SpringerLink
Log in

Rayleigh–Bénard Convection in a Bi-viscous Bingham Fluid with Weak Vertical Harmonic Oscillations: Linear and Non-linear Analyses

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

Abstract

Linear and weakly non-linear stability analyses of Rayleigh–Bénard convection in a bi-viscous Bingham fluid layer are performed in the presence of vertical harmonic vibrations. In the linear analysis, expression is obtained for the correction Rayleigh-number arising due to the vibrations. The non-linear-analysis based on the Ginzburg–Landau equation is used to compute the Nusselt-number in terms of the correction Rayleigh-number. The mean-Nusselt-number is then obtained as a function of the scaled-Rayleigh-number, the frequency and the amplitude of modulation, the Prandtl number, and the bi-viscous Bingham fluid parameter. The non-autonomous amplitude-equation is numerically solved using the Runge–Kutta–Fehlberg45 method. It is found that the influence of increasing the amplitude of modulation is to result in a delayed-onset situation and thereby to an enhanced-heat-transport situation. For small and moderate frequencies, the influence of increasing the frequency of oscillations is to decrease the critical Rayleigh-number. However, the mean-Nusselt-number decreases with increase in the frequency of oscillations only in the case of small frequencies. An increase in the value of the bi-viscous Bingham fluid parameter leads to advanced-onset and thereby to an enhanced-heat-transport situation. At very large frequencies, the effect of modulation on onset and heat-transport ceases.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Not applicable.

Code Availability

Not applicable.

References

  1. Bénard, H.: Les tourbillons cellulaires dans une nappe liquide.—Méthodes optiques d’observation et d’enregistrement. J. Phys. Théor. Appl. 23, 254–266 (1901)

  2. Rayleigh, L.: On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side. Philos. Mag. 32, 529–546 (1916)

    Article  MATH  Google Scholar 

  3. Siddheshwar, P.G.: A series solution for the Ginzburg-Landau equation with a time-periodic coefficient. Appl. Math. 3, 542–554 (2010)

    Article  Google Scholar 

  4. Bhadauria, B.S., Kiran, P.: Weak nonlinear analysis of magneto-convection under magnetic field modulation. Physica Scr. 89, 1–10 (2014)

    Article  Google Scholar 

  5. Malashetty, M.S., Padmavathi, V.: Effect of gravity modulation on the onset of convection in a fluid and porous layer. Int. J. Eng. Sci. 35, 829–840 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Siddheshwar, P.G., Bhadauria, B.S., Mishra, P., Srivastava, A.K.: Study of heat transport by stationary magneto-convection in a Newtonian liquid under temperature or gravity modulation using Ginzburg–Landau model. Int. J. Non-Linear Mech. 47, 418–425 (2012)

    Article  Google Scholar 

  7. Siddheshwar, P.G., Meenakshi, N.: Amplitude equation and heat transport for Rayleigh Bénard convection in Newtonian liquids with nanoparticles. Int. J. Appl. Comput. Math. 3, 271–292 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gresho, P.M., Sani, R.L.: The effects of gravity modulation on the stability of a heated fluid layer. J. Fluid Mech. 40, 783–806 (1970)

    Article  MATH  Google Scholar 

  9. Kiran, P., Narasimhulu, Y.: Weakly non-linear oscillatory convection in an electrically conduction fluid layer under gravity modulation. Int. J. Appl. Comput. Math. 3, 1969–1983 (2016)

    Article  MATH  Google Scholar 

  10. Frigaard, I.A., Poole, R.J.: Comment on “Bejan’s flow visualization of buoyancy-driven flow of a hydromagnetic Casson fluid from an isothermal wavy surface’’. Phys. Fluids 33, 1–2 (2021)

    Article  Google Scholar 

  11. Hayat, T., Shehzad, S.A., Alsaedi, A.: Soret and Dufour effects on magnetohydrodynamic (MHD) flow of Casson fluid. Appl. Math. Mech. (Engl. Edition) 33, 1301–1312 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Raza, J., Rohni, A.M., Omar, Z.: Multiple solutions of mixed convective MHD Casson fluid flow in a channel. J. Appl. Math. 2016, 1–10 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Shehzad, S.A., Hayat, T., Alsaedi, A.: Three-dimensional MHD flow of Casson fluid in porous medium with heat generation. J. Appl. Fluid Mech. 9, 215–223 (2016)

    Article  Google Scholar 

  14. Bilal Ashraf, M., Hayat, T., Alsaedi, A.: Mixed convection flow of Casson fluid over a stretching sheet with convective boundary conditions and Hall effect. Bound. Value Probl. 137, 1–17 (2017)

    MathSciNet  MATH  Google Scholar 

  15. Ganesh Kumar, K., Ramesh, G.K., Gireesha, B.J.: Impact of thermal radiation on double-diffusive convection flow of Casson fluid over a stretching vertical surface. Front. Heat Mass Transf. 5, 1–8 (2016)

    Google Scholar 

  16. Parmar, A., Jain, S.: Exploration of heat and mass transfer in the convective slip flow of non-Newtonian Casson fluid. Int. J. Appl. Comput. Math. 4, 67–79 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mukhopadhyay, S., De Ranjan, P., Bhattacharyya, K., Layek, G.C.: Casson fluid flow over an unsteady stretching surface. Ain Shams Eng. J. 4, 933–938 (2013)

    Article  Google Scholar 

  18. Mustafa, M., Hayat, T., Pop, I., Aziz, A.: Unsteady boundary layer flow of a Casson fluid due to an impulsively started moving flat plate. Heat Transf.-Asian Res. 40, 563–576 (2011)

    Article  Google Scholar 

  19. Aghighi, M.S., Ammar, A., Metivier, C., Gharagozlu, M.: Rayleigh–Bénard convection of Casson fluids. Int. J. Therm. Sci. 127, 79–90 (2018)

    Article  Google Scholar 

  20. Turan, O., Chakraborty, N., Poole, R.J.: Laminar natural convection of Bingham fluids in a square enclosure with differentially heated side walls. J. Non-Newton. Fluid Mech. 165, 901–913 (2010)

    Article  MATH  Google Scholar 

  21. Aghighi, M.S., Metivier, C., Masoumi, H.: Natural convection in a partially heated porous cavity to Casson fluid. Multidiscip. Model. Mater. Struct. 16, 1245–1259 (2020)

    Article  Google Scholar 

  22. Aneja, M., Chandra, A., Sharma, S.: Natural convection in a partially heated porous cavity to Casson fluid. Int. Commun. Heat Mass Transf. 114, 1–15 (2020)

    Article  Google Scholar 

  23. Das, M., Mahanta, G., Shaw, S., Parida, S.B.: Unsteady MHD chemically reactive double-diffusive Casson fluid past a flat plate in porous medium with heat and mass transfer. Heat Transf.-Asian Res. 48, 1761–1777 (2019)

    Article  Google Scholar 

  24. Shruthy, M., Mahanthesh, B.: Rayleigh–Bénard convection in Casson and hybrid nanofluids: an analytical investigation. J. Nanofluids 8, 222–229 (2019)

    Article  Google Scholar 

  25. Nakamura, M., Sawada, T.: Numerical study on the flow of a non-Newtonian fluid through an axisymmetric stenosis. J. Biomech. Eng. 110, 137–143 (1988)

    Article  Google Scholar 

Download references

Acknowledgements

We would like to acknowledge the support of CHRIST (Deemed to be University) and Universiti Malaysia Terengganu through the International Partnership Research Grant (IPRG). The authors are grateful to the referees for most useful comments that improved our paper immensely.

Funding

This work was supported by CHRIST (Deemed to be University) and Universiti Malaysia Terengganu through the International Partnership Research Grant (IPRG).

Author information

Authors and Affiliations

  1. Department of Mathematics, Centre for Mathematical Needs, CHRIST (Deemed to be University), Bengaluru, 560029, India

    S. Keerthana & P. G. Siddheshwar

  2. Department of Professional Studies, CHRIST (Deemed to be University), Bengaluru, 560029, India

    Sameena Tarannum

  3. Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu, 21030, Kuala Nerus, Terengganu, Malaysia

    Ruwaidiah Idris

Authors
  1. S. Keerthana
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. G. Siddheshwar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sameena Tarannum
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ruwaidiah Idris
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All the authors in the paper contributed equally to this work.

Corresponding author

Correspondence to S. Keerthana.

Ethics declarations

Conflict of interest

The authors declare that they have 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

Keerthana, S., Siddheshwar, P.G., Tarannum, S. et al. Rayleigh–Bénard Convection in a Bi-viscous Bingham Fluid with Weak Vertical Harmonic Oscillations: Linear and Non-linear Analyses. Int. J. Appl. Comput. Math 9, 30 (2023). https://doi.org/10.1007/s40819-023-01495-6

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40819-023-01495-6

Keywords

Search

Navigation