New Instability of a Thin Vortex Ring in an Ideal Fluid

Abstract—The problem of stability of steady-state thin vortex ring flow in an ideal fluid is studied in the linear approximation. The case of the isochronous vortex ring in which the liquid-particle rotation periods are identical is considered. In such a flow there are no perturbations of the continuous spectrum. This makes considerably easier to solve this complex problem. The instability of longwave oscillations related to the interaction between the perturbations with energy of different signs, namely, the oscillations with positive and negative energy, is revealed.

Appendices

APPENDIX A

The form of the base deformation ${\boldsymbol{\varepsilon }_{{\left( 1 \right)}}}$

$$\begin{gathered} \varepsilon _{{\left( 1 \right)}}^{\sigma } = i\left( {\left( { - \mu \frac{{a\omega {\kern 1pt} '}}{{n\sigma }} + {{\mu }^{3}}\left( {\frac{{\omega {\kern 1pt} '\left( {96{{a}^{2}}\omega {\kern 1pt} '\; + 107{{\sigma }^{2}}} \right)}}{{96an\sigma }}} \right)} \right){{J}_{1}}\left( {{{a}_{0}}} \right)} \right. \\ \, - \left. {\left( {{{\mu }^{2}}\omega {\kern 1pt} '\; + {{\mu }^{4}}\frac{1}{{192}}\omega {\kern 1pt} '\left( {\frac{{252\omega {\kern 1pt} '}}{{{{n}^{2}}}} + \frac{{125{{\sigma }^{2}}}}{{{{a}^{2}}}}} \right)} \right){{J}_{2}}\left( {{{a}_{0}}} \right)} \right){{e}^{{i1\psi }}} \\ \, + i\left( {{{\mu }^{4}}\frac{{{{\omega }^{{'2}}}}}{n}{{J}_{1}}\left( {{{a}_{0}}} \right) + {{\mu }^{5}}\left( { - \frac{{10a{{\omega }^{{'3}}}}}{{{{n}^{2}}\sigma }} + \frac{{2{{\omega }^{{'2}}}\sigma }}{a}} \right){{J}_{2}}\left( {{{a}_{0}}} \right)} \right){{e}^{{i2\psi }}} \\ \end{gathered} $$

$$\begin{gathered} \, - {{\mu }^{5}}i\frac{{3{{\omega }^{{'2}}}}}{{64an\sigma }}\left( { - 15{{a}^{2}} - 16{{\sigma }^{2}} + 24{{a}^{2}}\ln \frac{8}{\mu }} \right){{J}_{1}}\left( {{{a}_{0}}} \right){{e}^{{i3\psi }}} \\ \, - {{\mu }^{5}}\frac{{2i{{\omega }^{{'2}}}\sigma }}{a}{{J}_{2}}\left( {{{a}_{0}}} \right){{e}^{{i0\psi }}} - {{\mu }^{5}}\frac{{i{{\omega }^{{'2}}}}}{{64an\sigma }}\left( { - 15{{a}^{2}} - 16{{\sigma }^{2}} + 24{{a}^{2}}\ln \frac{8}{\mu }} \right){{J}_{1}}\left( {{{a}_{0}}} \right){{e}^{{ - i\psi }}} + O({{\mu }^{5}}){{e}^{{i\psi }}} + O({{\mu }^{6}}), \\ \end{gathered} $$

$$\begin{gathered} \varepsilon _{{\left( 1 \right)}}^{\psi } = \left( { - \frac{1}{\sigma } + {{\mu }^{2}}\frac{1}{{192\sigma }}\left( { - \frac{{252\omega {\kern 1pt} '}}{{{{n}^{2}}}} - \frac{{275{{\sigma }^{2}}}}{{{{a}^{2}}}}} \right) + {{\mu }^{4}}\frac{\sigma }{{1024{{a}^{2}}{{n}^{2}}}}( - 3021\omega {\kern 1pt} '\; - 1385{{n}^{2}})} \right){{J}_{2}}\left( {{{a}_{0}}} \right){{e}^{{i1\psi }}} \\ \, - {{\mu }^{4}}\frac{1}{{368\,640{{a}^{4}}{{n}^{4}}\sigma }}(1\,809\,487{{n}^{4}}{{\sigma }^{4}} + 720{{a}^{4}}\omega {\kern 1pt} '( - 441\omega {\kern 1pt} '\; + 16{{n}^{2}}(45 + 76\omega {\kern 1pt} '\; + 16{{n}^{2}}\omega {\kern 1pt} '))){{J}_{2}}\left( {{{a}_{0}}} \right){{e}^{{i1\psi }}} \\ \, + {{\mu }^{4}}\frac{1}{{128{{a}^{4}}{{n}^{4}}\sigma }}(288{{a}^{4}}{{n}^{2}}\omega {\kern 1pt} '\; + 277{{a}^{2}}{{n}^{4}}{{\sigma }^{2}})\ln \frac{8}{\mu }{{J}_{2}}\left( {{{a}_{0}}} \right){{e}^{{i1\psi }}} + \left( {{{\mu }^{3}}\frac{{\omega {\kern 1pt} '}}{{2a}}{{J}_{2}}\left( {{{a}_{0}}} \right) + {{\mu }^{4}}\frac{{4{{\omega }^{{'2}}}}}{{n\sigma }}{{J}_{1}}\left( {{{a}_{0}}} \right)} \right){{e}^{{i2\psi }}} \\ \end{gathered} $$

$$\begin{gathered} \, + {{\mu }^{5}}\frac{{\omega {\kern 1pt} '}}{{192{{a}^{3}}{{n}^{2}}{{\sigma }^{2}}}}( - 1920{{a}^{4}}{{\omega }^{{'2}}} + 331{{n}^{2}}{{\sigma }^{4}} - 9{{a}^{2}}{{\sigma }^{2}}( - 14\omega {\kern 1pt} '\; + {{n}^{2}}\left( { - 15 + 32\omega {\kern 1pt} '} \right))){{J}_{2}}\left( {{{a}_{0}}} \right){{e}^{{i2\psi }}} \\ \, - {{\mu }^{5}}\frac{{9\omega {\kern 1pt} '}}{{8a}}\ln \frac{8}{\mu }{{J}_{2}}\left( {{{a}_{0}}} \right){{e}^{{i2\psi }}} + {{\mu }^{4}}\frac{{\omega {\kern 1pt} '}}{{64{{a}^{2}}\sigma }}\left( {16{{\sigma }^{2}} + 15{{a}^{2}} - 24{{a}^{2}}\ln \frac{8}{\mu }} \right){{J}_{2}}\left( {{{a}_{0}}} \right){{e}^{{i3\psi }}} \\ \end{gathered} $$

$$\begin{gathered} \, + {{\mu }^{5}}\frac{{{{\omega }^{{'2}}}}}{{64an{{\sigma }^{2}}}}\left( {176{{\sigma }^{2}} + 45{{a}^{2}} - 72{{a}^{2}}\ln \frac{8}{\mu }} \right){{J}_{1}}\left( {{{a}_{0}}} \right){{e}^{{i3\psi }}} \\ \, - {{\mu }^{5}}\frac{{\omega {\kern 1pt} '}}{{768{{a}^{3}}}}\left( { - 141{{a}^{2}} - 116{{\sigma }^{2}} + 168{{a}^{2}}\ln \frac{8}{\mu }} \right){{J}_{1}}\left( {{{a}_{0}}} \right){{e}^{{i4\psi }}} \\ \end{gathered} $$

$$\begin{gathered} \, + \left( {{{\mu }^{3}}\frac{{\omega {\kern 1pt} '}}{{2a}}{{J}_{2}}\left( {{{a}_{0}}} \right) + {{\mu }^{4}}\frac{{2{{\omega }^{{'2}}}}}{{n\sigma }}{{J}_{1}}\left( {{{a}_{0}}} \right)} \right){{e}^{{i0\psi }}} \\ \, - {{\mu }^{5}}\frac{{\omega {\kern 1pt} '}}{{192{{a}^{3}}{{n}^{2}}}}\left( {9{{a}^{2}}(14\omega {\kern 1pt} '\; + {{n}^{2}}\left( {15 + 32\omega {\kern 1pt} '} \right)) + 331{{n}^{2}}{{\sigma }^{2}} - 216{{a}^{2}}{{n}^{2}}\ln \frac{8}{\mu }} \right){{J}_{2}}\left( {{{a}_{0}}} \right){{e}^{{i0\psi }}} \\ {\text{ }} \\ \end{gathered} $$

$$\begin{gathered} \, + \left( {{{\mu }^{4}}\frac{{\omega {\kern 1pt} '}}{{64{{a}^{2}}\sigma }}\left( { - 15{{a}^{2}} - 16{{\sigma }^{2}} + 24{{a}^{2}}\ln \frac{8}{\mu }} \right){{J}_{2}}\left( {{{a}_{0}}} \right)} \right. \\ \, + \left. {{{\mu }^{5}}\frac{{{{\omega }^{{'2}}}}}{{64an{{\sigma }^{2}}}}\left( {15{{a}^{2}} + 16{{\sigma }^{2}} - 24{{a}^{2}}\ln \frac{8}{\mu }} \right){{J}_{1}}\left( {{{a}_{0}}} \right)} \right){{e}^{{ - i\psi }}} \\ \, + {{\mu }^{5}}\frac{{\omega {\kern 1pt} '}}{{768{{a}^{3}}}}\left( { - 141{{a}^{2}} - 116{{\sigma }^{2}} + 168{{a}^{2}}\ln \frac{8}{\mu }} \right){{J}_{2}}\left( {{{a}_{0}}} \right){{e}^{{ - i2\psi }}} + O({{\mu }^{5}}){{e}^{{i1\psi }}} + O({{\mu }^{6}}), \\ \end{gathered} $$

$$\begin{gathered} \varepsilon _{{\left( 1 \right)}}^{s} = \left( {1 + {{\mu }^{2}}\frac{{307{{\sigma }^{2}}}}{{96{{a}^{2}}}} + {{\mu }^{4}}\frac{{{{\sigma }^{2}}}}{{92\,160{{a}^{4}}{{n}^{2}}}}} \right. \\ \, \times \left. {\left( {1\,192\,213{{n}^{2}}{{\sigma }^{2}} - 180{{a}^{2}}\left( {274\omega {\kern 1pt} '\; + 239{{n}^{2}}\left( { - 5 + 8\ln \frac{8}{\mu }} \right)} \right)} \right)} \right){{J}_{1}}\left( {{{a}_{0}}} \right){{e}^{{i1\psi }}} \\ \, - \left( {{{\mu }^{3}}\frac{{2\omega {\kern 1pt} '\sigma }}{a} + {{\mu }^{5}}\frac{{\omega {\kern 1pt} '\sigma }}{{96{{a}^{3}}}}\left( {833{{\sigma }^{2}} + 9{{a}^{2}}\left( {5 + 16\omega {\kern 1pt} '\; - 8\ln \frac{8}{\mu }} \right)} \right)} \right){{J}_{1}}\left( {{{a}_{0}}} \right){{e}^{{i2\psi }}} - {{\mu }^{4}}\frac{{5\omega {\kern 1pt} '{{\sigma }^{2}}}}{{8{{a}^{2}}}}{{J}_{1}}\left( {{{a}_{0}}} \right){{e}^{{i3\psi }}} \\ \end{gathered} $$

$$\begin{gathered} \, - {{\mu }^{5}}\frac{{11\omega {\kern 1pt} '{{\sigma }^{3}}}}{{48{{a}^{3}}}}{{J}_{1}}\left( {{{a}_{0}}} \right){{e}^{{i4\psi }}} + \left( {{{\mu }^{3}}\frac{{2\omega {\kern 1pt} '\sigma }}{a} + {{\mu }^{5}}\frac{{\omega {\kern 1pt} '\sigma }}{{96{{a}^{3}}}}\left( {833{{\sigma }^{2}} - 9{{a}^{2}}\left( { - 5 + 16\omega {\kern 1pt} '\; + 8\ln \frac{8}{\mu }} \right)} \right)} \right){{J}_{1}}\left( {{{a}_{0}}} \right){{e}^{{i0\psi }}} \\ \, + {{\mu }^{4}}\frac{{5\omega {\kern 1pt} '{{\sigma }^{2}}}}{{8{{a}^{2}}}}{{J}_{1}}\left( {{{a}_{0}}} \right){{e}^{{ - i\psi }}} + {{\mu }^{5}}\frac{{11\omega {\kern 1pt} '{{\sigma }^{3}}}}{{48{{a}^{3}}}}{{J}_{1}}\left( {{{a}_{0}}} \right){{e}^{{ - i2\psi }}} + O({{\mu }^{6}}), \\ \end{gathered} $$

$$\begin{gathered} {{a}_{0}} = \frac{{n\sigma }}{{a\omega {\kern 1pt} '\mu }}\left( {1 + {{\mu }^{2}}\left( {\frac{{21\omega {\kern 1pt} '}}{{16{{n}^{2}}}} + \frac{{163{{\sigma }^{2}}}}{{192{{a}^{2}}}}} \right) + {{\mu }^{4}}X + O({{\mu }^{6}})} \right), \\ X = \frac{1}{{40\,960{{a}^{4}}{{n}^{4}}}}(82\,611{{n}^{4}}{{\sigma }^{4}} + 40{{a}^{2}}{{n}^{2}}{{\sigma }^{2}}(1689\omega {\kern 1pt} '\; + 745{{n}^{2}}) - 20{{a}^{4}}(1764{{\omega }^{{'2}}} - 64{{n}^{2}}\omega {\kern 1pt} '(45 + 76\omega {\kern 1pt} '))) \\ \, + \frac{{225}}{{2048}} - \frac{{{{\omega }^{{'2}}}}}{2} - \frac{1}{{128{{a}^{2}}{{n}^{2}}}}\ln \frac{8}{\mu }\left( {149{{n}^{2}}{{\sigma }^{2}} + 9{{a}^{2}}(32\omega {\kern 1pt} '\, + 5{{n}^{2}}) - 36{{a}^{2}}{{n}^{2}}\ln \frac{8}{\mu }} \right). \\ \end{gathered} $$

The form of the base deformation ${\boldsymbol{\varepsilon }_{{\left( 2 \right)}}}$

$$\begin{gathered} \varepsilon _{{\left( 2 \right)}}^{\sigma } = \left( {\frac{\sigma }{a} - {{\mu }^{2}}\frac{{{{\sigma }^{3}}}}{{192{{a}^{3}}}}( - 26 + 64{{n}^{2}})} \right){{e}^{{i2\psi }}} + {{\mu }^{4}}\frac{{{{\sigma }^{3}}}}{{6144{{a}^{5}}}} \\ \, \times \left( { - 10\,240{{a}^{2}}( - 2 + {{n}^{2}})\omega {\kern 1pt} '\; + \left( {( - 5899 - 4164{{n}^{2}} + 240{{n}^{4}}){{\sigma }^{2}} + 3870{{a}^{2}} - 6192{{a}^{2}}\ln \frac{8}{\mu }} \right)} \right){{e}^{{i2\psi }}} \\ \, - \left( {\mu \frac{{5{{\sigma }^{2}}}}{{8{{a}^{2}}}} + {{\mu }^{3}}\frac{{{{\sigma }^{4}}}}{{384{{a}^{4}}}}(606 + 17{{n}^{2}}) + O({{\mu }^{5}})} \right){{e}^{{i3\psi }}} + O({{\mu }^{4}}){{e}^{{i4\psi }}} + \mu \left( { - \frac{{3{{\sigma }^{2}}}}{{4{{a}^{2}}}} - \frac{{5\omega {\kern 1pt} '}}{{2{{n}^{2}}}}} \right){{e}^{{i\psi }}} \\ \end{gathered} $$

$$\begin{gathered} \, + {{\mu }^{3}}\frac{1}{{768{{a}^{4}}{{n}^{4}}}}\left( {144{{a}^{2}}(35 + 32{{n}^{2}}){{{\omega '}}^{2}} + 6{{a}^{2}}{{n}^{2}}\omega {\kern 1pt} '\left( {(631 + 192{{n}^{2}}){{\sigma }^{2}} - 150{{a}^{2}} + 240{{a}^{2}}\ln \frac{8}{\mu }} \right)} \right){{e}^{{i\psi }}} \\ \, + {{\mu }^{3}}\frac{{{{\sigma }^{2}}}}{{768{{a}^{4}}}}\left( {( - 499 + 184{{n}^{2}}){{\sigma }^{2}} + 45{{a}^{2}} - 72{{a}^{2}}\ln \frac{8}{\mu }} \right){{e}^{{i\psi }}} + O({{\mu }^{5}}){{e}^{{i\psi }}} + O({{\mu }^{4}}){{e}^{{i0\psi }}} + O({{\mu }^{3}}){{e}^{{ - i1\psi }}} + O({{\mu }^{6}}){{e}^{{i2\psi }}}, \\ \end{gathered} $$

$$\begin{gathered} \varepsilon _{{\left( 2 \right)}}^{\psi } = i\left( {\frac{1}{a} + {{\mu }^{2}}\frac{{{{\sigma }^{2}}}}{{48{{a}^{3}}}}(13 - 20{{n}^{2}})} \right){{e}^{{i2\psi }}} + i{{\mu }^{4}}\frac{{{{\sigma }^{2}}}}{{6144{{a}^{5}}}} \\ \, \times \left( {1024{{a}^{2}}(25 - 14{{n}^{2}})\omega {\kern 1pt} '\; + \left( {( - 17\,697 - 8828{{n}^{2}} + 336{{n}^{4}}){{\sigma }^{2}} + 7740{{a}^{2}} - 12\,384{{a}^{2}}\ln \frac{8}{\mu }} \right)} \right){{e}^{{i2\psi }}} \\ \, - i\left( {\mu \frac{{5\sigma }}{{8{{a}^{2}}}} + {{\mu }^{3}}\frac{{{{\sigma }^{3}}}}{{384{{a}^{4}}}}(1010 + 7{{n}^{2}}) + O({{\mu }^{5}})} \right){{e}^{{i3\psi }}} + O({{\mu }^{4}}){{e}^{{i4\psi }}} + O({{\mu }^{3}}){{e}^{{i5\psi }}} + i\mu \left( {\frac{\sigma }{{4{{a}^{2}}}} - \frac{{5\omega {\kern 1pt} '}}{{2{{n}^{2}}\sigma }}} \right){{e}^{{i\psi }}} \\ \end{gathered} $$

$$\begin{gathered} \, + i{{\mu }^{3}}\frac{1}{{768{{a}^{4}}{{n}^{4}}\sigma }}\left( {144{{a}^{4}}(35 + 32{{n}^{2}})\omega {\kern 1pt} {{'}^{2}}\, + 6{{a}^{2}}{{n}^{2}}\omega {\kern 1pt} '\left( {(1053 + 128{{n}^{2}}){{\sigma }^{2}} - 150{{a}^{2}} + 240{{a}^{2}}\ln \frac{8}{\mu }} \right)} \right){{e}^{{i\psi }}} \\ \, + i{{\mu }^{3}}\frac{\sigma }{{768{{a}^{4}}}}{{e}^{{i\psi }}} + O({{\mu }^{5}}){{e}^{{i\psi }}} + i{{\mu }^{2}}\frac{3}{{32{{a}^{3}}}}\left( { - 6{{\sigma }^{2}} - 5{{a}^{2}} + 8{{a}^{2}}\ln \frac{8}{\mu }} \right){{e}^{{i0\psi }}} + O({{\mu }^{4}}){{e}^{{i0\psi }}} + O({{\mu }^{3}}){{e}^{{ - i\psi }}}, \\ \end{gathered} $$

$$\begin{gathered} \varepsilon _{{\left( 2 \right)}}^{s} = i\left( { - \mu \frac{{n{{\sigma }^{2}}}}{{2{{a}^{2}}}} + {{\mu }^{3}}\left( {\frac{{n{{\sigma }^{4}}}}{{192{{a}^{4}}}}( - 229 + 24{{n}^{2}}) + \frac{{\omega {\kern 1pt} '{{\sigma }^{2}}}}{{{{a}^{2}}n}}(5 - 2{{n}^{2}})} \right) + O({{\mu }^{5}})} \right){{e}^{{i2\psi }}} - i{{\mu }^{2}}\frac{{n{{\sigma }^{3}}}}{{6{{a}^{3}}}}{{e}^{{i3\psi }}} \\ \, + i{{\mu }^{4}}\frac{{{{\sigma }^{3}}}}{{256{{a}^{5}}n}}\left( {80{{a}^{2}}(5 - 2{{n}^{2}})\omega {\kern 1pt} '\; + {{n}^{2}}\left( {( - 179 + 12{{n}^{2}}){{\sigma }^{2}} + 5{{a}^{2}} - 8{{a}^{2}}\ln \frac{8}{\mu }} \right)} \right){{e}^{{i3\psi }}} + O({{\mu }^{6}}){{e}^{{i3\psi }}} \\ \, - i{{\mu }^{3}}\frac{{49n{{\sigma }^{4}}}}{{576{{a}^{4}}}}{{e}^{{i4\psi }}} + O({{\mu }^{5}}){{e}^{{i4\psi }}} + O({{\mu }^{3}}){{e}^{{i5\psi }}} + i{{\mu }^{2}}\frac{\sigma }{{128{{a}^{3}}{{n}^{3}}}} \\ \end{gathered} $$

$$\begin{gathered} \, \times \left( {56{{a}^{2}}(15 + 8{{n}^{2}})\omega {\kern 1pt} '\; + {{n}^{2}}\left( {( - 749 + 160{{n}^{2}}){{\sigma }^{2}} - 150{{a}^{2}} + 240{{a}^{2}}\ln \frac{8}{\mu }} \right)} \right){{e}^{{i1\psi }}} + O({{\mu }^{4}}){{e}^{{i1\psi }}} \\ \, - i{{\mu }^{3}}\frac{{{{\sigma }^{2}}}}{{64{{a}^{4}}n}}\left( {320{{a}^{2}}\omega {\kern 1pt} '\; + 9{{n}^{2}}\left( {5{{\sigma }^{2}} + 5{{a}^{2}} - 8{{a}^{2}}\ln \frac{8}{\mu }} \right)} \right){{e}^{{i0\psi }}} + O({{\mu }^{5}}){{e}^{{i0\psi }}} + O({{\mu }^{3}}){{e}^{{ - i\psi }}}. \\ \end{gathered} $$

The form of the base deformation ${\boldsymbol{\varepsilon }_{{\left( 3 \right)}}}$

$$\begin{gathered} \varepsilon _{{\left( 3 \right)}}^{\sigma } = \left( {{{\sigma }^{2}} + {{\mu }^{2}}\frac{{215{{\sigma }^{4}}}}{{128{{a}^{2}}}} + O({{\mu }^{4}})} \right){{e}^{{i3\psi }}} + \left( {\mu \frac{{8{{\sigma }^{3}}}}{{3a}} + O({{\mu }^{3}})} \right){{e}^{{i2\psi }}} + O({{\mu }^{2}}), \\ \varepsilon _{{\left( 3 \right)}}^{\psi } = i\left( {\sigma + {{\mu }^{2}}\frac{{1075{{\sigma }^{3}}}}{{384{{a}^{2}}}} + O({{\mu }^{4}})} \right){{e}^{{i3\psi }}} + i\left( {\mu \frac{{16{{\sigma }^{2}}}}{{3a}} + O({{\mu }^{3}})} \right){{e}^{{i2\psi }}} + O({{\mu }^{2}}), \\ \varepsilon _{{\left( 3 \right)}}^{s} = O({{\mu }^{2}}). \\ \end{gathered} $$

The form of the base deformation ${\boldsymbol{\varepsilon }_{{\left( 4 \right)}}}$

$$\begin{gathered} \varepsilon _{{\left( 4 \right)}}^{\sigma } = \left( {\frac{{{{\sigma }^{3}}}}{{{{a}^{2}}}} + {{\mu }^{2}}\left( {\frac{{{{\sigma }^{5}}}}{{720{{a}^{4}}}}(1449 + 26{{n}^{2}})} \right) + O({{\mu }^{4}})} \right){{e}^{{i4\psi }}} + \left( {\mu \frac{{99{{\sigma }^{4}}}}{{32{{a}^{3}}}} + O({{\mu }^{3}})} \right){{e}^{{i3\psi }}} \\ + \,{{\mu }^{2}}\frac{{9{{\sigma }^{3}}}}{{32{{a}^{4}}}}\left( {5{{a}^{2}} + 20{{\sigma }^{2}} - 8{{a}^{2}}\ln \frac{8}{\mu }} \right){{e}^{{i2\psi }}} + O({{\mu }^{2}}), \\ \end{gathered} $$

$$\begin{gathered} \varepsilon _{{\left( 4 \right)}}^{\psi } = i\left( {\frac{{{{\sigma }^{2}}}}{{{{a}^{2}}}} + {{\mu }^{2}}\frac{{{{\sigma }^{4}}}}{{480{{a}^{4}}}}(1449 + 16{{n}^{2}})} \right){{e}^{{i4\psi }}} + i\left( {\mu \frac{{165{{\sigma }^{3}}}}{{32{{a}^{3}}}} + O({{\mu }^{3}})} \right){{e}^{{i3\psi }}} \\ \, + i{{\mu }^{2}}\frac{{9{{\sigma }^{2}}}}{{16{{a}^{4}}}}\left( {30{{\sigma }^{2}} + 5{{a}^{2}} - 8{{a}^{2}}\ln \frac{8}{\mu }} \right){{e}^{{i2\psi }}} + O({{\mu }^{2}}), \\ \varepsilon _{{\left( 4 \right)}}^{s} = \left( {i\mu \frac{{n{{\sigma }^{4}}}}{{12{{a}^{3}}}} + O({{\mu }^{3}})} \right){{e}^{{i4\psi }}} + O({{\mu }^{2}}). \\ \end{gathered} $$

The form of the base deformation ${\boldsymbol{\varepsilon }_{{\left( 5 \right)}}}$

$$\begin{gathered} \varepsilon _{{(5)}}^{\sigma } = {{\sigma }^{4}}{{e}^{{i5\psi }}} + O(\mu ), \\ \varepsilon _{{(5)}}^{\psi } = i{{\sigma }^{3}}{{e}^{{i5\psi }}} + O(\mu ), \\ \varepsilon _{{(5)}}^{s} = O(\mu ). \\ \end{gathered} $$

The form of the base deformation ${\boldsymbol{\varepsilon }_{{\left( 0 \right)}}}$

$$\begin{gathered} \varepsilon _{{\left( 0 \right)}}^{\sigma } = i\left( { - {{\mu }^{2}}\frac{{n\sigma }}{{2a}} + {{\mu }^{4}}\frac{{n{{\sigma }^{3}}}}{{16{{a}^{3}}}}( - 8 + 3{{n}^{2}})} \right){{e}^{{i0\psi }}} + i\mu \frac{1}{{2n}}{{e}^{{i1\psi }}} \\ \, + i{{\mu }^{3}}\frac{1}{{128{{a}^{2}}{{n}^{3}}}}\left( { - 168{{a}^{2}}\omega {\kern 1pt} '\; + 96{{a}^{2}}{{n}^{2}}\omega {\kern 1pt} '\; + 15{{a}^{2}}{{n}^{2}} - 11{{n}^{2}}{{\sigma }^{2}} - 8{{n}^{4}}{{\sigma }^{2}} - 24{{a}^{2}}{{n}^{2}}\ln \frac{8}{\mu }} \right){{e}^{{i1\psi }}} \\ \, - i{{\mu }^{3}}\frac{{7n{{\sigma }^{2}}}}{{32{{a}^{2}}}}{{e}^{{ - i\psi }}} + O({{\mu }^{4}}), \\ \end{gathered} $$

$$\begin{gathered} \varepsilon _{{\left( 0 \right)}}^{\psi } = \left( { - {{\mu }^{2}}\frac{n}{a} + {{\mu }^{4}}\frac{{16{{a}^{2}}( - 1 + 2{{n}^{2}})\omega {\kern 1pt} '{{\sigma }^{2}} + 3{{n}^{2}}( - 11 + 2{{n}^{2}}){{\sigma }^{4}}}}{{16{{a}^{3}}n{{\sigma }^{2}}}}} \right){{e}^{{i0\psi }}} \\ + \left( { - \mu \frac{1}{{2n\sigma }} + {{\mu }^{3}}\frac{1}{{128{{a}^{2}}{{n}^{3}}\sigma }}\left( {168{{a}^{2}}\omega {\kern 1pt} '\; - 96{{a}^{2}}{{n}^{2}}\omega {\kern 1pt} '\; - 15{{a}^{2}}{{n}^{2}} - 135{{n}^{2}}{{\sigma }^{2}} + 56{{n}^{4}}{{\sigma }^{2}} + 24{{a}^{2}}{{n}^{2}}\ln \frac{8}{\mu }} \right)} \right){{e}^{{i\psi }}} \\ \, - {{\mu }^{3}}\frac{{5n\sigma }}{{32{{a}^{2}}}}{{e}^{{ - i\psi }}} + O({{\mu }^{4}}), \\ \varepsilon _{{\left( 0 \right)}}^{s} = (1 + O({{\mu }^{3}})){{e}^{{i0\psi }}} + O({{\mu }^{2}}). \\ \end{gathered} $$

The form of the base deformation ${\boldsymbol{\varepsilon }_{{\left( { - 1} \right)}}}$

$$\begin{gathered} \varepsilon _{{\left( { - 1} \right)}}^{\sigma } = {{e}^{{ - i\psi }}} + O\left( \mu \right), \\ \varepsilon _{{\left( { - 1} \right)}}^{\psi } = - i\frac{1}{\sigma }{{e}^{{ - i\psi }}} + O\left( \mu \right), \\ \varepsilon _{{\left( { - 1} \right)}}^{s} = O\left( \mu \right). \\ \end{gathered} $$

APPENDIX B

Let us find the velocity field ${\mathbf{v}}$ for a source with the density

$$\begin{gathered} Q\left( {\sigma ,\psi {\text{,}}\theta } \right) = \delta \left( {\sigma - {{\sigma }_{{{\text{bound}}}}}} \right){{e}^{{im\psi + in\theta }}}, \\ {{\sigma }_{{{\text{bound}}}}} = a - {{\mu }^{2}}\frac{5}{{16}}a + {{\mu }^{4}}\frac{1}{{1024}}\left( { - 469 + 312\ln \frac{8}{\mu }} \right)a + O({{\mu }^{6}}). \\ \end{gathered} $$

In the domain $\sigma > {{\sigma }_{{{\text{bound}}}}}$ the velocity field can be expressed in terms of $Q$ by the formula

$$\begin{gathered} {\mathbf{v}} = \nabla \Phi ,\quad \Phi \,{\text{ = }} - \frac{1}{{4\pi }}\int {\frac{{Q\left( {r{\kern 1pt} '} \right)dr{\kern 1pt} '}}{{\left| {r - r{\kern 1pt} '} \right|}}} , \\ {{\left| {r - r{\kern 1pt} '} \right|}^{2}} = {{R}^{2}}\left( {1 - \mu \frac{\rho }{a}\cos \phi } \right)\left( {1 - \mu \frac{{\rho {\kern 1pt} '}}{a}\cos \phi {\kern 1pt} '} \right)(2(1 - \cos \left( {\theta - \theta {\kern 1pt} '} \right) + {{\gamma }^{2}})), \\ {{\gamma }^{2}} = {{\mu }^{2}}\frac{{{{{\left( {\rho \sin \phi - \rho {\kern 1pt} '\sin \phi {\kern 1pt} '} \right)}}^{2}} + {{{\left( {\rho \cos \phi - \rho {\kern 1pt} '\cos \phi {\kern 1pt} '} \right)}}^{2}}}}{{a\left( {a - \mu \rho \cos \phi } \right)\left( {a - \mu \rho {\kern 1pt} '\cos \phi {\kern 1pt} '} \right)}}. \\ \end{gathered} $$

(B.1)

With regard to (B.1) we obtain the expression for $\Phi$

$$\Phi = - \frac{{{{e}^{{in\theta }}}}}{{2\pi {{{\left( {1 - \mu \frac{\rho }{a}\cos \phi } \right)}}^{{1/2}}}}}\int\limits_0^{{{\sigma }_{{bound}}}} {\int\limits_0^{2\pi } {\frac{{\delta (\sigma {\kern 1pt} '\, - {{\sigma }_{{bound}}}){{e}^{{im\psi '}}}\sigma {\kern 1pt} '}}{{{{{\left( {1 - \mu \frac{{\rho {\kern 1pt} '}}{a}\cos \phi {\kern 1pt} '} \right)}}^{{1/2}}}}}P} } \left( {\sigma ,\psi ,\sigma {\kern 1pt} ',\psi {\kern 1pt} '} \right)d\sigma {\kern 1pt} 'd\psi {\kern 1pt} ',$$

(B.2)

where

$$P = \int\limits_0^\pi {\frac{{\cos n\alpha d\alpha }}{{{{{(2\left( {1 - \cos \alpha } \right) + {{\gamma }^{2}})}}^{{1/2}}}}}} .$$

All the complexity consists in calculating the asymptotics of the integral P, while the further integration in (B.2) can be carried out elementary. In accordance with [6], we obtain

$$\begin{gathered} P = - 2{{S}_{n}} + \ln \frac{8}{\gamma } + {{\gamma }^{2}}\left( {\frac{{4{{n}^{2}} - 1}}{{16}}\ln \frac{8}{\gamma } + \frac{{4{{n}^{2}} + 1}}{{16}} - \frac{{4{{n}^{2}} - 1}}{8}{{S}_{n}}} \right) - \frac{1}{{2048}}{{\gamma }^{4}}\left( {21 + 36{{S}_{n}} + 8{{n}^{2}}_{{_{{_{{_{{_{{_{{_{{_{{_{{}}}}}}}}}}}}}}}}}}} \right. \\ \, \times \left. {\left( {7 - 20{{S}_{n}} + 2{{n}^{2}}\left( { - 3 + 4{{S}_{n}}} \right) + ( - 18 + 80{{n}^{2}} - 32{{n}^{4}})\ln \frac{8}{\gamma }} \right)} \right) + O({{\gamma }^{6}}), \\ {{S}_{n}} = \sum\limits_{l = 1}^n {\frac{1}{{2l - 1}}\,} . \\ \end{gathered} $$

(B.3)

For brevity, in what follows the multiplier ${{e}^{{in\theta }}}$ will be omitted. Integrating in (B.2) with regard to (B.3), from (B.1) we obtain

$$\begin{gathered} v_{{}}^{\sigma } = 1 + {{\mu }^{2}}\left( {\frac{1}{4}{{n}^{2}}\left( {4{{S}_{n}} - 2\ln \frac{8}{\mu }} \right)} \right) + \mu \frac{1}{8}\left( {1 + 8{{S}_{n}} - 4\ln \frac{8}{\mu }} \right)\cos \psi \\ \, + {{\mu }^{3}}\frac{1}{{512}}\left( {101 + 56{{n}^{2}}\left( { - 1 + 8{{S}_{n}}} \right) + 192{{S}_{n}} - 8( - 9 + 28{{n}^{2}} - 24{{S}_{n}})\ln \frac{8}{\mu }} \right)\cos \psi \\ \end{gathered} $$

$$\begin{gathered} \, - {{\mu }^{3}}\frac{1}{{64}}\left( {108\ln 2\ln 2 + \ln \mu \left( { - 72\ln 2 + 12\ln \mu } \right)} \right)\cos \psi + {{\mu }^{2}}\left( {\frac{7}{{64}}} \right)\cos 2\psi \\ \, + {{\mu }^{3}}\frac{1}{{256}}\left( {11 + 24{{S}_{n}} - 12\ln \frac{8}{\mu }} \right)\cos 3\psi + O({{\mu }^{4}}),\quad m = 0, \\ \end{gathered} $$

$$\begin{gathered} v_{{}}^{\sigma } = \left( {\frac{1}{2} + {{\mu }^{2}}\frac{1}{{16}}\left( {2 + 4{{S}_{n}} + {{n}^{2}}\left( { - 1 - 8{{S}_{n}}} \right) - (2 - 4{{n}^{2}})\ln \frac{8}{\mu }} \right) + O({{\mu }^{4}})} \right){{e}^{{i\psi }}} \\ \, + \left( { - \mu \frac{1}{8} + {{\mu }^{3}}\frac{1}{{1536}}\left( {61 + 144{{S}_{n}} - 8{{n}^{2}}\left( {7 + 72{{S}_{n}}} \right) - 144(1 - 2{{n}^{2}})\ln \frac{8}{\mu }} \right) + O({{\mu }^{5}})} \right){{e}^{{i2\psi }}} \\ \, + \left( { - {{\mu }^{2}}\frac{1}{{64}} + {{\mu }^{4}}\frac{1}{{12288}}\left( {1409 + 720{{S}_{n}} - 4{{n}^{2}}\left( {103 + 576{{S}_{n}}} \right) + 144( - 9 + 8{{n}^{2}})\ln \frac{8}{\mu }} \right) + O({{\mu }^{6}})} \right){{e}^{{i3\psi }}} \\ \end{gathered} $$

$$\begin{gathered} \, + \left( { - {{\mu }^{3}}\frac{{11}}{{768}} + O({{\mu }^{5}})} \right){{e}^{{i4\psi }}} - \left( {\frac{{59}}{{12288}}{{\mu }^{4}} + O({{\mu }^{6}})} \right){{e}^{{i5\psi }}} + \left( {\frac{1}{{32}}{{n}^{2}}{{\mu }^{3}}\left( {1 + 8{{S}_{n}} - 4\ln \frac{8}{\mu }} \right) + O({{\mu }^{5}})} \right){{e}^{{i0\psi }}} \\ \, + \left( {{{\mu }^{2}}\frac{1}{{64}}\left( {21 + 8{{S}_{n}} - 16\ln \frac{8}{\mu }} \right) + {{\mu }^{4}}\frac{1}{{2048}}\left( {1435 + 440{{S}_{n}} + 8{{n}^{2}}\left( { - 39 + 40{{S}_{n}}} \right)\mathop {}\limits_{_{{}}}^{^{{}}} } \right.} \right. \\ \end{gathered} $$

$$\begin{gathered} \left. {\left. {\, + 32( - 26 + {{n}^{2}})\ln \frac{8}{\mu }} \right)} \right){{e}^{{ - i\psi }}} + \left( { - \frac{3}{{1024}}{{\mu }^{3}}\left( { - 25 + 8\ln \frac{8}{\mu }} \right) + O({{\mu }^{5}})} \right){{e}^{{ - 2i\psi }}} \\ \, + \left( {{{\mu }^{4}}\frac{1}{{122880}}\left( {6233 + 1440{{S}_{n}} - 3240\ln \frac{8}{\mu }} \right) + O({{\mu }^{6}})} \right){{e}^{{ - 3i\psi }}} \\ \, + O({{\mu }^{6}}){{e}^{{ - i\psi }}} + O({{\mu }^{5}}){{e}^{{ - 4i\psi }}},\quad m = 1, \\ \end{gathered} $$

$$\begin{gathered} v_{{}}^{\sigma } = \left( {\frac{1}{2} + {{\mu }^{2}}\frac{1}{{24}}( - 2 + {{n}^{2}}) + {{\mu }^{4}}\frac{1}{{6144}}( - 343 + 576{{S}_{n}} + 8{{n}^{4}}\left( {48{{S}_{n}} - 5} \right) + 10{{n}^{2}}\left( {13 - 240{{S}_{n}}} \right))} \right){{e}^{{i2\psi }}} \\ \, - {{\mu }^{4}}\frac{1}{{128}}(6 - 25{{n}^{2}} + 4{{n}^{4}})\ln \frac{8}{\mu }{{e}^{{i2\psi }}} - \left( {\frac{1}{{16}}\mu - {{\mu }^{3}}\frac{1}{{3072}}\left( {116{{n}^{2}} - 165 - 72\ln \frac{8}{\mu }} \right) + O({{\mu }^{5}})} \right){{e}^{{i3\psi }}} \\ \, - \left( {\frac{1}{{96}}{{\mu }^{2}} - {{\mu }^{4}}\frac{1}{{122880}}\left( {1233 + 2720{{n}^{2}} - 4680\ln \frac{8}{\mu }} \right) + O({{\mu }^{6}})} \right){{e}^{{i4\psi }}} \\ \end{gathered} $$

$$\begin{gathered} \, - \left( {\mu \frac{1}{{16}} - {{\mu }^{3}}\frac{1}{{1024}}\left( {67 + 112{{S}_{n}} - 8{{n}^{2}}\left( {7 + 40{{S}_{n}}} \right) - 80\left( {1 - 2{{n}^{2}}} \right)\ln \frac{8}{\mu }} \right) + O({{\mu }^{5}})} \right){{e}^{{i\psi }}} \\ \, + \left( {{{\mu }^{4}}\frac{1}{{256}}{{n}^{2}}\left( {21 + 16{{S}_{n}} - 20\ln \frac{8}{\mu }} \right) + O({{\mu }^{6}})} \right){{e}^{{i0\psi }}} \\ \, + \left( {{{\mu }^{3}}\frac{1}{{2048}}\left( {383 + 64{{S}_{n}} - 248\ln \frac{8}{\mu }} \right) + O({{\mu }^{5}})} \right){{e}^{{ - i\psi }}} \\ \, + O({{\mu }^{3}}){{e}^{{i5\psi }}} + O({{\mu }^{4}}){{e}^{{ - 2i\psi }}} + O({{\mu }^{5}}),\quad m = 2, \\ \end{gathered} $$

$$\begin{gathered} v_{{}}^{\sigma } = \left( {\frac{1}{2} + {{\mu }^{2}}\left( { - \frac{3}{{64}} + \frac{{{{n}^{2}}}}{{96}}} \right) + O({{\mu }^{4}})} \right){{e}^{{i3\psi }}} - \left( {\frac{1}{{24}}\mu - {{\mu }^{3}}\frac{1}{{1920}}\left( { - 63 + 20{{n}^{2}} - 30\ln \frac{8}{\mu }} \right) + O({{\mu }^{5}})} \right){{e}^{{i4\psi }}} \\ \, - \left( {{{\mu }^{2}}\frac{1}{{128}} + O({{\mu }^{4}})} \right){{e}^{{i5\psi }}} - \left( {{{\mu }^{3}}\frac{7}{{1280}} + O({{\mu }^{5}})} \right){{e}^{{i6\psi }}} \\ \, - \left( {\mu \frac{1}{{24}} - {{\mu }^{3}}\frac{1}{{1536}}\left( { - 75 + 52{{n}^{2}} - 24\ln \frac{8}{\mu }} \right) + O({{\mu }^{5}})} \right){{e}^{{i2\psi }}} \\ \, - \left( {{{\mu }^{2}}\frac{1}{{64}} + O({{\mu }^{4}})} \right){{e}^{{i\psi }}} + O({{\mu }^{4}}){{e}^{{i0\psi }}},\quad m = 3, \\ \end{gathered} $$

$$\begin{gathered} v_{{}}^{\sigma } = \left( {\frac{1}{2} + {{\mu }^{2}}\frac{1}{{240}}( - 8 + {{n}^{2}})} \right){{e}^{{i4\psi }}} - \mu \frac{1}{{32}}{{e}^{{i5\psi }}} - {{\mu }^{2}}\frac{1}{{160}}{{e}^{{i6\psi }}} - \mu \frac{1}{{32}}{{e}^{{i3\psi }}} - {{\mu }^{2}}\frac{1}{{96}}{{e}^{{i2\psi }}} + O({{\mu }^{3}}),\quad m = 4, \\ v_{{}}^{\sigma } = \frac{1}{2}{{e}^{{i5\psi }}} - \frac{1}{{40}}\mu {{e}^{{i4\psi }}} - \frac{1}{{40}}\mu {{e}^{{i6\psi }}} + O({{\mu }^{2}}),\quad m = 5. \\ \end{gathered} $$

Here, $v_{{}}^{\sigma }$ is the expression for the $\sigma$-component of the velocity field v on the boundary $\sigma = {{\sigma }_{{{\text{bound}}}}}$. The velocities $v_{{}}^{\sigma }$, m < 0 can be obtained by means of complex conjugation.