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Linear Instability of Electromagnetic Viscoelastic Nanofluid: Analytical and Numerical Study

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Abstract

In this study, the effect of the induced electromagnetic fields on the linear instability of a viscoelastic nanofluid liquid layer is discussed. Two physical cases are considered for the channel flow. The first one is considered for the free–free boundaries and the second for the rigid–rigid boundaries. The instability is investigated analytically due to the normal mode analysis and numerically according to the relation between the energy density function and time. The Routh–Hurwitz criteria are applied. The dispersion relation is obtained as a sixth-degree equation of the growth rate. Also, the stationary, as well as oscillatory states of the thermal Rayleigh number, are obtained. The numerical solution is compared with the exact solution for a special choice of the parameters and the results show higher match between the results. The main important result of the study concludes that the Brownian motion of the nanoparticles destabilizes the nanofluid layer. This means that the pure fluid layer is more stable than the nanofluid layer. Also, the results confirm that the flow between the rigid–rigid boundaries is more stable than the free–free boundaries. Finally, due to the Lorentz force, the induced magnetic field destabilizes the motion due to the temperature rise.

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

    Mohamed A. Hassan

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Appendix

Appendix

The Constants of the Free–Free Boundaries

The elements of the determinant that appear in Eq. (34), are written as follows

\(a_{11} = \left( {1 - i\,\omega \,\lambda } \right)\,\alpha^{4} + i\,\omega \,{\text{Re}} \,\alpha^{2} + Ha^{2} \,\pi^{2}\), \(a_{12} = - \frac{i\,k\,Ra}{{{\text{Re}} \,\Pr }}\), \(a_{13} = \frac{{i\,k\,R_{N} }}{{{\text{Re}} \,\Pr }}\), \(a_{21} = i\,k\,{\text{Re}} \,\Pr\), \(a_{22} = \alpha^{2} + i\,\omega \,{\text{Re}} \,\Pr\), \(a_{31} = - i\,k\,N_{A} \,{\text{Re}} \,\Pr \,Le\), \(a_{32} = N_{A} \,\alpha^{2}\), \(a_{33} = \alpha^{2} + i\,\omega \,{\text{Re}} \,\Pr \,Le\), \(a_{44} = \alpha^{2} - \omega^{2} + i\,\omega \,{\text{Re}}_{m}\), \(a_{54} = - 1\), \(a_{55} = {\text{Re}}_{m} + i\,\omega\).

The dispersion relation coefficients in Eq. (35) \(c_{0} - c_{6}\) and \(d_{0} - d_{6}\) are written as follows

$$ c_{f1} = c_{f3} = c_{f5} = d_{f2} = d_{f4} = d_{f6} = 0, $$
$$ c_{f0} = Le\,\alpha^{2} \,\Pr^{2} {\text{Re}}^{2} (\alpha^{2} \lambda - {\text{Re}} ), $$
$$ \begin{aligned} c_{f2} & = - \alpha^{8} \lambda + {\text{Re}} \,(Ha^{2} (1 + Le)\,\pi^{2} \,\alpha^{2} \,\Pr + (1 + \Pr + Le\,\Pr )\,\alpha^{6} + \Pr \,( - Le\,a_{12} \,a_{21} - a_{13} \,a_{31} \\ & \quad + Le\,\Pr \,\alpha^{4} \,{\text{Re}} \,( - \alpha^{2} \lambda + {\text{Re}} ) + 2( - (1 + Le)\,\alpha^{6} \,\lambda + (Ha^{2} Le\,\pi^{2} \Pr \\ & \quad + (1 + Le + Le\,\Pr )\alpha^{4} ){\text{Re}} ){\text{Re}}_{m} + Le\,\Pr \,\alpha^{2} {\text{Re}} \,{\text{Re}}_{m}^{2} ( - \alpha^{2} \lambda + {\text{Re}} )\,)), \\ \end{aligned} $$
$$ \begin{aligned} c_{f4} & = \alpha^{10} \lambda - \Pr \,\alpha^{2} \,(Ha^{2} Le\,\pi^{2} \Pr + (1 + Le + Le\,\Pr )\,\alpha^{4} )\,{\text{Re}}^{2} {\text{Re}}_{m} \\ & \quad + {\text{Re}}_{m} (2\alpha^{2} a_{12} a_{21} - 2\,(Ha^{2} \,\pi^{2} \alpha^{4} + \alpha^{8} + a_{13} ( - \alpha^{2} a_{31} + a_{21} a_{32} )) + \alpha^{8} \lambda \,{\text{Re}}_{m} ) \\ & \quad + {\text{Re}} ( - Ha^{2} \,(1 + Le)\,\pi^{2} \Pr \alpha^{4} - (1 + Le + Le\,\Pr )\,\alpha^{8} + (1 + Le)\Pr \alpha^{8} \,\lambda \,{\text{Re}}_{m} \\ & \quad - \alpha^{2} (\alpha^{4} + (1 + Le)\,\Pr \,(Ha^{2} \pi^{2} + \alpha^{4} )){\text{Re}}_{m}^{2} + Le\,\Pr \,a_{12} a_{21} (\alpha^{2} + {\text{Re}}_{m}^{2} ) + \Pr a_{13} a_{31} (\alpha^{2} + {\text{Re}}_{m}^{2} )), \\ \end{aligned} $$
$$ c_{f6} = \frac{{{\text{Re}}_{m} \alpha^{4} (Ha^{2} \pi^{2} - k^{2} Ra + \alpha^{6} \, - k^{2} \,R_{N} N_{A} (1 + Le))}}{Le}, $$
$$ \begin{aligned} d_{f1} & = \Pr \,{\text{Re}} \,( - (1 + Le)\,\alpha^{6} \lambda + {\text{Re}} \,(Ha^{2} Le\,\pi^{2} \,\Pr + (1 + Le + Le\,\Pr )\,\alpha^{4} \\ & \quad + 2\,Le\,\Pr \,\alpha^{2} \,{\text{Re}}_{m} \,( - \alpha^{2} \,\lambda + {\text{Re}} ))), \\ \end{aligned} $$
$$ \begin{aligned} d_{f3} & = - (Ha^{2} \,\pi^{2} \alpha^{4} + \alpha^{8} + \Pr \,\alpha^{2} \,{\text{Re}} \,( - (1 + Le)\alpha^{6} \lambda + (Ha^{2} Le\,\pi^{2} \Pr + (1 + Le + Le\,\Pr )\alpha^{4} ){\text{Re}} ) \\ & \quad + \alpha^{2} ( - 2\alpha^{6} \lambda + {\text{Re}} \,((2(\alpha^{4} + (1 + Le)\Pr \,(Ha^{2} \pi^{2} + \alpha^{4} )) + Le\,\Pr^{2} \alpha^{2} {\text{Re}} ( - \alpha^{2} \lambda + {\text{Re}} )))){\text{Re}}_{m} \\ & \quad + \Pr \,{\text{Re}} \,( - (1 + Le)\alpha^{6} \lambda + (Ha^{2} Le\,\pi^{2} \Pr + (1 + Le + Le\,\Pr )\alpha^{4} ){\text{Re}} ){\text{Re}}_{m}^{2} \\ & \quad - a_{12} a_{21} (\alpha^{2} + 2\,Le\Pr {\text{Re}} {\text{Re}}_{m} ) + a_{13} (a_{21} a_{32} - a_{31} (\alpha^{2} + 2\,\Pr \,{\text{Re}} \,{\text{Re}}_{m} ))), \\ \end{aligned} $$
$$ \begin{aligned} d_{f5} & = - \alpha^{2} \,a_{12} a_{21} (\alpha^{2} + Le\,\Pr \,{\text{Re}} \,{\text{Re}}_{m} + {\text{Re}}_{m}^{2} ) - a_{13} ( - a_{21} a_{32} (\alpha^{2} + {\text{Re}}_{m}^{2} ) \\ & \quad + \alpha^{2} a_{31} (\alpha^{2} + \Pr \,{\text{Re}} \,{\text{Re}}_{m} + {\text{Re}}_{m}^{2} )) + \alpha^{4} (Ha^{2} \pi^{2} \alpha^{2} + \alpha^{6} + {\text{Re}}_{m} ( - \alpha^{6} \lambda \\ & \quad + (\alpha^{4} + (1 + Le)\Pr (Ha^{2} \pi^{2} + \alpha^{4} )){\text{Re}} + (Ha^{2} \pi^{2} + \alpha^{4} ){\text{Re}}_{m} )), \\ \end{aligned} $$

The constants \(m_{f1} - m_{f5}\) and \(n_{f1} - n_{f5}\) that appear in Eqs. (39) are defined as

$$ m_{f1} = i\,k\,Le\,a_{21} ,\quad m_{f2} = \frac{{i\,k\,a_{21} (\alpha^{2} + 2\,Le\,\Pr \,{\text{Re}} \,{\text{Re}}_{m} )}}{{\Pr \,{\text{Re}} }}, $$
$$ m_{f3} = \frac{{i\,k\,a_{21} (2\alpha^{2} {\text{Re}}_{m} + Le\,\Pr \,{\text{Re}} \,\alpha^{2} {\text{Re}}_{m}^{3} )}}{{\Pr \,{\text{Re}} }}, $$
$$ m_{f4} = \frac{{ - i\,k\,\alpha^{2} a_{21} (\alpha^{2} + Le\,\Pr \,{\text{Re}} \,{\text{Re}}_{m} + {\text{Re}}_{m}^{2} )}}{{\Pr \,{\text{Re}} }}, $$
$$ m_{f4} = \frac{{ - i\,k\,\alpha^{2} a_{21} (\alpha^{2} + Le\,\Pr \,{\text{Re}} \,{\text{Re}}_{m} + {\text{Re}}_{m}^{2} )}}{{\Pr \,{\text{Re}} }},\quad m_{f5} = \frac{{i\,k\,\alpha^{4} a_{21} \,{\text{Re}}_{m} }}{{\Pr \,{\text{Re}} }}, $$
$$ \begin{gathered} n_{f1} = - \alpha^{8} \lambda + {\text{Re}} \,(Ha^{2} (1 + Le)\,\pi^{2} \,\alpha^{2} \,\Pr + (1 + \Pr + Le\,\Pr )\,\alpha^{6} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, + \Pr \,( - a_{13} \,a_{31} - 2(1 + Le)\,\alpha^{6} \,\lambda {\text{Re}}_{m} + Le\,\Pr \,\alpha^{2} {\text{Re}}^{2} (\alpha^{2} + {\text{Re}}_{m}^{2} ) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, + {\text{Re}} ( - Le\,\Pr \,\alpha^{6} \,\lambda + {\text{Re}}_{m\,} (2Ha^{2} Le\,\pi^{2} \Pr + 2\left( {1 + Le + Le\,\Pr } \right)\alpha^{4} - Le\,\Pr \,\alpha^{4} \lambda \,{\text{Re}}_{m} )))), \hfill \\ \end{gathered} $$
$$ \begin{gathered} n_{f2} = Ha^{2} \,\pi^{2} \alpha^{4} + \alpha^{8} + \Pr \,\alpha^{2} \,{\text{Re}} \,( - (1 + Le)\alpha^{6} \lambda + (Ha^{2} Le\,\pi^{2} \Pr + \left( {1 + Le + Le\,\Pr } \right)\alpha^{4} ){\text{Re}} ) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, + \alpha^{2} ( - 2\alpha^{6} \lambda + {\text{Re}} \,(2(\alpha^{4} + \left( {1 + Le} \right)\Pr \,(Ha^{2} \pi^{2} + \alpha^{4} )) + Le\,\Pr^{2} \alpha^{2} {\text{Re}} ( - \alpha^{2} \lambda + {\text{Re}} ))){\text{Re}}_{m} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, + \Pr \,{\text{Re}} \,( - (1 + Le)\alpha^{6} \lambda + (Ha^{2} Le\,\pi^{2} \Pr + \left( {1 + Le + Le\,\Pr } \right)\alpha^{4} ){\text{Re}} ){\text{Re}}_{m}^{2} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, + a_{13} (a_{21} a_{32} - a_{31} (\alpha^{2} + 2\,\Pr \,{\text{Re}} \,{\text{Re}}_{m} )), \hfill \\ \end{gathered} $$
$$ \begin{gathered} n_{f3} = - \alpha^{10} \lambda + 2(Ha^{2} \,\pi^{2} \,\alpha^{4} + \alpha^{8} + a_{13} \,( - \alpha^{2} a_{31} + a_{21} a_{32} )){\text{Re}}_{m} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, + \Pr \,\alpha^{2} (Ha^{2} Le\,\pi^{2} \Pr + (1 + Le + Le\,\Pr )\alpha^{4} ){\text{Re}}^{2} {\text{Re}}_{m} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, - \alpha^{8} \lambda {\text{Re}}_{m}^{2} + {\text{Re}} (Ha^{2} \pi^{2} \Pr \alpha^{4} (1 + Le) + \alpha^{8} (1 + \Pr + Le\,\Pr ) - \Pr \alpha^{8} \lambda {\text{Re}}_{m} (1 + Le) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, + \alpha^{2} (\alpha^{4} + (1 + Le)\Pr (Ha^{2} + \pi^{2} + \alpha^{4} )){\text{Re}}_{m}^{2} - \Pr \,a_{13} a_{31} (\alpha^{2} + {\text{Re}}_{m}^{2} )), \hfill \\ \end{gathered} $$
$$ \begin{gathered} n_{f4} = - \alpha^{4} (Ha^{2} \pi^{2} \,\alpha^{2} + \alpha^{6} + {\text{Re}}_{m} \,( - \alpha^{6} \lambda + (\alpha^{4} + (1 + Le)\Pr (Ha^{2} \pi^{2} + \alpha^{4} )){\text{Re}} + {\text{Re}}_{m} (Ha^{2} \pi^{2} + \alpha^{4} )) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, + \,a_{13} ( - a_{21} a_{32} (\alpha^{2} + {\text{Re}}_{m}^{2} ) + \alpha^{2} a_{31} (\alpha^{2} + \Pr \,{\text{Re}} \,{\text{Re}}_{m} + {\text{Re}}_{m}^{2} ))), \hfill \\ \end{gathered} $$
$$ n_{f5} = \alpha^{2} (Ha^{2} \,\pi^{2} \,\alpha^{4} + \alpha^{8} + a_{31} ( - \alpha^{2} a_{31} + a_{21} a_{32} )){\text{Re}}_{m} . $$

The Constants of the Rigid–Rigid Boundaries

The elements of the determinant that appear in Eq. (42), are written as follows

\(\begin{gathered} b_{11} = \left( {\frac{{k^{4} }}{30} + 24} \right)\left( {1 - i\,\omega \,\lambda } \right)\, + \frac{{i\,\omega \,{\text{Re}} \,k^{2} }}{30},b_{12} = - \frac{i\,k\,Ra}{{6{\text{Re}} \,\Pr }},b_{13} = \frac{{i\,k\,R_{N} }}{{6{\text{Re}} \,\Pr }}, \hfill \\ b_{21} = \frac{1}{30}i\,k\,{\text{Re}} \,\Pr ,b_{22} = \frac{1}{6}k^{2} + \frac{1}{6}i\,\omega \,{\text{Re}} \,\Pr + 2,b_{31} = \frac{{ - i\,k\,N_{A} \,{\text{Re}} \,\Pr }}{30}, \hfill \\ b_{32} = \frac{{N_{A} \,k^{2} }}{6\,Le} + \frac{{2\,N_{A} }}{Le},b_{33} = \frac{{k^{2} }}{6\,Le} + \frac{{i\,\omega \,{\text{Re}} \,\Pr }}{6} + \frac{2}{Le},b_{44} = \frac{1}{6}(\omega^{2} - k^{2} - i\,\omega \,{\text{Re}}_{m} ) - 2, \hfill \\ b_{54} = - \frac{1}{6},b_{55} = \frac{1}{6}({\text{Re}}_{m} + i\,\omega ). \hfill \\ \end{gathered}\)

The dispersion relation coefficients in Eq. (43) are

$$ c_{r1} = c_{r3} = c_{r5} = d_{r2} = d_{r4} = d_{r6} = 0, $$
$$ c_{r0} = \left( {{1 \mathord{\left/ {\vphantom {1 {38880\,Le}}} \right. \kern-\nulldelimiterspace} {38880\,Le}}} \right)(\Pr^{2} {\text{Re}}^{2} (k^{2} {\text{Re}} - (720 + k^{4} ))), $$
$$ \begin{gathered} c_{r2} = \left( {{1 \mathord{\left/ {\vphantom {1 {38880\,Le}}} \right. \kern-\nulldelimiterspace} {38880\,Le}}} \right)\,((12 + k^{2} )^{2} (720 + k^{4} )\lambda - k^{2} Le\,\Pr^{2} {\text{Re}}^{3} (12 + k^{2} + {\text{Re}}_{m}^{2} ) \hfill \\ \,\,\,\,\,\,\,\,\, + \Pr^{2} {\text{Re}}^{2} (12 + k^{2} )(720 + k^{4} )Le\,\,\lambda - 2\Pr {\text{Re}}^{2} \,{\text{Re}}_{m} (k^{2} (12 + k^{2} )(1 + Le) \hfill \\ \,\,\,\,\,\,\,\,\, + (720 + k^{4} )Le\,\Pr + (720 + k^{4} )\,Le\,\Pr \,\lambda \,{\text{Re}}_{m}^{2} ) + \Pr^{2} {\text{Re}}^{3} (180Le\Pr b_{12} b_{21} + 180Le\Pr b_{13} b_{31} \hfill \\ \,\,\,\,\,\,\,\,\, + (12 + k^{2} )( - k^{2} (12 + k^{2} ) - (720 + k^{4} )(1 + Le)\Pr \,\lambda \,{\text{Re}}_{m} ))), \hfill \\ \end{gathered} $$
$$ \begin{gathered} c_{r4} = \left( {{1 \mathord{\left/ {\vphantom {1 {38880\,Le}}} \right. \kern-\nulldelimiterspace} {38880\,Le}}} \right)\,( - (12 + k^{2} )^{3} (720 + k^{4} )\lambda + 2((12 + k^{2} )^{2} (720 + k^{4} ) - 180(12 + k^{2} )b_{12} b_{21} \hfill \\ \,\,\,\,\,\,\,\,\,\,\, - 180\,Le\,b_{13} ((12 + k^{2} )b_{31} - 6b_{21} b_{32} )){\text{Re}}_{m} + (12 + k^{2} )\Pr (k^{2} (12 + k^{2} )(1 + Le) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, + (720 + k^{4} )Le\,\Pr ){\text{Re}}^{2} {\text{Re}}_{m} - (12 + k^{2} )^{2} (720 + k^{4} )\lambda {\text{Re}}_{m}^{2} \backslash \hfill \\ \,\,\,\,\,\,\,\,\,\,\, + {\text{Re}} ( - 180Le\,\Pr b_{21} b_{12} (12 + k^{2} + {\text{Re}}_{m}^{2} ) - 180Le\,\Pr b_{31} b_{13} (12 + k^{2} + {\text{Re}}_{m}^{2} )\, \hfill \\ \,\,\,\,\,\,\,\,\,\, + (12 + k^{2} )((12 + k^{2} )(k^{2} (12 + k^{2} ) + (720 + k^{4} )(1 + Le)\Pr ) \hfill \\ \,\,\,\,\,\,\,\,\,\,\, - (12 + k^{2} )(720 + k^{4} )(1 + Le)\Pr \,\lambda \,{\text{Re}}_{m} + (k^{2} (12 + k^{2} ) + (720 + k^{4} )(1 + Le)\Pr ){\text{Re}}_{m}^{2} \,), \hfill \\ \end{gathered} $$
$$ \begin{gathered} c_{r6} = - \left( {{1 \mathord{\left/ {\vphantom {1 {38880\,Le}}} \right. \kern-\nulldelimiterspace} {38880\,Le}}} \right)\,\,((12 + k^{2} )(\,(12 + k^{2} )((12 + k^{2} )(720 + k^{4} ) - 180b_{12} b_{21} - 180Leb_{31} b_{13} ) \hfill \\ \,\,\,\,\,\,\, + 1080Le\,b_{31} b_{21} b_{32} ){\text{Re}}_{m} ), \hfill \\ \end{gathered} $$

\(\begin{gathered} d_{r1} = \,\left( {{{ - 1} \mathord{\left/ {\vphantom {{ - 1} {38880\,}}} \right. \kern-\nulldelimiterspace} {38880\,}}} \right)\,({\text{Re}} \Pr ( - \lambda (12 + k^{2} )(720 + k^{4} )(1 + Le) + {\text{Re}} k^{2} (12 + k^{2} )(1 + Le) \hfill \\ \,\,\,\,\,\,\,\, + (720 + k^{4} )Le\,\Pr {\text{Re}} - 2\,Le\,\Pr \,{\text{Re}} ((720 + k^{4} )\lambda - k^{2} {\text{Re}} ){\text{Re}}_{m} )), \hfill \\ \end{gathered}\)

\( \begin{aligned} d_{{r3}} & = (1/38880\,Le)\,((12 + k^{2} )^{2} (720 + k^{4} ) - 103,680\,\,\Pr \lambda \text{Re} - 17280\,\,k^{2} \Pr \lambda \text{Re} \\ & - \quad 864\,k^{4} \,\Pr \,\lambda \text{Re} - 24\,\,k^{6} \,\Pr \,\lambda \text{Re} - \,k^{8} \,\Pr \,\lambda \text{Re} - 103680Le\,\Pr \,\lambda \text{Re} \\ & - \quad 17,280\,k^{2} Le\,\Pr \,\lambda \text{Re} - 864\,k^{4} Le\,\Pr \,\lambda \text{Re} - 24\,k^{6} Le\,\Pr \,\lambda \text{Re} - k^{8} Le\,\Pr \,\lambda \text{Re} \\ & + \quad 144\,k^{2} \Pr \,\text{Re} ^{2} + 24\,k^{4} \Pr \,\text{Re} ^{2} + k^{6} \Pr \,\text{Re} ^{2} + 144\,k^{2} Le\Pr \,\text{Re} ^{2} + 24\,k^{4} Le\Pr \,\text{Re} ^{2} \\ & + \quad k^{6} Le\Pr \,\text{Re} ^{2} + 8640Le\Pr ^{2} \,\text{Re} ^{2} + 720k^{2} Le\Pr ^{2} \,\text{Re} ^{2} + 12k^{4} Le\Pr ^{2} \,\text{Re} ^{2} + k^{6} Le\Pr ^{2} \,\text{Re} ^{2} \\ & - \quad (12 + k^{2} )(2(12 + k^{2} )(720 + k^{4} )\lambda + \text{Re} ( - 2k^{2} (12 + k^{2} ) - 2(720 + k^{4} )(1 + Le)\Pr \\ & + \quad Le\,\Pr ^{2} \text{Re} ((720 + k^{4} )\lambda - k^{2} \text{Re} )))\text{Re} _{m} + \Pr \,\text{Re} ( - (12 + k^{2} )(720 + k^{4} )(1 + Le)\lambda \\ & + \quad (k^{2} (12 + k^{2} )(1 + Le) + (720 + k^{4} )Le\,\Pr )\text{Re} )\text{Re} _{m}^{2} - 180\,b_{{12}} b_{{21}} (12 + k^{2} + 2\,Le\Pr \,\text{Re} \text{Re} _{m} ) \\ & - \quad 180\,Le\,b_{{13}} ( - 6b_{{21}} b_{{32}} + b_{{31}} (12 + k^{2} + 2\,\Pr \,\text{Re} \,\text{Re} _{m} ))), \\ \end{aligned} \)

$$ \begin{aligned} d_{r5} & = \left( {{1 \mathord{\left/ {\vphantom {1 {38880\,Le}}} \right. \kern-\nulldelimiterspace} {38880\,Le}}} \right)\,\,((12 + k^{2} )^{2} ( - (12 + k^{2} )(720 + k^{4} ) - ((k^{2} (12 + k^{2} ) + (720 + k^{4} )(1 + Le)\Pr ){\text{Re}} \\ & \quad - (720 + k^{4} )((12 + k^{2} )\lambda - {\text{Re}}_{m} )){\text{Re}}_{m} \,) + 180(12 + k^{2} )b_{12} b_{21} (12 + k^{2} + Le\Pr {\text{Re}} {\text{Re}}_{m} + {\text{Re}}_{m}^{2} ) \\ & \quad + 180Le\,b_{13} ( - 6b_{21} b_{32} (12 + k^{2} + {\text{Re}}_{m}^{2} ) + (12 + k^{2} )b_{31} (12 + k^{2} + \Pr {\text{Re}} {\text{Re}}_{m} + {\text{Re}}_{m}^{2} ))), \\ \end{aligned} $$

The constants \(m_{r1} - m_{r5}\) and \(n_{r1} - n_{r5}\) that appear in Eq. (46) are defined as

$$ m_{r1} = - {{i\,k\,b_{21} } \mathord{\left/ {\vphantom {{i\,k\,b_{21} } {1296}}} \right. \kern-\nulldelimiterspace} {1296}},m_{r2} = - {{i\,k\,b_{21} (12 + k^{2} + 2Le\,\Pr \,{\text{Re}} \,{\text{Re}}_{m} )} \mathord{\left/ {\vphantom {{i\,k\,b_{21} (12 + k^{2} + 2Le\,\Pr \,{\text{Re}} \,{\text{Re}}_{m} )} {1296Le\,\Pr \,{\text{Re}} }}} \right. \kern-\nulldelimiterspace} {1296Le\,\Pr \,{\text{Re}} }}, $$
$$ m_{r3} = \left( {{i \mathord{\left/ {\vphantom {i {1296Le\,\Pr \,{\text{Re}} }}} \right. \kern-\nulldelimiterspace} {1296Le\,\Pr \,{\text{Re}} }}} \right)(k\,b_{21} (2(12 + k^{2} ){\text{Re}}_{m} + Le\,\Pr \,{\text{Re}} (12 + k^{2} + {\text{Re}}_{m}^{2} ))), $$
$$ m_{r4} = - \left( {{i \mathord{\left/ {\vphantom {i {1296Le\,\Pr \,{\text{Re}} }}} \right. \kern-\nulldelimiterspace} {1296Le\,\Pr \,{\text{Re}} }}} \right)(\,k\,(12 + k^{2} )b_{21} (12 + k^{2} + Le\,\Pr \,{\text{Re}} {\text{Re}}_{m} + {\text{Re}}_{m}^{2} )), $$
$$ m_{r5} = - {{i\,k\,b_{21} (12 + k^{2} )^{2} {\text{Re}}_{m} } \mathord{\left/ {\vphantom {{i\,k\,b_{21} (12 + k^{2} )^{2} {\text{Re}}_{m} } {1296Le\,\Pr \,{\text{Re}} }}} \right. \kern-\nulldelimiterspace} {1296Le\,\Pr \,{\text{Re}} }}, $$
$$ \begin{aligned} n_{r1} & = \left( {{1 \mathord{\left/ {\vphantom {1 {38880\,Le}}} \right. \kern-\nulldelimiterspace} {38880\,Le}}} \right)\,((12 + k^{2} )^{2} (720 + k^{4} )\lambda + {\text{Re}} (\,180\,Le\,\Pr \,b_{13} b_{31} \\ & \quad + (12 + k^{2} )( - k^{2} (12 + k^{2} ) - (720 + k^{4} )(1 + Le)\Pr + 2(720 + k^{4} )(1 + Le)\Pr \,\lambda \,{\text{Re}}_{m} ) \\ & \quad - k^{2} Le\,\Pr^{2} {\text{Re}}^{2} (12 + k^{2} + {\text{Re}}_{m}^{2} ) + \Pr \,{\text{Re}} ((12 + k^{2} )^{2} (720 + k^{4} )\,Le\,\Pr \lambda \\ & \quad - 2(k^{2} (12 + k^{2} )(1 + Le) + (720 + k^{4} )Le\,\Pr {\text{Re}}_{m} + (720 + k^{4} )Le\,\Pr \lambda {\text{Re}}_{m}^{2} )), \\ \end{aligned} $$
$$ \begin{aligned} n_{r2} & = \left( {{1 \mathord{\left/ {\vphantom {1 {38880\,Le}}} \right. \kern-\nulldelimiterspace} {38880\,Le}}} \right)(k^{2} (12 + k^{2} )Le\,\Pr \,{\text{Re}}^{3} {\text{Re}}_{m} - (12 + k^{2} )^{2} (720 + k^{4} )( - 1 + 2\lambda \,{\text{Re}}_{m} ) \\ & \quad + \Pr \,{\text{Re}}^{2} ((12\,k + k^{3} )^{2} (1 + Le) + (12 + k^{2} )(720 + k^{4} )Le\,\Pr - (12 + k^{2} )(720 + k^{4} )Le\,\Pr \lambda \,{\text{Re}}_{m} \\ & \quad + (k^{2} (12 + k^{2} )(1 + Le) + (720 + k^{4} )Le\,\Pr ){\text{Re}}_{m}^{2} ) \\ & \quad + (12 + k^{2} ){\text{Re}} ( - (12 + k^{2} )(720 + k^{4} )(1 + Le)\,\Pr \,\lambda \\ & \quad + {\text{Re}}_{m} (2k^{2} (12 + k^{2} ) + 2(720 + k^{4} )(1 + Le)\,\Pr - (720 + k^{4} )(1 + Le)\,\Pr \lambda \,{\text{Re}}_{m} )) \\ & \quad - 180\,Le\,b_{13} ( - 6\,b_{21} b_{32} + b_{31} (12 + k^{2} + 2\,\Pr \,{\text{Re}} \,{\text{Re}}_{m} ))), \\ \end{aligned} $$
$$ \begin{aligned} n_{r3} & = \left( {{1 \mathord{\left/ {\vphantom {1 {38880\,Le}}} \right. \kern-\nulldelimiterspace} {38880\,Le}}} \right)\,( - (12 + k^{2} )^{3} (720 + k^{4} )\lambda + (12 + k^{2} )\Pr (k^{2} (12 + k^{2} )(1 + Le) \\ & \quad + (720 + k^{4} )Le\,\Pr ){\text{Re}}^{2} {\text{Re}}_{m} + {\text{Re}}_{m} ( - 360Leb_{13} ((12 + k^{2} )b_{31} - 6b_{21} b_{32} ) \\ & \quad - (12 + k^{2} )^{2} (720 + k^{4} )( - 2 + \lambda {\text{Re}}_{m} )) + {\text{Re}} ( - 180\,Le\,\Pr \,b_{13} \,b_{31} (12 + k^{2} + {\text{Re}}_{m}^{2} ) \\ & \quad + (12 + k^{2} )((12 + k^{2} )(k^{2} (12 + k^{2} ) + (720 + k^{4} )(1 + Le)\Pr ) \\ & \quad - (12 + k^{2} )(1 + Le)(720 + k^{4} )\Pr \,\lambda \,{\text{Re}}_{m} + (k^{2} (12 + k^{2} ) + (720 + k^{4} )(1 + Le)\Pr ){\text{Re}}_{m}^{2} ))),\, \\ \end{aligned} $$
$$ \begin{aligned} n_{r4} & = \left( {{1 \mathord{\left/ {\vphantom {1 {38880\,Le}}} \right. \kern-\nulldelimiterspace} {38880\,Le}}} \right)\,((12 + k^{2} )^{2} ( - (12 + k^{2} )(720 + k^{4} ) \\ & \quad - ((k^{2} (12 + k^{2} ) + (720 + k^{4} )(1 + Le)\,\Pr ){\text{Re}} - (720 + k^{4} )((12 + k^{2} )\lambda - \,{\text{Re}}_{m} )){\text{Re}}_{m} ) \\ & \quad + 180Le\,b_{13} ( - 6\,b_{21} b_{32} (12 + k^{2} + {\text{Re}}_{m}^{2} ) + (12 + k^{2} )b_{31} (12 + k^{2} + \Pr {\text{Re}} {\text{Re}}_{m} + {\text{Re}}_{m}^{2} ))), \\ \end{aligned} $$
$$ n_{r5} = \left( {{1 \mathord{\left/ {\vphantom {1 {38880\,Le}}} \right. \kern-\nulldelimiterspace} {38880\,Le}}} \right)((12 + k^{2} )((12 + k^{2} )^{2} (720 + k^{4} ) - 180Le\,b_{13} ((12 + k^{2} )b_{31} - 6b_{21} b_{32} )){\text{Re}}_{m} ). $$

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Hassan, M.A. Linear Instability of Electromagnetic Viscoelastic Nanofluid: Analytical and Numerical Study. Differ Equ Dyn Syst 31, 427–455 (2023). https://doi.org/10.1007/s12591-020-00541-9

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