1.2150834

    Article  Google Scholar 

  6. R. Dhanai, P. Rana, L. Kumar, MHD mixed convection nanofluid flow and heat transfer over an inclined cylinder due to velocity and thermal slip effects: Buongiorno’s model. Powder Technol. 288, 140–150 (2016). https://doi.org/10.1016/j.powtec.2015.11.004

    Article  Google Scholar 

  7. P. Rana, R. Bhargava, O. Anwar Bég, Numerical solution for mixed convection boundary layer flow of a nanofluid along an inclined plate embedded in a porous medium. Comput. Math. Appl. 64, 2816–2832 (2012). https://doi.org/10.1016/j.camwa.2012.04.014

    Article  MathSciNet  MATH  Google Scholar 

  8. P. Rana, R. Bhargava, O. Anwar Bég, Finite element modeling of conjugate mixed convection flow of Al2O3–water nanofluid from an inclined slender hollow cylinder. Phys. Scr. 87, 1–15 (2013). https://doi.org/10.1088/0031-8949/87/05/055005

    Article  Google Scholar 

  9. B. Sahoo, Effects of slip, viscous dissipation and Joule heating on the MHD flow and heat transfer of a second-grade fluid past a radially stretching sheet. Appl. Math. Mech. 31, 159–173 (2010). https://doi.org/10.1007/s10483-010-0204-7

    Article  MathSciNet  MATH  Google Scholar 

  10. H. Yarmand, S. Gharehkhani, S.F.S. Shirazi, A. Amiri, E. Montazer, H.K. Arzani, R. Sadri, M. Dahari, S.N. Kazi, Nanofluid based on activated hybrid of biomass carbon/graphene oxide: synthesis, thermo-physical and electrical properties. Int. Commun. Heat Mass Transf. 72, 10–15 (2016). https://doi.org/10.1016/j.icheatmasstransfer.2016.01.004

    Article  Google Scholar 

  11. S. Suresh, K.P. Venkitaraj, P. Selvakumar, M. Chandrasekar, Effect of Al2O3–Cu/water hybrid nanofluid in heat transfer. Exp. Therm. Fluid Sci. 38, 54–60 (2012). https://doi.org/10.1016/j.expthermflusci.2011.11.007

    Article  Google Scholar 

  12. A. Moghadassi, E. Ghomi, F. Parvizian, A numerical study of water based Al2O3 and Al2O3–Cu hybrid nanofluid effect on forced convective heat transfer. Int. J. Therm. Sci. 92, 50–57 (2015). https://doi.org/10.1016/j.ijthermalsci.2015.01.025

    Article  Google Scholar 

  13. S. Kumar, P.K. Sharma, P. Rana, Critical values in transport phenomena for curved power-law sheet utilizing Al2O3–Cu/water hybrid nanoliquid: model prediction and stability analysis. Adv. Powder Technol. 30, 2787–2800 (2019). https://doi.org/10.1016/j.apt.2019.08.026

    Article  Google Scholar 

  14. H. Sadaf, S.I. Abdelsalam, Adverse effects of a hybrid nanofluid in a wavy non-uniform annulus with convective boundary conditions. RSC Adv. 10, 15035–15043 (2020). https://doi.org/10.1039/D0RA01134G

    Article  Google Scholar 

  15. U. Khan, A. Shafiq, A. Zaib, D. Baleanu, Hybrid nanofluid on mixed convective radiative flow from an irregular variably thick moving surface with convex and concave effects. Case Stud. Therm. Eng. 21, 100660 (2020). https://doi.org/10.1016/j.csite.2020.100660

    Article  Google Scholar 

  16. A. Bejan, Method of entropy generation minimization, or modeling and optimization based on combined heat transfer and thermodynamics. Rev. Générale Therm. 35, 637–646 (1996). https://doi.org/10.1016/S0035-3159(96)80059-6

    Article  Google Scholar 

  17. S. Rehman, R. Haq, Z.H. Khan, C. Lee, Entropy generation analysis for non-Newtonian nanofluid with zero normal flux of nanoparticles at the stretching surface. J. Taiwan Inst. Chem. Eng. 63, 226–235 (2016). https://doi.org/10.1016/j.jtice.2016.03.006

    Article  Google Scholar 

  18. M. Bhatti, T. Abbas, M. Rashidi, M. Ali, Z. Yang, M.M. Bhatti, T. Abbas, M.M. Rashidi, M.E.-S. Ali, Z. Yang, Entropy generation on MHD Eyring–Powell nanofluid through a permeable stretching surface. Entropy 18, 224 (2016). https://doi.org/10.3390/e18060224

    Article  ADS  MathSciNet  Google Scholar 

  19. P. Rana, N. Shukla, O. Anwar Bég, A. Kadir, B. Singh, Unsteady electromagnetic radiative nanofluid stagnation-point flow from a stretching sheet with chemically reactive nanoparticles, Stefan blowing effect and entropy generation. Proc. Inst. Mech. Eng. Part N. J. Nanomater. Nanoeng. Nanosyst. (2018). https://doi.org/10.1177/2397791418782030

    Article  Google Scholar 

  20. N. Shukla, P. Rana, O. Anwar Bég, Unsteady MHD non-Newtonian heat transfer nanofluids with entropy generation analysis. Nonlinear Eng. 8, 630–644 (2019). https://doi.org/10.1515/nleng-2017-0177

    Article  ADS  Google Scholar 

  21. M.I. Afridi, T.A. Alkanhal, M. Qasim, I. Tlili, Entropy generation in Cu–Al2O3–H2O Hybrid nanofluid flow over a curved surface with thermal dissipation. Entropy 21, 941 (2019). https://doi.org/10.3390/e21100941

    Article  ADS  Google Scholar 

  22. G.B. Jeffery, The two-dimensional steady motion of a viscous fluid. Philos. Mag. 29, 455–465 (1915). https://doi.org/10.1080/14786440408635327

    Article  MATH  Google Scholar 

  23. G. Hamel, Spiralförmige Bewegungen zäher Flüssigkeiten. Jahresber. Dtsch. Math.-Ver. 25, 34–60 (1917)

    MATH  Google Scholar 

  24. M.R. Sari, M. Kezzar, R. Adjabi, Heat transfer of copper/water nanofluid flow through converging-diverging channel. J. Cent. South Univ. 23, 484–496 (2016). https://doi.org/10.1007/s11771-016-3094-0

    Article  Google Scholar 

  25. J. Nagler, Jeffery–Hamel flow of non-Newtonian fluid with nonlinear viscosity and wall friction. Appl. Math. Mech. 38, 815–830 (2017). https://doi.org/10.1007/s10483-017-2206-8

    Article  MathSciNet  MATH  Google Scholar 

  26. A.S. Dogonchi, D.D. Ganji, Investigation of MHD nanofluid flow and heat transfer in a stretching/shrinking convergent/divergent channel considering thermal radiation. J. Mol. Liq. 220, 592–603 (2016). https://doi.org/10.1016/j.molliq.2016.05.022

    Article  Google Scholar 

  27. M. Adnan, U. Asadullah, N. Khan, S.T. Ahmed, Mohyud–Din, analytical and numerical investigation of thermal radiation effects on flow of viscous incompressible fluid with stretchable convergent/divergent channels. J. Mol. Liq. 224, 768–775 (2016). https://doi.org/10.1016/j.molliq.2016.10.073

    Article  Google Scholar 

  28. S.A. Shehzad, T. Hayat, A. Alsaedi, M.A. Obid, Nonlinear thermal radiation in three-dimensional flow of Jeffrey nanofluid: a model for solar energy. Appl. Math. Comput. 248, 273–286 (2014). https://doi.org/10.1016/j.amc.2014.09.091

    Article  MathSciNet  MATH  Google Scholar 

  29. E.R. Onyango, M.N. Kinyanjui, M. Kimathi, S.M. Uppal, Heat and mass transfer on MHD Jeffrey–Hamel flow in presence of inclined magnetic field. Appl. Comput. Math. 9, 102 (2020). https://doi.org/10.11648/j.acm.20200904.11

    Article  Google Scholar 

  30. J. Nagler, Jeffery–Hamel flow of nano fluid influenced by wall slip conditions. J. Nanofluids 5, 960–967 (2016). https://doi.org/10.1166/jon.2016.1282

    Article  Google Scholar 

  31. N. Freidoonimehr, M.M. Rashidi, Dual solutions for MHD Jeffery–Hamel nano-fluid flow in non-parallel walls using predictor homotopy analysis method. J. Appl. Fluid Mech. 8, 911–919 (2015)

    Article  Google Scholar 

  32. P. Rana, N. Shukla, Y. Gupta, I. Pop, Analytical prediction of multiple solutions for MHD Jeffery–Hamel flow and heat transfer utilizing KKL nanofluid model. Phys. Lett. A (2018). https://doi.org/10.1016/j.physleta.2018.10.026

    Article  MATH  Google Scholar 

  33. S. Abbasbandy, E. Shivanian, Prediction of multiplicity of solutions of nonlinear boundary value problems: novel application of homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul. 15, 3830–3846 (2010). https://doi.org/10.1016/j.cnsns.2010.01.030

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method (Chapman & Hall/CRC Press, London/Boca Ratton, 2003). https://www.crcpress.com/Beyond-Perturbation-Introduction-to-the-Homotopy-Analysis-Method/Liao/p/book/9781584884071. Accessed 28 June 2017

  35. M. Barzegar Gerdroodbary, M. Rahimi Takami, D.D. Ganji, Investigation of thermal radiation on traditional Jeffery–Hamel flow to stretchable convergent/divergent channels. Case Stud. Therm. Eng. 6, 28–39 (2015). https://doi.org/10.1016/j.csite.2015.04.002

    Article  Google Scholar 

  36. S. Suresh, K.P. Venkitaraj, P. Selvakumar, M. Chandrasekar, Synthesis of Al2O3–Cu/water hybrid nanofluids using two step method and its thermo physical properties. Colloids Surf. Physicochem. Eng. Asp. 388, 41–48 (2011). https://doi.org/10.1016/j.colsurfa.2011.08.005

    Article  Google Scholar 

  37. S.J. Liao, A new branch of solutions of boundary-layer flows over a permeable stretching plate. Int. J. Non-Linear Mech. 42, 819–830 (2007). https://doi.org/10.1016/j.ijnonlinmec.2007.03.007

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. S. Abbasbandy, E. Shivanian, Predictor homotopy analysis method and its application to some nonlinear problems. Commun. Nonlinear Sci. Numer. Simul. 16, 2456–2468 (2011). https://doi.org/10.1016/j.cnsns.2010.09.027

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. A. Moradi, A. Alsaedi, T. Hayat, Investigation of nanoparticles effect on the Jeffery–Hamel flow. Arab. J. Sci. Eng. 38, 2845–2853 (2013). https://doi.org/10.1007/s13369-012-0472-2

    Article  MathSciNet  Google Scholar 

  40. S.S. Motsa, P. Sibanda, G.T. Marewo, On a new analytical method for flow between two inclined walls. Numer. Algorithms. 61, 499–514 (2012). https://doi.org/10.1007/s11075-012-9545-2

    Article  MathSciNet  MATH  Google Scholar 

  41. M. Turkyilmazoglu, Extending the traditional Jeffery–Hamel flow to stretchable convergent/divergent channels. Comput. Fluids 100, 196–203 (2014). https://doi.org/10.1016/j.compfluid.2014.05.016

    Article  MathSciNet  MATH  Google Scholar 

  42. M. Esmaeilpour, D.D. Ganji, Solution of the Jeffery–Hamel flow problem by optimal homotopy asymptotic method. Comput. Math. Appl. 59, 3405–3411 (2010). https://doi.org/10.1016/j.camwa.2010.03.024

    Article  MathSciNet  MATH  Google Scholar 

  43. S.S. Motsa, P. Sibanda, F.G. Awad, S. Shateyi, A new spectral-homotopy analysis method for the MHD Jeffery–Hamel problem. Comput. Fluids 39, 1219–1225 (2010). https://doi.org/10.1016/j.compfluid.2010.03.004

    Article  MathSciNet  MATH  Google Scholar 

  44. O. AnwarBég, Multi-physical electro-magnetic propulsion fluid dynamics: mathematical modelling and computation, in Mathematical Modelling: Methods, Application and Research, ed. by W. Willis, S. Sparks (Nova Science, New York, 2018), p. 88

    Google Scholar 

  45. O. Anwar Bég, S.S. Motsa, M.N. Islam, M. Lockwood, Pseudo-spectral and variational iteration simulation of exothermically reacting Rivlin–Ericksen viscoelastic flow and heat transfer in a rocket propulsion duct. Comput. Therm. Sci. 6(2), 91–102 (2014)

    Article  Google Scholar 

  46. S.I. Abdelsalam, M.M. Bhatti, A. Zeeshan, A. Riaz, O. Anwar Bég, Metachronal propulsion of magnetized particle-fluid suspension in a ciliated channel with heat and mass transfer. Physica Scr. 94, 115301 (2019)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

The authors wish to express their sincere thanks to the editor and the reviewers for their valuable comments and suggestions which have considerably improved the clarity of the manuscript. In addition, the contribution of our collaborators (Prof. O. Anwar Bég, UK and Dr. Gaurav Gupta, China) to the manuscript is highly appreciated.

Author information

Authors and Affiliations

  1. Department of Mathematics, Institute of Applied Science and Humanities, GLA University, Mathura, 281406, Uttar Pradesh, India

    Nisha Shukla

  2. School of Mathematical Sciences, College of Science and Technology, Wenzhou-Kean University, Wenzhou, China

    Puneet Rana

  3. Department of Applied Mathematics, Babeş-Bolyai University, 400084, Cluj-Napoca, Romania

    Ioan Pop

Authors
  1. Nisha Shukla
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Puneet Rana
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ioan Pop
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Puneet Rana.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shukla, N., Rana, P. & Pop, I. Second law thermodynamic analysis of thermo-magnetic Jeffery–Hamel dissipative radiative hybrid nanofluid slip flow: existence of multiple solutions. Eur. Phys. J. Plus 135, 849 (2020). https://doi.org/10.1140/epjp/s13360-020-00822-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/s13360-020-00822-w

Search

Navigation