APPENDIX
The dispersion equation of the problem and its transformation in the limits of minor viscosity:
$$\begin{gathered} {{\alpha }^{2}} + \frac{{2\alpha \nu {{k}^{2}}}}{{{{I}_{0}}(kR)}}\frac{{\left( {I_{1}^{'}(kR) - \frac{{2kl}}{{({{k}^{2}} + {{l}^{2}})}}\frac{{{{I}_{1}}(kR)}}{{{{I}_{1}}(lR)}}I_{1}^{'}(lR) + {{F}_{2}}(k,l,{{R}_{0}},R)} \right)}}{{(1 + {{F}_{1}}(k,l,{{R}_{0}},R))}} \\ = \left[ {\frac{{\sigma k}}{{\rho {{R}^{2}}}}(1 - {{k}^{2}}{{R}^{2}}) + \frac{{4\pi {{\chi }^{2}}}}{\rho }\frac{k}{R}\left( {kR\frac{{{{K}_{1}}(kR)}}{{{{K}_{0}}(kR)}} - 1} \right)} \right]\frac{{{{I}_{1}}(kR)}}{{{{I}_{0}}(kR)}}\frac{{({{l}^{2}} - {{k}^{2}})}}{{({{l}^{2}} + {{k}^{2}})}}\frac{{F(k,l,{{R}_{0}},R)}}{{(1 + {{F}_{1}}(k,l,{{R}_{0}},R))}}; \\ \end{gathered} $$
(A.1)
where
$$\begin{gathered} F(k,l,{{R}_{0}},R) \\ = \left( {1 + {{G}_{{11}}}(k,l,{{R}_{0}})\frac{{{{K}_{1}}(kR)}}{{{{I}_{1}}(kR)}}} \right. - \frac{{{{K}_{1}}(lR)}}{{{{I}_{1}}(lR)}}({{G}_{{22}}}(k,l,{{R}_{0}}) \\ - \,\,({{G}_{{12}}}(k,l,{{R}_{0}}){{G}_{{21}}}(k,l,{{R}_{0}}) \\ \left. { - \,\,{{G}_{{11}}}(k,l,{{R}_{0}}){{G}_{{22}}}(k,l,{{R}_{0}}))\left. {\frac{{{{K}_{1}}(kR)}}{{{{I}_{1}}(kR)}}} \right)} \right); \\ {{F}_{1}}(k,l,{{R}_{0}},R) = \left( {\frac{{2{{k}^{2}}}}{{\left( {{{l}^{2}} + {{k}^{2}}} \right)}}} \right.\left( {{{K}_{1}}(kR) + \frac{{{{I}_{1}}(kR)}}{{{{I}_{0}}(kR)}}{{K}_{0}}(kR)} \right) \\ \times \,\,\frac{1}{{{{I}_{1}}(lR)}}{{G}_{{12}}}(k,l,{{R}_{0}}) - \frac{{{{K}_{0}}(kR)}}{{{{I}_{0}}(kR)}} \\ \times \,\,\left( {{{G}_{{11}}}(k,l,{{R}_{0}}) + \frac{{{{K}_{1}}(lR)}}{{{{I}_{1}}(lR)}}\left( {{{G}_{{12}}}(k,l,{{R}_{0}}){{G}_{{21}}}(k,l,{{R}_{0}})} \right.} \right. \\ + \,\,\left. {\left. {\left. {\left( {\frac{{{{I}_{0}}(kR)}}{{{{K}_{0}}(kR)}} - {{G}_{{11}}}(k,l,{{R}_{0}}){{G}_{{22}}}(k,l,{{R}_{0}})} \right)} \right)} \right)} \right); \\ \end{gathered} $$
$$\begin{gathered} {{F}_{2}}(k,l,{{R}_{0}},R) = \frac{{{{k}^{2}}}}{{\left( {{{l}^{2}} + {{k}^{2}}} \right)}} \\ \times \,\,\left( {2{{I}_{1}}(kR){{K}_{1}}(kR)\left( {\frac{{I_{1}^{'}(kR)}}{{{{I}_{1}}(kR)}} - \frac{{K_{1}^{'}(kR)}}{{{{K}_{1}}(kR)}}} \right)\frac{{{{G}_{{12}}}(k,l,{{R}_{0}})}}{{{{I}_{1}}(lR)}}} \right. \\ + \,\,\left( {1 + \frac{{{{l}^{2}}}}{{{{k}^{2}}}}} \right)\frac{l}{k}{{K}_{1}}(lR)\left( {\frac{{I_{1}^{'}(lR)}}{{{{I}_{1}}(lR)}} - \frac{{K_{1}^{'}(lR)}}{{{{K}_{1}}(lR)}}} \right){{G}_{{21}}}(k,l,{{R}_{0}}) \\ \end{gathered} $$
$$\begin{gathered} + \,\,{{I}_{1}}(kR)\frac{{{{K}_{1}}(lR)}}{{{{I}_{1}}(lR)}}\left( {2\frac{l}{k}\frac{{K_{1}^{'}(lR)}}{{{{K}_{1}}(lR)}} - \left( {1 + \frac{{{{l}^{2}}}}{{{{k}^{2}}}}} \right)\frac{{I_{1}^{'}(kR)}}{{{{I}_{1}}(kR)}}} \right) \\ \times \,\,{{G}_{{22}}}(k,l,{{R}_{0}}) - {{K}_{1}}(lR) \\ \end{gathered} $$
$$ \times \,\,\left( {2\frac{l}{k}\frac{{I_{1}^{'}(lR)}}{{{{I}_{1}}(lR)}} - \left( {1 + \frac{{{{l}^{2}}}}{{{{k}^{2}}}}} \right)\frac{{K_{1}^{'}(kR)}}{{{{K}_{1}}(kR)}}} \right){{G}_{{11}}}(k,l,{{R}_{0}})$$
$$\begin{gathered} + \,\,{{K}_{1}}(kR)\frac{{{{K}_{1}}(lR)}}{{{{I}_{1}}(lR)}}\left( {2\frac{l}{k}\frac{{K_{1}^{'}(lR)}}{{{{K}_{1}}(lR)}} - \left( {1 + \frac{{{{l}^{2}}}}{{{{k}^{2}}}}} \right)\frac{{K_{1}^{'}(kR)}}{{{{K}_{1}}(kR)}}} \right) \\ \left. {\frac{{^{{}}}}{{_{{}}}} \times \,\,\left( {{{G}_{{11}}}{{G}_{{22}}} - {{G}_{{12}}}{{G}_{{21}}}} \right)} \right). \\ \end{gathered} $$
$$\begin{gathered} {{G}_{{11}}}(k,l,{{R}_{0}}) = \frac{{{{I}_{0}}(k{{R}_{0}})}}{{{{K}_{0}}(k{{R}_{0}})H(k,l,{{R}_{0}})}} \\ \times \,\,\left( {1 + \frac{l}{k}\frac{{{{K}_{0}}(l{{R}_{0}})}}{{{{K}_{1}}(l{{R}_{0}})}}\frac{{{{I}_{1}}(k{{R}_{0}})}}{{{{I}_{0}}(k{{R}_{0}})}}} \right); \\ \end{gathered} $$
$$\begin{gathered} {{G}_{{12}}}(k,l,{{R}_{0}}) = \frac{l}{k}\frac{{{{I}_{0}}(l{{R}_{0}})}}{{{{K}_{0}}(k{{R}_{0}})H(k,l,{{R}_{0}})}} \\ \times \,\,\left( {1 + \frac{{{{K}_{0}}(l{{R}_{0}})}}{{{{K}_{1}}(l{{R}_{0}})}}\frac{{{{I}_{1}}(k{{R}_{0}})}}{{{{I}_{0}}(k{{R}_{0}})}}} \right); \\ \end{gathered} $$
$$\begin{gathered} {{G}_{{21}}}(k,l,{{r}_{0}}) = \frac{{{{I}_{1}}(k{{R}_{0}})}}{{{{K}_{1}}(l{{R}_{0}})H(k,l,{{R}_{0}})}} \\ \times \,\,\left( {1 + \frac{{{{K}_{1}}(k{{R}_{0}})}}{{{{K}_{0}}(k{{R}_{0}})}}\frac{{{{I}_{0}}(k{{R}_{0}})}}{{{{I}_{1}}(k{{R}_{0}})}}} \right); \\ \end{gathered} $$
$$\begin{gathered} {{G}_{{22}}}(k,l,{{R}_{0}}) = \frac{{{{I}_{1}}(l{{R}_{0}})}}{{{{K}_{1}}(l{{R}_{0}})H(k,l,{{R}_{0}})}} \\ \times \,\,\left( {1 + \frac{l}{k}\frac{{{{K}_{1}}(k{{R}_{0}})}}{{{{K}_{0}}(k{{R}_{0}})}}\frac{{{{I}_{0}}(l{{R}_{0}})}}{{{{I}_{1}}(l{{R}_{0}})}}} \right); \\ \end{gathered} $$
$$H(k,l,{{R}_{0}}) = \left( {1 - \frac{l}{k}\frac{{{{K}_{0}}(l{{R}_{0}})}}{{{{K}_{1}}(l{{R}_{0}})}}\frac{{{{K}_{1}}(k{{R}_{0}})}}{{{{K}_{0}}(k{{R}_{0}})}}} \right).$$
Since the dispersion equation (5) is too complicated for the numerical solution (and impossible for the analytical) we will find the limit for the minor viscosity of this equation. For this purpose, let us use expanding formula
$$\begin{gathered} l = \sqrt {{{k}^{2}} + \frac{\alpha }{\nu }} \\ = k\sqrt {\frac{\alpha }{{\nu {{k}^{2}}}}} \left( {1 + \frac{1}{2}\frac{{\nu {{k}^{2}}}}{\alpha } - \frac{1}{8}{{{\left( {\frac{{\nu {{k}^{2}}}}{\alpha }} \right)}}^{2}} + {\text{O}}({{\nu }^{3}})} \right) \\ \end{gathered} $$
and the asymptotic formulas for the modified Bessel functions [14]:
$$\begin{gathered} {{I}_{n}}(z) = \frac{{\exp (z)}}{{\sqrt {2\pi z} }} \\ \times \,\,\left( {1 - \frac{{4{{n}^{2}} - 1}}{{8z}} + \frac{{\left( {4{{n}^{2}} - 1} \right)\left( {4{{n}^{2}} - 9} \right)}}{{2!{{{\left( {8z} \right)}}^{2}}}} + {\text{O}}\left( {\frac{1}{{{{z}^{3}}}}} \right)} \right); \\ {{K}_{n}}(z) = \sqrt {\frac{\pi }{{2z}}} \exp ( - z) \\ \times \,\,\left( {1 + \frac{{4{{n}^{2}} - 1}}{{8z}} + \frac{{\left( {4{{n}^{2}} - 1} \right)\left( {4{{n}^{2}} - 9} \right)}}{{2!{{{\left( {8z} \right)}}^{2}}}} + {\text{O}}\left( {\frac{1}{{{{z}^{3}}}}} \right)} \right); \\ \end{gathered} $$
$$\begin{gathered} I_{n}^{'}(z) = \frac{{\exp (z)}}{{\sqrt {2\pi z} }} \\ \times \,\,\left( {1 - \frac{{4{{n}^{2}} + 3}}{{8z}} + \frac{{\left( {4{{n}^{2}} - 1} \right)\left( {4{{n}^{2}} + 15} \right)}}{{2!{{{\left( {8z} \right)}}^{2}}}} + {\text{O}}\left( {\frac{1}{{{{z}^{3}}}}} \right)} \right); \\ \end{gathered} $$
$$\begin{gathered} K_{n}^{'}(z) = - \sqrt {\frac{\pi }{{2z}}} \exp ( - z) \\ \times \,\,\left( {1 + \frac{{4{{n}^{2}} + 3}}{{8z}} + \frac{{\left( {4{{n}^{2}} - 1} \right)\left( {4{{n}^{2}} + 15} \right)}}{{2!{{{\left( {8z} \right)}}^{2}}}} + {\text{O}}\left( {\frac{1}{{{{z}^{3}}}}} \right)} \right); \\ \end{gathered} $$
where О is the order symbol [16]. As a result, for the Bessel functions that enter dispersion equation (5), we will obtain the following expansions:
$$\begin{gathered} {{I}_{0}}(l{{R}_{0}}) \sim \frac{1}{{\sqrt {2\pi } }}\frac{1}{{\sqrt {k{{R}_{0}}} }}\exp \left( {\sqrt {\frac{\alpha }{{\nu {{k}^{2}}}}} } \right){{\left( {\frac{{\nu {{k}^{2}}}}{\alpha }} \right)}^{{\frac{1}{4}}}} \\ \times \,\,\left( {1 + \frac{1}{8}\frac{1}{{k{{R}_{0}}}}\sqrt {\frac{{\nu {{k}^{2}}}}{\alpha }} - \frac{1}{4}\left( {1 - \frac{9}{{32}}\frac{1}{{{{k}^{2}}R_{0}^{2}}}} \right)\frac{{\nu {{k}^{2}}}}{\alpha } + {\text{O}}\left( {{{\nu }^{{\frac{3}{2}}}}} \right)} \right); \\ \end{gathered} $$
$$\begin{gathered} {{I}_{1}}(l{{R}_{0}}) \sim \frac{1}{{\sqrt {2\pi } }}\frac{1}{{\sqrt {k{{r}_{0}}} }}\exp \left( {k{{R}_{0}}\sqrt {\frac{\alpha }{{\nu {{k}^{2}}}}} } \right){{\left( {\frac{{\nu {{k}^{2}}}}{\alpha }} \right)}^{{\frac{1}{4}}}} \\ \times \,\,\left( {1 - \frac{3}{8}\frac{1}{{k{{R}_{0}}}}\sqrt {\frac{{\nu {{k}^{2}}}}{\alpha }} - \frac{1}{4}\left( {1 + \frac{{15}}{{32}}\frac{1}{{{{k}^{2}}R_{0}^{2}}}} \right)\frac{{\nu {{k}^{2}}}}{\alpha } + {\text{O}}\left( {{{\nu }^{{\frac{3}{2}}}}} \right)} \right); \\ \end{gathered} $$
$$\begin{gathered} {{K}_{0}}(l{{R}_{0}}) \sim \sqrt {\frac{\pi }{2}} \frac{1}{{\sqrt {k{{R}_{0}}} }}\exp \left( { - k{{R}_{0}}\sqrt {\frac{\alpha }{{\nu {{k}^{2}}}}} } \right){{\left( {\frac{{\nu {{k}^{2}}}}{\alpha }} \right)}^{{\frac{1}{4}}}} \\ \times \,\,\left( {1 - \frac{1}{8}\frac{1}{{k{{R}_{0}}}}\sqrt {\frac{{\nu {{k}^{2}}}}{\alpha }} - \frac{1}{4}\left( {1 - \frac{9}{{32}}\frac{1}{{{{k}^{2}}R_{0}^{2}}}} \right)\frac{{\nu {{k}^{2}}}}{\alpha } + {\text{O}}\left( {{{\nu }^{{\frac{3}{2}}}}} \right)} \right); \\ \end{gathered} $$
$$\begin{gathered} {{K}_{1}}(l{{R}_{0}}) \sim \sqrt {\frac{\pi }{2}} \frac{1}{{\sqrt {k{{R}_{0}}} }}\exp \left( { - k{{R}_{0}}\sqrt {\frac{\alpha }{{\nu {{k}^{2}}}}} } \right){{\left( {\frac{{\nu {{k}^{2}}}}{\alpha }} \right)}^{{\frac{1}{4}}}} \\ \times \,\,\left( {1 + \frac{3}{8}\frac{1}{{k{{R}_{0}}}}\sqrt {\frac{{\nu {{k}^{2}}}}{\alpha }} - \frac{1}{4}\left( {1 + \frac{{15}}{{32}}\frac{1}{{{{k}^{2}}R_{0}^{2}}}} \right)\frac{{\nu {{k}^{2}}}}{\alpha } + {\text{O}}\left( {{{\nu }^{{\frac{3}{2}}}}} \right)} \right); \\ \end{gathered} $$
$$\begin{gathered} I_{1}^{'}(l{{R}_{0}}) \sim \frac{1}{{\sqrt {2\pi } }}\frac{1}{{\sqrt {k{{R}_{0}}} }}\exp \left( {k{{R}_{0}}\sqrt {\frac{\alpha }{{\nu {{k}^{2}}}}} } \right){{\left( {\frac{{\nu {{k}^{2}}}}{\alpha }} \right)}^{{\frac{1}{4}}}} \\ \times \,\,\left( {1 - \frac{7}{8}\frac{1}{{k{{R}_{0}}}}\sqrt {\frac{{\nu {{k}^{2}}}}{\alpha }} - \frac{1}{4}\left( {1 + \frac{{57}}{{32}}\frac{1}{{{{k}^{2}}R_{0}^{2}}}} \right)\frac{{\nu {{k}^{2}}}}{\alpha } + {\text{O}}\left( {{{\nu }^{{\frac{3}{2}}}}} \right)} \right); \\ \end{gathered} $$
$$\begin{gathered} K_{1}^{'}(l{{R}_{0}}) \sim - \sqrt {\frac{\pi }{2}} \frac{1}{{\sqrt {k{{R}_{0}}} }}\exp \left( { - k{{R}_{0}}\sqrt {\frac{\alpha }{{\nu {{k}^{2}}}}} } \right){{\left( {\frac{{\nu {{k}^{2}}}}{\alpha }} \right)}^{{\frac{1}{4}}}} \\ \times \,\,\left( {1 + \frac{7}{8}\frac{1}{{k{{R}_{0}}}}\sqrt {\frac{{\nu {{k}^{2}}}}{\alpha }} - \frac{1}{4}\left( {1 + \frac{{57}}{{32}}\frac{1}{{{{k}^{2}}R_{0}^{2}}}} \right)\frac{{\nu {{k}^{2}}}}{\alpha } + {\text{O}}\left( {{{\nu }^{{\frac{3}{2}}}}} \right)} \right). \\ \end{gathered} $$
Using the written expansions, we will get the following dispersion equation in the limit of the minor viscosity:
$$\begin{gathered} {{\alpha }^{2}} + 2\alpha \nu {{k}^{2}}\left[ {1 - L(k,{{R}_{0}},R)} \right] - L(k,{{R}_{0}},R) \\ \times \,\,\left[ {\frac{{\sigma {{k}^{2}}}}{{\rho R}}(1 - {{k}^{2}}{{R}^{2}}) + 4\pi \mu _{0}^{2}\frac{{{{k}^{2}}}}{\rho }\left( {kR\frac{{{{K}_{1}}(kR)}}{{{{K}_{0}}(kR)}} - 1} \right)} \right] \\ \times \,\,\left( {1 - \sqrt {\frac{{\nu {{k}^{2}}}}{{\alpha }}} } \right.\left( {{{g}_{1}} - {{d}_{1}}} \right) - \frac{{\nu {{k}^{2}}}}{{\alpha }}\left( {{{g}_{2}}\left( {k,{{R}_{0}},R} \right)} \right. \\ \left. {\frac{{^{{^{{}}}}}}{{_{{_{{}}}}}} + \,\,\left. {{{g}_{1}}\left( {k,{{R}_{0}},R} \right){{d}_{1}}\left( {k,{{R}_{0}},R} \right) - {{d}_{2}}\left( {k,{{R}_{0}},R} \right)} \right)} \right) = 0; \\ \end{gathered} $$
$$L(k,{{R}_{0}},R) \equiv \frac{1}{{kR}}\frac{{{{I}_{1}}(kR)}}{{{{I}_{0}}(kR)}}\frac{{\left( {1 - \frac{{{{I}_{1}}(k{{R}_{0}})}}{{{{I}_{1}}(kR)}}\frac{{{{K}_{1}}(kR)}}{{{{K}_{1}}(k{{R}_{0}})}}} \right)}}{{\left( {1 + \frac{{{{I}_{1}}(k{{R}_{0}})}}{{{{I}_{0}}(kR)}}\frac{{{{K}_{0}}(kR)}}{{{{K}_{1}}(k{{R}_{0}})}}} \right)}};$$
$${{g}_{1}}(k,{{R}_{0}},R) \equiv \frac{{{{I}_{0}}(k{{R}_{0}})}}{{{{I}_{1}}(kR)}}\frac{{{{K}_{1}}(kR)}}{{{{K}_{1}}(k{{R}_{0}})}}\frac{{\left( {1 + \frac{{{{I}_{1}}(k{{R}_{0}})}}{{{{I}_{0}}(k{{R}_{0}})}}\frac{{{{K}_{0}}(k{{R}_{0}})}}{{{{K}_{1}}(k{{R}_{0}})}}} \right)}}{{\left( {1 - \frac{{{{I}_{1}}(k{{R}_{0}})}}{{{{I}_{1}}(kR)}}\frac{{{{K}_{1}}(kR)}}{{{{K}_{1}}(k{{R}_{0}})}}} \right)}};$$
$${{g}_{2}}(k,{{R}_{0}},R) = 2 + \frac{{{{g}_{1}}(k,{{R}_{0}},R)}}{{h(k{{R}_{0}})}};$$
$$h(k{{R}_{0}}) \equiv \left( {\frac{{{{K}_{0}}(k{{R}_{0}})}}{{{{K}_{1}}(k{{R}_{0}})}} + \frac{1}{{2k{{R}_{0}}}}} \right);$$
$${{d}_{1}}(k,{{R}_{0}},R) = \frac{{{{I}_{1}}(k{{R}_{0}})}}{{{{I}_{0}}(kR)}}\frac{{{{K}_{0}}(kR)}}{{{{K}_{1}}(k{{R}_{0}})}}\frac{{\left( {\frac{{{{I}_{0}}(k{{R}_{0}})}}{{{{I}_{1}}(k{{R}_{0}})}} + \frac{{{{K}_{0}}(k{{R}_{0}})}}{{{{K}_{1}}(k{{R}_{0}})}}} \right)}}{{\left( {1 + \frac{{{{I}_{1}}(k{{R}_{0}})}}{{{{I}_{0}}(kR)}}\frac{{{{K}_{0}}(kR)}}{{{{K}_{1}}(k{{R}_{0}})}}} \right)}};$$
$${{d}_{2}}(k,{{R}_{0}},R) = d_{1}^{2}(k,{{R}_{0}},R)\left( {1 - h(k{{R}_{0}})} \right).$$
