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Nonlinear stability analysis of coupled azimuthal interfaces between three rotating magnetic fluids

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Abstract

The current work deals with the nonlinear azimuthal stability analysis of coupled interfaces between three magnetic fluids. The considered system consists of three incompressible rotating magnetic fluids throughout the porous media. Additionally, the system is pervaded by a uniform azimuthal magnetic field. Therefore, for simplicity, the problem is considered in a planar configuration. The adopted nonlinear approach depends mainly on solving the linear governing equations of motion with the implication of the corresponding convenient nonlinear boundary conditions. The linear stability analysis resulted in a quadratic algebraic equation in the frequency of the surface waves. Consequently, the stability criteria are theoretically analysed. A set of diagrams is plotted to discuss the implication of various physical parameters on the stability profile. On the other hand, the nonlinear stability approach revealed two nonlinear partial differential equations of the Schrödinger type. With the aid of these equations, the stability of the interface deflections is achieved. Subsequently, the stability criteria are theoretically accomplished and numerically confirmed. Regions of stability/instability are addressed to illustrate the implication of various parameters on the stability profile.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt

    Galal M Moatimid

  2. Department of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt

    Marwa H Zekry

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  1. Galal M Moatimid
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  2. Marwa H Zekry
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Correspondence to Marwa H Zekry.

Appendix

Appendix

The coefficients appearing in eqs (27) and (28) may be listed as follows:

$$\begin{aligned} L_{1}\equiv & {} a_{1}\frac{\partial ^{2}}{\partial t^{2}}+b_{1}\frac{\partial ^{2}}{\partial \theta ^{2}}+c_{1}\frac{\partial ^{2}}{\partial \theta \partial t}\\&+(h_{1}+ik_{1})\frac{\partial }{\partial t}+(g_{1}+if_{1})\frac{\partial }{\partial \theta }\\ a_{1}= & {} 2m^{2}R_{1}^{13}R_{2}^{4}\left( (\mu _{1}-\mu _{2})^{2}(\mu _{2}-\mu _{3})^{2}R_{1}^{4m}\right. \\&\left. +2(\mu _{1}^{2}-\mu _{2}^{2})(\mu _{2}^{2}-\mu _{3}^{2})R_{1}^{2m}R_{2}^{2m}\right. \\&\left. +(\mu _{1}+\mu _{2})^{2}(\mu _{2}+\mu _{3})^{2}R_{2}^{4m} \right) \\&\times \left( (\rho _{1}-\rho _{2})R_{1}^{4m}\right. \\&\left. -2\rho _{1}R_{1}^{2m}R_{2}^{2m}+(\rho _{1}+\rho _{2})R_{2}^{4m} \right) ,\\ b_{1}= & {} -2T_{1} m^{2}R_{1}^{10} R_{2}^{4} (R_{1}^{2m} -R_{2}^{2m} )\left( (\mu _{1} -\mu _{2} )^{2}(\mu _{2}\right. \\&\left. -\mu _{3} )^{2}R_{1}^{4m} +2(\mu _{1}^{2} -\mu _{2}^{2} )(\mu _{2}^{2} -\mu _{3}^{2} )R_{1}^{2m} R_{2}^{2m}\right. \\&\left. +(\mu _{1} +\mu _{2} )^{2}(\mu _{2} +\mu _{3} )^{2}R_{2}^{4m} \right) ,\\ c_{1}&{=}&2m^{2}R_{1}^{13}R_{2}^{4}\left( (\mu _{1}{-}\mu _{2})^{2}(\mu _{2}{-}\mu _{3})^{2}R_{1}^{4m}\right. \\&\left. {+}2(\mu _{1}^{2}{-}\mu _{2}^{2})(\mu _{2}^{2}{-}\mu _{3}^{2})R_{1}^{2m}R_{2}^{2m}\right. \\&\left. +(\mu _{1}{+}\mu _{2})^{2}(\mu _{2}{+}\mu _{3})^{2}R_{2}^{4m} \right) \\&\times \left( (\rho _{1}{\Omega }_{1}-\rho _{2}{\Omega }_{2})R_{1}^{4m}-2\rho _{1}{\Omega }_{\mathrm {1}}R_{1}^{2m}R_{2}^{2m}\right. \\&\left. +(\rho _{1}{\Omega }_{\mathrm {1}}+\rho _{2}{\Omega }_{\mathrm {2}})R_{2}^{4m} \right) {,}\\ h_{1}&{=}&2m^{2}R_{1}^{13} R_{2}^{4} (R_{1}^{2m} -R_{2}^{2m} )\\&\times \left( (\mu _{1} -\mu _{2} )(\mu _{2} -\mu _{3} )R_{1}^{2m}\right. \\&\left. +(\mu _{1} +\mu _{2} )(\mu _{2} +\mu _{3} )R_{2}^{2m} \right) ^{2}\\&\times \left( {(\rho _{1} \nu _{1} -\rho _{2} \nu _{2} )R_{1}^{2m} -(\rho _{1} \nu _{1}} \right) \\&+\rho _{2} \nu _{2} )R_{2}^{2m} \\ k_{1}= & {} 2m^{2}R_{1}^{13}R_{2}^{4}(R_{1}^{2m}-R_{2}^{2m})\left( (\mu _{1}-\mu _{2})\right. \\&\left. \times (\mu _{2}-\mu _{3})R_{1}^{2m}+(\mu _{1}+\mu _{2})(\mu _{2}+\mu _{3})R_{2}^{2m} \right) ^{2}\\&\mathbf {\times } \left( (m-2)(\rho _{1}{\Omega }_{\mathrm {1}}-\rho _{2}{\Omega }_{\mathrm {2}})R_{1}^{2m}\right. \\&\left. -((m-2)\rho _{1}{\Omega }_{\mathrm {1}}{+}(m{+}2)\rho _{2}{\Omega }_{\mathrm {2}})R_{2}^{2m} \right) ,\\ g_{1}= & {} 2m^{2}R_{1}^{13}R_{2}^{4}(R_{1}^{2m}-R_{2}^{2m})\left( (\mu _{1}-\mu _{2})\right. \\&\left. \times (\mu _{2}-\mu _{3})R_{1}^{2m}+(\mu _{1}+\mu _{2})(\mu _{2}+\mu _{3})R_{2}^{2m} \right) ^{2}\\&\times \left( (\rho _{1}\nu _{1}{\Omega }_{1}-\rho _{2}\nu _{2}{\Omega }_{2})R_{1}^{2m}\right. \\&\left. -(\rho _{1}\nu _{1}{\Omega }_{1}+\rho _{2}\nu _{2}{\Omega }_{\mathrm {2}})R_{2}^{2m} \right) \\ f_{1}&{=}&-2m^{2}R_{1}^{11} R_{2}^{4} (R_{1}^{2m} {-}R_{2}^{2m} )\left( (\mu _{1} -\mu _{2} )\right. \\&\left. \times (\mu _{2}-\mu _{3} )R_{1}^{2m}+(\mu _{1} +\mu _{2} )(\mu _{2} +\mu _{3} )R_{2}^{2m} \right) \\&\,\times \left( {mH_{0}^{2} } (R_{1}^{2m} -R_{2}^{2m} )(\mu _{1}-\mu _{2} )^{2}\right. \\&\left. \times \left( (\mu _{2} -\mu _{3} )R_{1}^{2m} +(\mu _{2} +\mu _{3} )R_{2}^{2m} \right) \right. \\&\left. -R_{1}^{2} \left( (\mu _{1} -\mu _{2} )(\mu _{2} -\mu _{3} )R_{1}^{2m}\right. \right. \\&\left. \left. +(\mu _{1} +\mu _{2} )(\mu _{2} +\mu _{3} )R_{2}^{2m} \right) \,\left( (m-2)(\rho _{1} \Omega _{1}^{2}\right. \right. \\&\left. \left. -\rho _{2} \Omega _{2}^{2} )R_{1}^{2m} -((m-2)\rho _{1} \Omega _{1}^{2}\right. \right. \\&\left. \left. +(m+2)\rho _{2} \Omega _{2}^{2} )R_{2}^{2m} \right) \right) , \end{aligned}$$
$$\begin{aligned} L_{2}\equiv & {} a_{2}\frac{\partial ^{2}}{\partial t^{2}}+b_{2}\frac{\partial ^{2}}{\partial \theta ^{2}}\\&+c_{2}\frac{\partial ^{2}}{\partial \theta \partial t}+(h_{2}+ik_{2})\frac{\partial }{\partial t}+(g_{2}+if_{2})\frac{\partial }{\partial \theta }\\&a_{2} =2m^{2}R_{1}^{12+m} R_{2}^{5+m} \left( (\mu _{1} -\mu _{2} )^{2}(\mu _{2} -\mu _{3} )^{2}R_{1}^{4m}\right. \\&\left. +2(\mu _{1}^{2} -\mu _{2}^{2} )(\mu _{2}^{2} -\mu _{3}^{2} )R_{1}^{2m} R_{2}^{2m}\right. \\&\left. +(\mu _{1} +\mu _{2} )^{2}(\mu _{2} +\mu _{3} )^{2}R_{2}^{4m} \right) \rho _{2} ,\\ b_{2}= & {} 0,\\&c_{2} =4m^{2}R_{1}^{12+m} R_{2}^{5+m} (R_{1}^{2m} -R_{2}^{2m} )\left( (\mu _{1} -\mu _{2} )^{2}(\mu _{2}\right. \\&\left. -\mu _{3} )^{2}R_{1}^{4m} +2(\mu _{1}^{2} -\mu _{2}^{2} )(\mu _{2}^{2} -\mu _{3}^{2} )R_{1}^{2m} R_{2}^{2m}\right. \\&\left. +(\mu _{1} +\mu _{2} )^{2}(\mu _{2} +\mu _{3} )^{2}R_{2}^{4m} \right) \rho _{2} \Omega _{2},\\&h_{2}=\nu _{2}c_{2}/{\Omega }_{\mathrm {2}}\\&k_{2}=mc_{2}\\&g_{2} =4m^{2}R_{1}^{12+m} R_{2}^{5+m} (R_{1}^{2m} -R_{2}^{2m} )\\&\times \left( (\mu _{1} -\mu _{2} )(\mu _{2} -\mu _{3} )R_{1}^{2m}\right. \\&\left. +(\mu _{1} +\mu _{2} )(\mu _{2} +\mu _{3} )R_{2}^{2m} \right) ^{2}\rho _{2} \nu _{2} \Omega _{2} ,\\&f_{2}=-4m^{3}R_{1}^{11+m}R_{2}^{4+m}(R_{1}^{2m}-R_{2}^{2m})\\&\times \left( (\mu _{1}-\mu _{2})(\mu _{2}-\mu _{3})R_{1}^{2m}\right. \\&\left. +(\mu _{1}+\mu _{2})(\mu _{2}+\mu _{3})R_{2}^{2m} \right) \\&\times \left( H_{0}^{2} (R_{1}^{2m}-R_{2}^{2m})(\mu _{1}-\mu _{2})(\mu _{2}-\mu _{3})\right. \\&\left. - R_{1}R_{2}\left( (\mu _{1}-\mu _{2})(\mu _{2}-\mu _{3})R_{1}^{2m}\right. \right. \\&\left. \left. +(\mu _{1}+\mu _{2})(\mu _{2}+\mu _{3})R_{2}^{2m} \right) \rho _{2}\Omega _{2}^{2} \right) , \\ L_{3}\equiv & {} a_{3}\frac{\partial ^{2}}{\partial t^{2}}+b_{3}\frac{\partial ^{2}}{\partial \theta ^{2}}\\&+c_{3}\frac{\partial ^{2}}{\partial \theta \partial t}+(h_{3}+ik_{3})\frac{\partial }{\partial t}+(g_{3}+if_{3})\frac{\partial }{\partial \theta }\\ a_{3}= & {} 4m^{2}R_{1}^{5+m} R_{2}^{12+m} (R_{1}^{2m} -R_{2}^{2m} )\left( (\mu _{1} -\mu _{2} )^{2}(\mu _{2}\right. \\&\left. -\mu _{3} )^{2}R_{1}^{4m} +2(\mu _{1}^{2} -\mu _{2}^{2} )(\mu _{2}^{2} -\mu _{3}^{2} )R_{1}^{2m} R_{2}^{2m}\right. \\&\left. +(\mu _{1} +\mu _{2} )^{2}(\mu _{2} +\mu _{3} )^{2}R_{2}^{4m} \right) \rho _{2} ,\\ b_{3}= & {} 0,\\ c_{3}= & {} 4m^{2}R_{1}^{5+m} R_{2}^{12+m} (R_{1}^{2m} -R_{2}^{2m} )\left( (\mu _{1} -\mu _{2} )^{2}(\mu _{2}\right. \\&\left. -\mu _{3} )^{2}R_{1}^{4m} +2(\mu _{1}^{2} -\mu _{2}^{2} )(\mu _{2}^{2} -\mu _{3}^{2} )R_{1}^{2m} R_{2}^{2m}\right. \\&\left. +(\mu _{1} +\mu _{2} )^{2}(\mu _{2} +\mu _{3} )^{2}R_{2}^{4m} \right) \rho _{2} \Omega _{2},\\ h_{3}= & {} 4m^{2}R_{1}^{5+m}R_{2}^{12+m}(R_{1}^{2m}-R_{2}^{2m})\\&\times \left( (\mu _{1}-\mu _{2})(\mu _{2}-\mu _{3})R_{1}^{2m}\right. \\&\left. +(\mu _{1}+\mu _{2})(\mu _{2}+\mu _{3})R_{2}^{2m} \right) ^{2}\rho _{2}\nu _{2},\\ k_{3}= & {} m{\Omega }_{2}h_{3}/\nu _{2},\\ g_{3}= & {} h_{3}\Omega _{2},\\ f_{3}= & {} -4m^{3}R_{1}^{4+m}R_{2}^{11+m}(R_{1}^{2m}-R_{2}^{2m})\left( (\mu _{1}-\mu _{2})(\mu _{2}\right. \\&\left. -\mu _{3})R_{1}^{2m}+(\mu _{1}+\mu _{2})(\mu _{2}+\mu _{3})R_{2}^{2m} \right) \left( H_{0}^{2} (R_{1}^{2m}\right. \\&\left. -R_{2}^{2m})\mu _{2}(\mu _{1}-\mu _{2})(\mu _{2}-\mu _{3})\right. \\&\left. - R_{1}R_{2}\left( (\mu _{1}-\mu _{2})(\mu _{2}-\mu _{3})R_{1}^{2m}\right. \right. \\&\left. \left. +(\mu _{1}+\mu _{2})(\mu _{2}+\mu _{3})R_{2}^{2m} \right) \rho _{2}{\Omega }_{2}^{2} \right) , \end{aligned}$$
$$\begin{aligned} L_{4} \,\equiv & {} a_{4} \frac{\partial ^{2}}{\partial t^{2}}+b_{4} \frac{\partial ^{2}}{\partial \theta ^{2}}+c_{4} \frac{\partial ^{2}}{\partial \theta \,\partial t}\\&+(h_{4} +ik_{4} )\frac{\partial }{\partial t}+(g_{4} +if_{4}) \frac{\partial }{\partial \theta } \\ a_{4}= & {} 2m^{2}R_{1}^{4}R_{2}^{13}\left( (\mu _{1}-\mu _{2})^{2}(\mu _{2}-\mu _{3})^{2}R_{1}^{4m}\right. \\&\left. +2(\mu _{1}^{2}-\mu _{2}^{2})(\mu _{2}^{2}-\mu _{3}^{2})R_{1}^{2m}R_{2}^{2m}\right. \\&\left. +(\mu _{1}+\mu _{2})^{2}(\mu _{2}+\mu _{3})^{2}R_{2}^{4m} \right) \\&\times \left( (\rho _{3}-\rho _{2})R_{1}^{4m}-2\rho _{3}R_{1}^{2m}R_{2}^{2m}\right. \\&\left. +(\rho _{1}+\rho _{3})R_{2}^{4m} \right) , \\ b_{4}= & {} -2T_{2} m^{2}R_{1}^{4} R_{2}^{10} (R_{1}^{2m} -R_{2}^{2m} )\left( (\mu _{1} -\mu _{2} )^{2}(\mu _{2}\right. \\&\left. -\mu _{3} )^{2}R_{1}^{4m} +2(\mu _{1}^{2} -\mu _{2}^{2} )(\mu _{2}^{2} -\mu _{3}^{2} )R_{1}^{2m} R_{2}^{2m}\right. \\&\left. +(\mu _{1} +\mu _{2} )^{2}(\mu _{2} +\mu _{3} )^{2}R_{2}^{4m} \right) ,\\ c_{4}= & {} 2m^{2}R_{1}^{4}R_{2}^{13}\left( (\mu _{1}-\mu _{2})^{2}(\mu _{2}-\mu _{3})^{2}R_{1}^{4m}\right. \\&\left. +2(\mu _{1}^{2}-\mu _{2}^{2})(\mu _{2}^{2}-\mu _{3}^{2})R_{1}^{2m}R_{2}^{2m}\right. \\&\left. +(\mu _{1}+\mu _{2})^{2}(\mu _{2}+\mu _{3})^{2}R_{2}^{4m} \right) \\&\times \left( (\rho _{3}{\Omega }_{3}-\rho _{2}{\Omega }_{2})R_{1}^{4m}-2\rho _{3}{\Omega }_{\mathrm {3}}R_{1}^{2m}R_{2}^{2m}\right. \\&\left. +(\rho _{3}{\Omega }_{3}+\rho _{2}{\Omega }_{2})R_{2}^{4m} \right) , \end{aligned}$$
$$\begin{aligned} h_{4}= & {} 2m^{2}R_{1}^{4} R_{2}^{13} (R_{1}^{2m} -R_{2}^{2m} )\\&\times \left( (\mu _{1} -\mu _{2} )(\mu _{2} -\mu _{3} )R_{1}^{2m} +(\mu _{1}\right. \\&\left. +\mu _{2} )(\mu _{2}+\mu _{3} )R_{2}^{2m} \right) ^{2}\left( (\rho _{2} \nu _{2}\right. \\&\left. -\rho _{3}\nu _{3} )R_{1}^{2m} +(\rho _{2} \nu _{2}+\rho _{3} \nu _{3} )R_{2}^{2m}\right) ,\\ k_{4}= & {} 2m^{2}R_{1}^{4}R_{2}^{13}(R_{1}^{2m}-R_{2}^{2m})\left( (\mu _{1}-\mu _{2})(\mu _{2}\right. \\&\left. -\mu _{3})R_{1}^{2m}+(\mu _{1}+\mu _{2})(\mu _{2}+\mu _{3})R_{2}^{2m} \right) ^{2}\\&\times \left( (m+2)(\rho _{2}{\Omega }_{2}-\rho _{3}{\Omega }_{3})R_{1}^{2m}\right. \\&\left. +((m-2)\rho _{2}{\Omega }_{\mathrm {2}}+(m+2)\rho _{3}{\Omega }_{\mathrm {3}})R_{2}^{2m} \right) ,\\ g_{4}= & {} -2m^{2}R_{1}^{4}R_{2}^{11}(R_{1}^{2m}-R_{2}^{2m})\\&\times \left( (\mu _{1}-\mu _{2})(\mu _{2}-\mu _{3})R_{1}^{2m}\right. \\&\left. +(\mu _{1}+\mu _{2})(\mu _{2}+\mu _{3})R_{2}^{2m} \right) ^{2}\\&\times \left( (\rho _{2}\nu _{2}{\Omega }_{2}-\rho _{3}\nu _{3}{\Omega }_{3})R_{1}^{2m}\right. \\&\left. +(\rho _{2}\nu _{2}{\Omega }_{2}+\rho _{3}\nu _{3}{\Omega }_{3})R_{2}^{2m} \right) , \end{aligned}$$

and

$$\begin{aligned} f_{4}= & {} -2m^{2}R_{1}^{4}R_{2}^{11}(R_{1}^{2m}-R_{2}^{2m})\\&\times \left( (\mu _{1}-\mu _{2})(\mu _{2}-\mu _{3})R_{1}^{2m}\right. \\&\left. +(\mu _{1}+\mu _{2})(\mu _{2}+\mu _{3})R_{2}^{2m} \right) \\&\times \left( mH_{0}^{2} (R_{1}^{2m}-R_{2}^{2m})(\mu _{2}\right. \\&\left. -\mu _{3})^{2}\left( (\mu _{1}-\mu _{2})R_{1}^{2m}(\mu _{1}+\mu _{2})R_{2}^{2m} \right) \right. \\&\left. -R_{2}^{2}\left( (\mu _{1}-\mu _{2})(\mu _{2}-\mu _{3})R_{1}^{2m}\right. \right. \\&\left. \left. +(\mu _{1}+\mu _{2})(\mu _{2}+\mu _{3})R_{2}^{2m} \right) \right. \\&\left. \times \left( (m+2)(\rho _{2}{\Omega }_{2}^{2}-\rho _{3}{\Omega }_{\mathrm {3}}^{\mathrm {2}})R_{1}^{2m}\right. \right. \\&\left. \left. +((m-2)\rho _{2}{\Omega }_{\mathrm {2}}^{\mathrm {2}}+(m+2)\rho _{3}{\Omega }_{3}^{2})R_{2}^{2m} \right) \right) . \end{aligned}$$

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Moatimid, G.M., Zekry, M.H. Nonlinear stability analysis of coupled azimuthal interfaces between three rotating magnetic fluids. Pramana - J Phys 94, 115 (2020). https://doi.org/10.1007/s12043-020-01962-5

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