Stability analysis of thermosolutal convection in a rotating Navier–Stokes–Voigt fluid

Sweta Sharma; Sunil; Poonam Sharma

doi:10.1515/zna-2023-0284

Article

Stability analysis of thermosolutal convection in a rotating Navier–Stokes–Voigt fluid

Sweta Sharma , Sunil and Poonam Sharma

Published/Copyright: March 14, 2024

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Zeitschrift für Naturforschung A Volume 79 Issue 7

Abstract

This work presents nonlinear and linear analyses of the rotating Navier–Stokes–Voigt fluid layer that is simultaneously heated and soluted from below, considering different boundary surfaces. The energy method is used to form the eigenvalue problem for nonlinear analysis, whereas the normal mode analysis is used for the linear analysis. The Rayleigh number is numerically calculated by employing the Galerkin technique. Both nonlinear and linear analyses yield the same Rayleigh number, indicating the absence of subcritical regions and implying global stability. The Kelvin–Voigt parameter doesn’t affect the Rayleigh number for stationary convection. However, the crucial role of this parameter is established through an energy argument. The presence of rotation, Kelvin–Voigt parameter, and solute gradient give rise to oscillatory modes. Also, the effects of rotation and solute gradient are stabilizing on the system, whereas the stabilizing effect of the Kelvin–Voigt parameter becomes evident when convection exhibits an oscillatory behavior.

Keywords: thermosolutal convection; rotation; Navier–Stokes–Voigt fluid; Kelvin–Voigt parameter; nonlinear and linear analyses

Corresponding author: Sunil, Department of Mathematics and Scientific Computing, National Institute of Technology Hamirpur, Hamirpur, HP 177005, India, E-mail: sunil@nith.ac.in

Acknowledgments

The authors express their gratitude to the reviewers for their invaluable comments and constructive suggestions, which have significantly contributed to the enhancement of the quality of this research work.

Research ethics: Not applicable.
Author contributions: All authors have contributed equally to this study. The authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Competing interests: The authors state no conflict of interest.
Research funding: None declared.
Data availability: Not applicable.

Appendix A

The real and imaginary parts of the Rayleigh number (Ra_L) for respective bounding surfaces are provided below.

A.1 For free-free bounding surfaces

Ra r = a 4 1 + b 1 + Pr ω 2 − 1 + b 1 Le Pr + π 2 Pr λ ω 2 × − 2 + b 2 + π 4 3 + b 1 + 2 b 2 + 1 a 2 π 2 − 1 + b 2 ( Pr ω 2 + π 2 Pr λ ω 2 + π 4 ) + a 2 − Pr λ ω 2 + π 2 3 + 2 b 1 + b 2 ,

Ra i = a 4 λ + a 2 1 + 3 π 2 λ + Pr + ( 1 − Le ) b 1 − π 2 λ b 2 + π 2 2 − b 2 + π 2 λ ( 3 − 2 b 2 ) + π 2 Pr 2 + ( 1 − Le ) b 1 + b 2 ) + π 4 a 2 1 + Pr − b 2 + Pr b 2 + π 2 ( λ − λ b 2 ) ,

where b 1 = Ra s ( a 2 + π 2 ) 2 + L e 2 P r 2 ω 2 , b 2 = Ta ( a 2 + π 2 ) 2 + ( ω + ( a 2 + π 2 ) λ ω ) 2 .

A.2 For rigid-free bounding surfaces

Ra r = 1 9025 ( 589 a 4 ( 16 + 14725 b 3 ) + 4 a 2 ( 76824 + 42805575 b 3 + 1225926 b 4 + 108 ( 40416 + 7824675 b 3 + 594880 b 4 ) + 8673025 b 3 Le Pr 2 ω 2 + 248 Pr ( − 38 + 9 ( − 96 + 2197 b 4 ) λ ) ω 2 − 2356 Pr λ ω 2 ) + 72 ( 12852 ( 24 + 169 b 4 ) + 31 Pr ( − 48 ( 1 + 21 λ ) + 169 b 4 ( 13 + 42 λ ) ) ω 2 ) a 2 ) ,

Ra i = 1 9025 a 2 ( 9424 a 4 λ + a 4 ( 9424 − 589 ( − 16 + 14725 b 3 × ( − 1 + Le ) ) Pr − 72 ( − 4268 + 68107 b 4 ) λ ) + 432 ( 217 Pr ( 24 + 169 b 4 ) + 2448 ( 1 + 21 λ ) − 8619 b 4 ( 13 + 42 λ ) ) − 18 a 2 ( 31 Pr ( 153425 b 3 × ( − 1 + Le ) − 4 ( 96 + 2197 b 4 ) ) + 676 b 4 × ( 403 + 5280 λ ) − 16 ( 695 + 15156 λ ) ) ) ,

where b 3 = Ra s ( 306 + 31 a 2 ) 2 + 961 Le 2 Pr 2 ω 2 , b 4 = Ta ( 42 + 13 a 2 ) 2 + ( 13 + 42 λ + 13 a 2 λ ) 2 ω 2 .

A.3 For rigid-rigid bounding surfaces

Ra r = 1 121 ( 31 a 4 ( 4 + 3751 b 3 ) + 396 ( 232 + 28611 b 3 + 2160 b 4 ) + 116281 Le Pr 2 ω 2 + 62 Pr ( − 2 + 3 ( − 16 + 351 b 4 ) λ ) ω 2 + 2 a 2 × 2100 + 1147806 b 3 + 32643 b 4 − 62 Pr λ ω 2 + 77112 ( 8 + 27 b 4 ) + 186 Pr ( − 8 + 351 b 4 + 42 ( − 8 + 27 b 4 ) λ ) ω 2 a 2 ) ,

Ra i = 1 121 ( 124 a 4 λ + a 2 ( 128 − 116281 b 3 ( − 1 + Le ) Pr + ( 4200 − 65286 b 4 ) λ ) + 6 ( 452 − 10881 b 4 + 31 × Pr ( 16 − 6171 b 3 ( − 1 + Le ) + 351 b 4 ) + 528 ( 29 − 270 b 4 ) λ ) + 36 ( 217 Pr ( 8 + 27 b 4 ) + 51 ( 9 + 336 λ − 27 b 4 ( 13 + 42 λ ) ) ) a 2 ) .

References

[1] G. Veronis, “On finite amplitude instability in thermohaline convection,” J. Mar. Res., vol. 23, no. 1, pp. 1–17, 1965.Search in Google Scholar

[2] J. S. Turner, “Double-diffusive phenomena,” Annu. Rev. Fluid. Mech., vol. 6, no. 1, pp. 37–54, 1974. https://doi.org/10.1146/annurev.fl.06.010174.000345.Search in Google Scholar

[3] H. E. Huppert and J. S. Turner, “Double-diffusive convection,” J. Fluid Mech., vol. 106, no. 1, pp. 299–329, 1981. https://doi.org/10.1017/S0022112081001614.Search in Google Scholar

[4] J. K. Platten and J. C. Legros, Convection in Liquids, Berlin, Springer Science & Business Media, 2012.Search in Google Scholar

[5] Sunil, P. K. Bharti, and R. C. Sharma, “Thermosolutal convection in ferromagnetic fluid,” Arch. Mech., vol. 56, no. 2, pp. 117–135, 2004.Search in Google Scholar

[6] Sunil, Y. D. Sharma, P. K. Bharti, and R. C. Sharma, “Thermosolutal instability of compressible Rivlin-Ericksen fluid with Hall currents,” Int. J. Appl. Mech. Eng., vol. 10, no. 2, pp. 329–343, 2005.Search in Google Scholar

[7] Sunil, P. Sharma, and A. Mahajan, “A nonlinear stability analysis of a rotating double-diffusive magnetized ferrofluid,” Appl. Math. Comput., vol. 218, no. 6, pp. 2785–2799, 2011. https://doi.org/10.1016/j.amc.2011.08.021.Search in Google Scholar

[8] B. Straughan, “Competitive double diffusive convection in a Kelvin–Voigt fluid of order one,” Appl. Math. Opt., vol. 84, no. S1, pp. 631–650, 2021. https://doi.org/10.1007/s00245-021-09781-9.Search in Google Scholar

[9] B. M. Shankar, J. Kumar, and I. S. Shivakumara, “Stability of double-diffusive natural convection in a vertical fluid layer,” Phys. Fluids, vol. 33, no. 9, p. 094113, 2021. https://doi.org/10.1063/5.0056350.Search in Google Scholar

[10] S. Sengupta and A. S. Gupta, “Thermohaline convection with finite amplitude in a rotating fluid,” Z. Angew. Math. Phys., vol. 22, no. 5, pp. 906–914, 1971. https://doi.org/10.1007/BF01591818.Search in Google Scholar

[11] P. Bhatia and J. Steiner, “Convective instability in a rotating viscoelastic fluid layer,” Appl. Math. Mech., vol. 52, no. 6, pp. 321–327, 1972. https://doi.org/10.1002/zamm.19720520601.Search in Google Scholar

[12] A. J. Pearlstein, “Effect of rotation on the stability of a doubly diffusive fluid layer,” J. Fluid Mech., vol. 103, no. 1, pp. 389–412, 1981. https://doi.org/10.1017/S0022112081001390.Search in Google Scholar

[13] U. Gupta and V. Kumar, “Thermosolutal instability of a compressible rotating Walters’ (model B′) elastico-viscous fluid in the presence of Hall currents,” Chem. Eng. Commun., vol. 197, no. 9, pp. 1225–1239, 2010. https://doi.org/10.1080/00986440903574925.Search in Google Scholar

[14] K. R. Raghunatha and I. S. Shivakumara, “Double-diffusive convection in a rotating viscoelastic fluid layer,” Appl. Math. Mech., vol. 101, no. 4, p. e201900025, 2021. https://doi.org/10.1002/zamm.201900025.Search in Google Scholar

[15] V. A. Pavlovskii, “On theoretical description of weak aqueous solutions of polymers,” Dokl. Akad. Nauk SSSR, vol. 200, no. 4, pp. 809–812, 1971.Search in Google Scholar

[16] A. P. Oskolkov, “Initial-boundary value problems for equations of motion of Kelvin--Voight fluids and Oldroyd fluids,” Tr. Mat. Inst. im. V. A. Steklova, vol. 179, pp. 126–164, 1988.Search in Google Scholar

[17] A. P. Oskolkov, “Nonlocal problems for the equations of motion of Kelvin–Voight fluids,” J. Math. Sci., vol. 75, no. 6, pp. 2058–2078, 1995. https://doi.org/10.1007/BF02362946.Search in Google Scholar

[18] V. G. Zvyagin and M. V. Turbin, “The study of initial-boundary value problems for mathematical models of the motion of Kelvin–Voigt fluids,” J. Math. Sci., vol. 168, no. 2, pp. 157–308, 2010. https://doi.org/10.1007/s10958-010-9981-2.Search in Google Scholar

[19] L. C. Berselli and L. Bisconti, “On the structural stability of the Euler–Voigt and Navier–Stokes–Voigt models,” Nonlinear Anal. Theor. Methods Appl., vol. 75, no. 1, pp. 117–130, 2012. https://doi.org/10.1016/j.na.2011.08.011.Search in Google Scholar

[20] B. Straughan, “Thermosolutal convection with a Navier–Stokes–Voigt fluid,” Appl. Math. Opt., vol. 84, no. 3, pp. 2587–2599, 2020. https://doi.org/10.1007/s00245-020-09719-7.Search in Google Scholar

[21] Sunil, P. Sharma, and A. Mahajan, “A nonlinear stability analysis of a rotating double-diffusive magnetized ferrofluid saturating a porous medium,” Heat Tran. Res., vol. 40, no. 4, pp. 351–378, 2009. https://doi.org/10.1615/HeatTransRes.v40.i4.60.Search in Google Scholar

[22] B. Straughan, The Energy Method, Stability, and Nonlinear Convection, New York, Springer Science & Business Media, 2004.10.1007/978-0-387-21740-6Search in Google Scholar

[23] B. Straughan, “Nonlinear stability for convection with temperature dependent viscosity in a Navier–Stokes–Voigt fluid,” Eur. Phys. J. Plus, vol. 138, p. 438, 2023. https://doi.org/10.1140/epjp/s13360-023-04055-5.Search in Google Scholar

[24] Z. Li and R. E. Khayat, “Three-dimensional thermal convection of viscoelastic fluids,” Phys. Rev. E, vol. 71, no. 6, p. 066305, 2005. https://doi.org/10.1103/physreve.71.066305.Search in Google Scholar

[25] Z. Li and R. E. Khayat, “Finite-amplitude Rayleigh–Bénard convection and pattern selection for viscoelastic fluids,” J. Fluid Mech., vol. 529, pp. 221–251, 2005, https://doi.org/10.1017/s0022112005003563.Search in Google Scholar

[26] M. S. Malashetty and M. Swamy, “The onset of double diffusive convection in a viscoelastic fluid layer,” J. Non-Newtonian Fluid Mech., vol. 165, no. 19–20, pp. 1129–1138, 2010. https://doi.org/10.1016/j.jnnfm.2010.05.011.Search in Google Scholar

[27] S. Choudhary and Sunil, “Global stability for double-diffusive convection in a couple-stress fluid saturating A porous medium,” Stud. Geotech. Mech., vol. 41, no. 1, pp. 13–20, 2019. https://doi.org/10.2478/sgem-2018-0044.Search in Google Scholar

[28] R. Yang, I. C. Ivan, I. M. Griffiths, and G. Z. Ramon, “Time-averaged transport in oscillatory squeeze flow of a viscoelastic fluid,” Phys. Rev. Fluids, vol. 5, no. 9, p. 094501, 2020. https://doi.org/10.1103/PhysRevFluids.5.094501.Search in Google Scholar

[29] F. Capone, R. D. Luca, and P. Vadasz, “Onset of thermosolutal convection in rotating horizontal nanofluid layers,” Acta Mech., vol. 233, no. 6, pp. 2237–2247, 2022. https://doi.org/10.1007/s00707-022-03217-3.Search in Google Scholar

[30] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, New York, Dover Publications, 1981.Search in Google Scholar

Received: 2023-10-19

Accepted: 2024-02-20

Published Online: 2024-03-14

Published in Print: 2024-07-26

You are currently not able to access this content.

Articles in the same Issue

https://doi.org/10.1515/zna-2023-0284

Keywords for this article

thermosolutal convection; rotation; Navier–Stokes–Voigt fluid; Kelvin–Voigt parameter; nonlinear and linear analyses