Nonlinear hydromagnetic instability of oscillatory rotating rigid-fluid columns

Abstract

The current paper is concerned with the nonlinear stability analysis of rotating magnetic fluid columns. The rotation sources are a mixture of both uniform and oscillating behavior. The motivation behind tackling this topic is the increasing interest in atmospheric and oceanic motions. The system consists of two magnetic phase fluid that fills two infinite vertical cylinders. An azimuthal uniform magnetic field is penetrated on the system. The governing equations of motion, in terms of the Coriolis force and reduced pressure, along with Maxwell’s equation in the quasi-static approximations are considered. Consequently, the disturbance of the interface has an azimuthal behavior. The fluids are fully saturated in porous media. In light of the implication of the nonlinear boundary conditions, the solutions of the linearized equations of motion resulted in a nonlinear characteristic dispersion equation. Utilizing the homotopy perturbation technique, this equation is analyzed. A modification of the latter equation is made to seem like a nonlinear Klein–Gordon equation. The stability criteria are realized in linear as well as nonlinear approaches. A set of diagrams is graphed to illustrate the effects of several non-dimensional numbers on the stability profile in resonance as well as non-resonance cases.

This is a preview of subscription content, access via your institution.

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

References

  1. [1]

    R. E. Rosensweig Ferrohydrodynamics. (Cambridge: Cambridge University Press) (1985)

    Google Scholar 

  2. [2]

    A. S. Lűbbe, C. Alexiou and C. Bergemann J. Surg. Res. 95 200 (2001).

    Article  Google Scholar 

  3. [3]

    M. Sankar, M. Venkatachalappa and I. S. Shivakumara Int. J. Eng. Sci. 44 1556 (2006).

    Article  Google Scholar 

  4. [4]

    P. Yecko Phys. Fluids 21 034102 (2009).

    ADS  Article  Google Scholar 

  5. [5]

    N. Girish, O. D. Makinde and M. Sankar Defect Diffus. Forum 387 442 (2018).

    Article  Google Scholar 

  6. [6]

    Y. O. El-Dib and A. A. Mady J. Comp. Appl. Mech. 49 261 (2018).

    Google Scholar 

  7. [7]

    M. Venkatachalappa, Y. Do and M. Sankar Int. J. Eng. Sci. 49 262 (2011).

    Article  Google Scholar 

  8. [8]

    M. Sankar and J. Park D Kim and Y Do Numer Heat Tr. 63 687 (2013).

    Article  Google Scholar 

  9. [9]

    R. H. Roberts and A. M. Soward Rotating fluids in geophysics. (New York: Academic Press) (1978)

    Google Scholar 

  10. [10]

    E. J. Hopfinger Rotating Fluids in Geophysical and Industrial Applications. (Wien: Springer) (1992)

    Google Scholar 

  11. [11]

    K. Neumann, K. Gladyszewski, K. Groß, H. Qammar, D. Wenzel, A. Górak and M. Skiborowski Chem. Eng. Res. Des. 134 443 (2018).

    Article  Google Scholar 

  12. [12]

    P. Vadasz Fluids 4 147 (2019).

    Article  Google Scholar 

  13. [13]

    M. Basta, V. Picciarelli and R. Stella Phys. Edu. 35 120 (2000).

    Article  Google Scholar 

  14. [14]

    M. Venkatachalappa, M. Sankar and A. A. Natarajan Acta Mech. 147 173 (2001).

    Article  Google Scholar 

  15. [15]

    R. W. Lenz and R. S. Stein Philos. Mag. J. Sci. 34 145 (1892).

    Article  Google Scholar 

  16. [16]

    L. M. Hocking and D. H. Michael Mathematica 6 25 (1959).

    Google Scholar 

  17. [17]

    D. D. Joseph, Y. Renardy, M. Renardy and K. Nguyen J. Fluid Mech. 153 151 (1985).

    ADS  MathSciNet  Article  Google Scholar 

  18. [18]

    A. H. Nayfeh Phys. Fluids 15 1751 (1972).

    ADS  Article  Google Scholar 

  19. [19]

    R. Raghavan and S. S. Marsden Q. J. Mech. Appl. Math. 26 205 (1973).

    Article  Google Scholar 

  20. [20]

    J. Bishnoi and S. C. Agrawal Indian J. Appl. Math. 22 611 (1991).

    Google Scholar 

  21. [21]

    R. C. Sharma and P. Kumar Indian J. Appl. Math. 24 563 (1993).

    Google Scholar 

  22. [22]

    G. M. Moatimid and M. H. Zekry Microsyst. Technol. 26 2013 (2020).

    Article  Google Scholar 

  23. [23]

    L. Xu and Z. Li Acta Math. Sci. 39B 119 (2019).

    Article  Google Scholar 

  24. [24]

    Y. O. El-Dib, G. M. Moatimid and A. M. Mady Chin. J. Phys. 66 285 (2020).

    Article  Google Scholar 

  25. [25]

    P. D. Weidman, M. Goto and A. Fridberg ZAMP 48 921 (1997).

    ADS  Google Scholar 

  26. [26]

    J. R. Melcher Field Coupled Surface Waves. (Cambridge: MIT Press) (1963)

    Google Scholar 

  27. [27]

    S. Chandrasekhar Hydrodynamic and Hydromagnetic Stability. (Cambridge: Cambrigde University Press) (1961)

    Google Scholar 

  28. [28]

    R Mousa PhD Thesis (University of Wisconsin-Milwaukee, Wisconsin) (2014)

  29. [29]

    J. H. He Comput. Method Appl. Mech. Eng. 178 257 (1999).

    ADS  Article  Google Scholar 

  30. [30]

    Y. O. El-Dib J. Appl. Comput. Mech. 4 260 (2018).

    Google Scholar 

  31. [31]

    G. M. Moatimid, F. M. F. Elsabaa and M. H. Zekry J. Appl. Comput. Mech. 6 1404 (2020).

    Google Scholar 

  32. [32]

    A. A. Fedorov, A. S. Berdnikov and V. E. Kurochkin J. Math. Chem. 57 971 (2019).

    MathSciNet  Article  Google Scholar 

  33. [33]

    Y. O. El-Dib, G. M. Moatimid and A. A. Mady Pramana-J. Phys. 93 82 (2019).

    ADS  Article  Google Scholar 

  34. [34]

    M. F. El-Sayed, G. M. Moatimid, F. M. F. Elsabaa and M. F. E. Amer Atomiz. Sprays 26 349 (2016).

    Article  Google Scholar 

  35. [35]

    Y. O. El-Dib and G. M. Moatimid Arab. J. Sci. Eng. 44 6581 (2019).

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Marwa H. Zekry.

Additional information

Publisher's Note

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

Appendix

Appendix

The coefficients that appear in the Eqs. (4) and (5) may be listed as follows:

$$\begin{aligned} & a_{0} = - \frac{{R^{2} }}{m}(1 + \rho ),\,\,\,a_{1} = - \frac{{Oh\,R^{2} }}{m}(1 + \lambda ),\,\,\,a_{2} = R(1 - \rho ),\,\,a_{3} = Oh\,R(1 - \lambda ), \\ & a_{4} = \frac{1}{{R^{2} }}\left( { - 2 + m(7m + 2(2\hat{W}e + We^{*} )\,R^{3} - m(2\hat{W}e + We^{*} )\,R^{3} + m(2 + m)(2\hat{W}e\hat{\Omega }^{2} + We^{*} \Omega^{*2} )\,R^{3} \rho )} \right), \\ & a_{5} = 2m(1 + \rho ),a_{6} = 2m\,Oh(1 + \lambda ), \\ & a_{7} = \frac{{m^{2} }}{{2R^{3} }}\left( { - 16m + 3m^{2} + 2(2 - m)(2\hat{W}e + We^{*} )R^{3} - 2(2 + m)(2\hat{W}e\hat{\Omega }^{2} + We^{*} \Omega^{*2} )\,R^{3} \rho } \right), \\ & b_{1} = - \frac{1}{m}(m - 1 + (1 + m)\rho \hat{\Omega })R^{2} Oh\sqrt {\hat{T}a} ,\,b_{2} = - \frac{1}{2}(Oh)^{2} \sqrt {\hat{T}a} \,R^{2} (1 + \lambda \,\hat{\Omega }), \\ & b_{3} = - \frac{1}{2}Oh\sqrt {\hat{T}a} \,R^{2} (2 - 3m + (2 + 3m)\rho \,\hat{\Omega }), \\ & b_{4} = \frac{m}{2}(Oh)^{2} \sqrt {\hat{T}a} \,R^{2} (1 - \lambda \,\hat{\Omega }),\,\,\,b_{5} = - \frac{1}{2}Oh\sqrt {\hat{T}a} \,R^{2} ( - 2 + m + (2 + m)\rho \,\hat{\Omega }), \\ & b_{6} = m^{2} (Oh)^{2} \sqrt {\hat{T}a} \,(1 + \lambda \,\hat{\Omega }), \\ & c_{0} = c_{0a} - c_{0b} H_{0}^{2} , \\ \end{aligned}$$
$$\begin{aligned} & c_{0a} = \frac{1}{8R}\left( {8 - 8m^{2} + \left( {8\hat{W}e + Ta^{*} \,(Oh)^{2} } \right)R^{3} (m - 2) + (2 + m)\left( {8\hat{W}e\,\hat{\Omega }^{2} + Ta^{*} \,(Oh)^{2} \Omega^{*2} } \right)} \right), \\ & c_{0b} = \frac{{m(1 - \mu )^{2} }}{(1 + \mu )}, \\ & c_{1} = - \frac{{Oh\,\sqrt {Ta^{*} } R^{2} }}{m}( - 1 + m + (1 + m)\,\rho \,\Omega^{*} ),\,\,c_{2} = \frac{{(Oh)^{2} \,\sqrt {Ta^{*} } R^{2} }}{8}( - 2 + m + (2 + m)\,\rho \,\Omega^{*} ), \\ & c_{3} = \frac{{(Oh)^{2} \,\sqrt {\hat{T}a\,Ta^{*} } R^{2} }}{2}( - 2 + m + (2 + m)\,\rho \,\hat{\Omega }\,\Omega^{*} ),\,\,c_{4} = - \frac{{(Oh)^{2} \,\sqrt {Ta^{*} } R^{2} }}{2}(1 + \lambda \,\Omega^{*} ),\, \\ & c_{5} = \sqrt {We^{*} } R^{2} (1 + \rho \,\Omega^{*} ), \\ & c_{6} = \frac{{ - Oh\sqrt {Ta^{*} } R}}{2}(2 - 3m + (2 + 3m)\rho \,\Omega^{*} ),\,\,c_{7} = \frac{{m\,R\,We^{*} }}{2}(2 - m + (2 + m)\rho \,\Omega^{*2} ), \\ & c_{8} = 2m\,R\,\sqrt {\hat{W}e\,We^{*} } (2 - m + (2 + m)\,\rho \,\hat{\Omega }\,\Omega^{*} ),\,\,c{}_{9} = \frac{{m(Oh)^{2} \,\sqrt {Ta^{*} } R}}{2}(1 + \lambda \,\Omega^{*} ), \\ & c_{10} = - m\,R\,\sqrt {We^{*} } (1 - \rho \,\Omega^{*} ),\,\,c_{11} = m\,Oh\sqrt {Ta^{*} } (m - 2 + (2 + m)\rho \,\Omega^{*} ), \\ & c_{12} = m^{2} \,\,We^{*} (m - 2 + (m + 2)\rho \,\Omega^{*2} ),\,c_{13} = - 4m^{2} \,\sqrt {\hat{W}e\,We^{*} } (2 - m + (2 + m)\,\rho \,\hat{\Omega }\,\Omega^{*} ), \\ & c_{14} = m^{2} \,(Oh)^{2} \sqrt {Ta^{*} } (1 + \lambda \,\Omega^{*} ),\,c_{15} = - 2m^{2} \,We^{*} (1 + \rho \,\Omega^{*} ),d_{1} = - \frac{{4m^{2} \mu (1 - \mu )H_{0}^{2} }}{{R(1 + \mu )^{2} }},\,\hat{d}_{2} = \frac{{2m^{3} (1 - \mu )^{2} }}{{R^{2} (1 + \mu )}}. \\ \end{aligned}$$

The coefficients that appear in Eq. (16) may be listed as follows:

$$\begin{aligned} & r_{0} = \frac{{A^{2} (a_{4} + d_{1} )}}{{2\left( {a_{0} + b_{1} } \right)}},\,\,r_{1} = - \frac{{A^{2} (c_{10} \omega^{2} + c_{8} )}}{{2(a_{0} + b_{1} )}},\,\,r_{2} = \frac{{A^{2} c_{9} \omega }}{{2\left( {a_{0} + b_{1} } \right)}},\,\,r_{3} = - \frac{{A^{2} c_{7} }}{{2\left( {a_{0} + b_{1} } \right)}}, \\ & r_{4} = \frac{{A\left( {4a_{5} A^{2} \varpi^{2} - 3a_{7} A^{2} + 4a_{0} K^{2} + A^{2} b_{5} \varpi^{2} - 3A^{2} d_{2} + 4b_{1} K^{2} } \right)}}{{4(a_{0} + b_{1} )}}, \\ & r_{5} = \frac{{A\varpi \left( {a_{6} A^{2} + 4a_{1} + A^{2} b_{6} + 4b_{2} } \right)}}{{4(a_{0} + b_{1} )}}, \\ & r_{6} = \frac{{A^{2} \left( {2a_{2} \varpi^{2} - a_{4} + 2b_{3} \varpi^{2} - d_{1} } \right)}}{{2(a_{0} + b_{1} )}},\,\,r_{7} = \frac{{\varpi \,A^{2} \left( {a_{3} + 2b_{4} } \right)}}{{2(a_{0} + b_{1} )}},\, \\ & r_{8} = - \frac{{A^{3} \left( {a_{7} - 3b_{5} \varpi^{2} + d_{2} } \right)}}{{4\left( {a_{0} + b_{1} } \right)}},\,\,r_{9} = \frac{{\varpi \,A^{3} \left( {a_{6} + 3b_{6} } \right)}}{{4(a_{0} + b_{1} )}} \\ & r_{10} = \frac{{A\left( {A^{2} \left( {(\omega + \varpi )\left( {c_{11} \varpi - 3c_{15} \omega } \right) - 3c_{13} } \right) + 4c_{1} \varpi (\omega + \varpi ) - 4c_{5} \omega (\omega + \varpi ) - 4c_{3} } \right)}}{{8\left( {a_{0} + b_{1} } \right)}}, \\ & r_{11} = \frac{{A\left( {3A^{2} c_{14} \omega + 3A^{2} c_{14} \varpi + 4c_{4} \omega + 4c_{4} \varpi } \right)}}{{8(a_{0} + b_{1} )}},\, \\ & r_{12} = \frac{{A\left( {A^{2} \left( {(\varpi - \omega )\left( {3c_{15} \omega + c_{11} \varpi } \right) - 3c_{13} } \right) + 4c_{1} \varpi (\varpi - \omega ) + 4c_{5} \omega (\varpi - \omega ) - 4c_{3} } \right)}}{{8\left( {a_{0} + b_{1} } \right)}}, \\ & r_{13} = \frac{{A\left( { - 3A^{2} c_{14} \omega + 3A^{2} c_{14} \varpi - 4c_{4} \omega + 4c_{4} \varpi } \right)}}{{8(a_{0} + b_{1} )}},\,\,r_{14} = \frac{{ - A\left( {3A^{2} c_{12} + 4c_{2} } \right)}}{{8(a_{0} + b_{1} )}}, \\ & r_{15} = \frac{{A^{2} \left( { - c_{10} \omega^{2} + 2c_{6} \varpi^{2} + c_{6} \omega \varpi - 2c_{10} \omega \varpi - c_{8} } \right)}}{{4(a_{0} + b_{1} )}},\,\,r_{16} = \frac{{A^{2} c_{9} \left( {\omega + 2\varpi } \right)}}{{4(a_{0} + b_{1} )}}, \\ & r_{17} = \frac{{A^{2} \left( { - c_{10} \omega^{2} + 2c_{6} \varpi^{2} - c_{6} \omega \varpi + 2c_{10} \omega \varpi - c_{8} } \right)}}{{4(a_{0} + b_{1} )}},\,\,\,r_{18} = \frac{{A^{2} c_{9} \left( {2\varpi - \omega } \right)}}{{4(a_{0} + b_{1} )}} \\ & r_{19} = - \frac{{A^{2} c_{7} }}{{4\left( {a_{0} + b_{1} } \right)}},\,\,\,r_{20} = \frac{{A^{3} \left( {(\omega + 3\varpi )\left( {c_{11} \varpi - c_{15} \omega } \right) - c_{13} } \right)}}{{8\left( {a_{0} + b_{1} } \right)}},\,\,r_{21} = \frac{{A^{3} c_{14} \left( {\omega + 3\varpi } \right)}}{{8(a_{0} + b_{1} )}}, \\ & r_{22} = \frac{{A^{3} \left( {(3\varpi - \omega )\left( {c_{15} \omega + c_{11} \varpi } \right) - c_{13} } \right)}}{{8\left( {a_{0} + b_{1} } \right)}},\,\,\,r_{23} = \frac{{A^{3} c_{14} \left( {3\varpi - \omega } \right)}}{{a_{0} + b_{1} }},\,\,r_{24} = - \frac{{A^{3} c_{12} }}{{8\left( {a_{0} + b_{1} } \right)}} \\ \end{aligned}$$

The coefficients that appear in Eq. (20) may be listed as follows:

$$\alpha = \frac{{3\hat{d}_{2} (a_{1} + b_{2} )}}{{(a_{1} + b_{2} )(4a_{5} + b_{5} ) - (a_{6} + 3b_{6} )(a_{0} + b_{1} )}},\,\,{\text{and}}\,\,\beta = \frac{{3a_{7} (a_{1} + b_{2} ) - (a_{6} + 3b_{6} )(a_{0} + b_{1} )\omega_{0}^{2} }}{{(a_{1} + b_{2} )(4a_{5} + b_{5} ) - (a_{6} + 3b_{6} )(a_{0} + b_{1} )}}.$$

The coefficients that appear in Eq. (31) may be listed as follows:

$$\begin{aligned} & \tilde{r}_{4} = \frac{{A\left( {4a_{5} A^{2} \omega^{2} - 3a_{7} A^{2} + 4a_{0} K^{2} + A^{2} b_{5} \omega^{2} - 3A^{2} d_{2} + 4b_{1} K^{2} + 4\sigma (a_{0} + b_{1} )} \right)}}{{4(a_{0} + b_{1} )}}, \\ & \tilde{r}_{10} = - \frac{{A(4c_{4} + 3A^{2} c_{12} )}}{{2\left( {a_{0} + b_{1} } \right)}},\,\,\tilde{r}_{11} = \frac{{A\left( {4\omega^{2} (c_{1} - c_{5} ) - 4c_{3} + A^{2} (2\omega^{2} c_{11} - 3c_{13} - 6A^{2} \omega^{2} c_{15} )} \right)}}{{8(a_{0} + b_{1} )}}, \\ & \tilde{r}_{12} = \frac{{A\omega (4c_{4} + 3A^{2} c_{12} )}}{{4\left( {a_{0} + b_{1} } \right)}},\,\,\tilde{r}_{13} = - \frac{{A(4c_{2} + 3A^{2} c_{12} )}}{{8\left( {a_{0} + b_{1} } \right)}},\,\,\,\tilde{r}_{14} = \frac{{A(\omega^{2} c_{6} - c_{8} + \omega^{2} c_{10} )}}{{4\left( {a_{0} + b_{1} } \right)}}, \\ & \tilde{r}_{15} = \frac{{A^{2} \omega \,c_{9} }}{{4\left( {a_{0} + b_{1} } \right)}},\,\,\,\tilde{r}_{16} = \frac{{A^{2} (3\omega^{2} c_{6} - c_{8} + 3\omega^{2} c_{10} )}}{{4\left( {a_{0} + b_{1} } \right)}},\,\,\,\tilde{r}_{17} = \frac{{3A^{2} \omega \,c_{9} }}{{4\left( {a_{0} + b_{1} } \right)}},\,\,\,\tilde{r}_{18} = - \frac{{A^{2} \,c_{7} }}{{4\left( {a_{0} + b_{1} } \right)}}, \\ & \tilde{r}_{19} = - \frac{{A^{3} \,c_{12} }}{{8\left( {a_{0} + b_{1} } \right)}},\,\,\,\tilde{r}_{20} = \frac{{A^{3} (2\omega^{2} c_{11} - c_{13} + 2\omega^{2} c_{15} )}}{{8\left( {a_{0} + b_{1} } \right)}},\,\,\,\tilde{r}_{21} = \frac{{A^{3} \omega \,c_{14} }}{{4\left( {a_{0} + b_{1} } \right)}}, \\ & \tilde{r}_{22} = \frac{{A^{3} (4\omega^{2} c_{11} - c_{13} + 4\omega^{2} c_{15} )}}{{8\left( {a_{0} + b_{1} } \right)}},\,\,\,\tilde{r}_{23} = \frac{{A^{3} \omega \,c_{14} }}{{2\left( {a_{0} + b_{1} } \right)}},\,\,\,\,\tilde{r}_{24} = - \frac{{A^{3} c_{12} }}{{8\left( {a_{0} + b_{1} } \right)}}, \\ & \tilde{r}_{25} = - \frac{{A(4c_{3} + 3A^{2} c_{13} )}}{{8\left( {a_{0} + b_{1} } \right)}},\,\,\,\,\,\tilde{r}_{26} = - \frac{{A^{2} c_{7} }}{{4\left( {a_{0} + b_{1} } \right)}}. \\ \end{aligned}$$

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

El-Dib, Y.O., Moatimid, G.M., Mady, A.A. et al. Nonlinear hydromagnetic instability of oscillatory rotating rigid-fluid columns. Indian J Phys (2021). https://doi.org/10.1007/s12648-021-02022-3

Download citation

Keywords

  • Azimuthal nonlinear instability
  • Rotating fluids
  • Porous media
  • Magnetic fluids
  • Homotopy perturbation method