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.

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}$$