Two-Dimensional Surface Periodic Flows of an Incompressible Fluid in Various Models of the Medium

Abstract

A comparative analysis of the properties of two-dimensional infinitesimal periodic perturbations propagating over the incompressible fluid surface in various representations of the medium density profiles is carried out. Viscous or ideal liquids stratified and homogeneous in density are considered. Calculations are carried out by methods of the theory of singular perturbations. Dispersion relations and dependences of phase and group velocities for surface waves in physically observed variables are given. The change in the meaning of dispersion relations during the transition from ideal liquids to viscous and from homogeneous to stratified is shown. Taking into account the influence of electric charge does not qualitatively change the nature of two-dimensional dispersion relations. An increase in the surface density of the electric charge leads to a decrease in the wavelength at a fixed frequency and has no noticeable effect on the fine structure of the periodic flow.

Appendices

Appendix A

Equation (25) breaks down into three independent equations:

$${{k}_{{*x}}} = 0,$$

(A.1)

$$\frac{{1 - i}}{{\sqrt {2\varepsilon } }}\sqrt {{{\omega }_{*}}} - {{k}_{{*x}}}\frac{{\sqrt {\omega _{*}^{2} - 1} }}{{{{\omega }_{*}}}} = 0,$$

(A.2)

$$\begin{gathered} \frac{{1 - i}}{{\sqrt {2\varepsilon } }}\sqrt {{{\omega }_{*}}} {{k}_{{*x}}} + \left( {\frac{{\sqrt {\omega _{*}^{2} - 1} }}{{{{\omega }_{*}}}} - \frac{{1 - i}}{{\sqrt 2 }}W\delta \sqrt {{{\omega }_{*}}} } \right)k_{{*x}}^{2} + \left( {\frac{{1 - i}}{{\sqrt 2 }}{{\delta }^{2}}\sqrt {{{\omega }_{*}}} k_{{*x}}^{3} - \frac{{\sqrt {\omega _{*}^{2} - 1} }}{{{{\omega }_{*}}}}\delta Wk_{{*x}}^{3} - \frac{{1 - i}}{{\sqrt 2 }}\omega _{*}^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2}}}\sqrt {\omega _{*}^{2} - 1} } \right) \\ \times \,\,\sqrt \varepsilon + \left( {{{k}_{{*x}}}\left( {1 - \omega _{*}^{2}} \right) + \frac{{{{\delta }^{2}}\sqrt {\omega _{*}^{2} - 1} }}{{{{\omega }_{*}}}}k_{{*x}}^{4}} \right)\varepsilon = 0. \\ \end{gathered} $$

(A.3)

Solutions of equations (A.1)–(A.3) have the form

$${{k}_{{*x}}} = 0,$$

(A.4)

$${{k}_{{*x}}} = \frac{{\left( {1 - i} \right)\omega _{*}^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2}}}}}{{\sqrt {2\varepsilon \left( {\omega _{*}^{2} - 1} \right)} }},$$

(A.5)

$${{k}_{{*x}}} = - \frac{1}{2}\sqrt {{{\eta }_{W}}} - \frac{{{{\mu }_{W}}{{\omega }_{*}}}}{{4{{\delta }^{2}}\varepsilon \sqrt {\omega _{*}^{2} - 1} }} - \frac{1}{2}\sqrt { - {{\eta }_{W}} + {{\chi }_{W}} - \frac{{{{\varpi }_{W}}}}{{4\sqrt {{{\eta }_{W}}} }}} ,$$

(A.6)

$${{k}_{{*x}}} = - \frac{1}{2}\sqrt {{{\eta }_{W}}} - \frac{{{{\mu }_{W}}{{\omega }_{*}}}}{{4{{\delta }^{2}}\varepsilon \sqrt {\omega _{*}^{2} - 1} }} + \frac{1}{2}\sqrt { - {{\eta }_{W}} + {{\chi }_{W}} - \frac{{{{\varpi }_{W}}}}{{4\sqrt {{{\eta }_{W}}} }}} ,$$

(A.7)

$${{k}_{{*x}}} = \frac{1}{2}\sqrt {{{\eta }_{W}}} - \frac{{{{\mu }_{W}}{{\omega }_{*}}}}{{4{{\delta }^{2}}\varepsilon \sqrt {\omega _{*}^{2} - 1} }} - \frac{1}{2}\sqrt { - {{\eta }_{W}} + {{\chi }_{W}} + \frac{{{{\varpi }_{W}}}}{{4\sqrt {{{\eta }_{W}}} }}} ,$$

(A.8)

$${{k}_{{*x}}} = \frac{1}{2}\sqrt {{{\eta }_{W}}} - \frac{{{{\mu }_{W}}{{\omega }_{*}}}}{{4{{\delta }^{2}}\varepsilon \sqrt {\omega _{*}^{2} - 1} }} + \frac{1}{2}\sqrt { - {{\eta }_{W}} + {{\chi }_{W}} + \frac{{{{\varpi }_{W}}}}{{4\sqrt {{{\eta }_{W}}} }}} ,$$

(A.9)

$$\begin{matrix} {{\eta }_{W}}=\frac{1}{12{{\left( {{\alpha }_{W}}+\sqrt{\alpha _{W}^{2}+4\beta _{W}^{3}} \right)}^{{1}/{3}\;}}{{\delta }^{4}}{{\varepsilon }^{2}}{{\left( \omega _{*}^{2}-1 \right)}^{{3}/{2}\;}}}\left[ {{W}^{2}}{{\delta }^{2}}\sqrt{\omega _{*}^{2}-1}\left( 3\varepsilon {{\left( {{\alpha }_{W}}+\sqrt{\alpha _{W}^{2}+4\beta _{W}^{3}} \right)}^{{1}/{3}\;}}\left( \omega _{*}^{2}-1 \right)-{{2}^{{1}/{3}\;}}\times 4i\omega _{*}^{3} \right) \right. \\ -\,\,\left( 1-i \right){{2}^{{1}/{3}\;}}W\delta {{\omega }_{*}}\left( \omega _{*}^{2}-1 \right)\left( -2\sqrt{2}\sqrt{{{\omega }_{*}}}-{{2}^{{1}/{6}\;}}{{\left( {{\alpha }_{W}}+\sqrt{\alpha _{W}^{2}+4\beta _{W}^{3}} \right)}^{{1}/{3}\;}}{{\delta }^{2}}\varepsilon \sqrt{{{\omega }_{*}}}+6{{\varepsilon }^{{3}/{2}\;}}\left( 1+i \right)\left( \omega _{*}^{2}-1 \right) \right) \\ +\,\,\sqrt{\omega _{*}^{2}-1}\left( {{2}^{{1}/{3}\;}}\times 4\left( \omega _{*}^{2}-1 \right)+{{\left( {{\alpha }_{W}}+\sqrt{\alpha _{W}^{2}+4\beta _{W}^{3}} \right)}^{{1}/{3}\;}}{{\delta }^{4}}\varepsilon \left( -3i+{{2}^{{2}/{3}\;}}\times 2\varepsilon {{\left( {{\alpha }_{W}}+\sqrt{\alpha _{W}^{2}+4\beta _{W}^{3}} \right)}^{{1}/{3}\;}}\left( \omega _{*}^{2}-1 \right) \right) \right. \\ \left. \left. +\,\,{{\delta }^{2}}\left( {{2}^{{1}/{3}\;}}\times 12i\omega _{*}^{3}-8\varepsilon {{\left( {{\alpha }_{W}}+\sqrt{\alpha _{W}^{2}+4\beta _{W}^{3}} \right)}^{{1}/{3}\;}}\left( \omega _{*}^{2}-1 \right)-18\left( 1-i \right){{2}^{{5}/{6}\;}}{{\varepsilon }^{{3}/{2}\;}}{{\omega }^{{5}/{2}\;}}\left( \omega _{*}^{2}-1 \right) \right) \right) \right], \\ {{\mu }_{W}}=\frac{\left( 1-i \right)}{\sqrt{2}}{{\delta }^{2}}\sqrt{\varepsilon }{{\omega }_{*}}-\frac{W\delta \sqrt{\varepsilon }\sqrt{\omega _{*}^{2}-1}}{{{\omega }_{*}}}, \\ \end{matrix}$$

$${{\chi }_{W}} = \frac{{3\mu _{W}^{2}\omega _{*}^{2}}}{{4{{\delta }^{4}}{{\varepsilon }^{2}}\left( {\omega _{*}^{2} - 1} \right)}} - \frac{{2{{\omega }_{*}}}}{{{{\delta }^{2}}\varepsilon \sqrt {\omega _{*}^{2} - 1} }}\left( {\frac{{\sqrt {\omega _{*}^{2} - 1} }}{{{{\omega }_{*}}}} - \frac{{\left( {1 - i} \right)W\delta \sqrt {{{\omega }_{*}}} }}{{\sqrt 2 }}} \right),$$

$${{\varpi }_{W}} = - \frac{{\mu _{W}^{3}\omega _{*}^{3}}}{{{{\delta }^{6}}{{\varepsilon }^{3}}{{{\left( {\omega _{*}^{2} - 1} \right)}}^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2}}}}}} - \frac{{2\sqrt 2 \left( {1 - i} \right)W{{\mu }_{W}}\omega _{*}^{{{5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-0em} 2}}}}}{{{{\delta }^{3}}{{\varepsilon }^{2}}\left( {\omega _{*}^{2} - 1} \right)}} + \frac{{4{{\mu }_{W}}{{\omega }_{*}}}}{{{{\delta }^{4}}{{\varepsilon }^{2}}\sqrt {\omega _{*}^{2} - 1} }} - \frac{{8{{\omega }_{*}}}}{{{{\delta }^{2}}\varepsilon \sqrt {\omega _{*}^{2} - 1} }}\left( {\varepsilon \left( {1 - \omega _{*}^{2}} \right) + \frac{{\left( {1 - i} \right)\sqrt {{{\omega }_{*}}} }}{{\sqrt 2 \sqrt \varepsilon }}} \right),$$

$${{\beta }_{W}} = - \frac{1}{{{{\delta }^{4}}{{\varepsilon }^{2}}}} + \frac{{9\left( {1 - i} \right)\omega _{*}^{{{5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-0em} 2}}}\left( {\omega _{*}^{2} - 1} \right)}}{{\sqrt 2 {{\delta }^{2}}\sqrt \varepsilon \left( {\omega _{*}^{2} - 1} \right)}} + \frac{{i{{W}^{2}}\omega _{*}^{3} - 3i\omega _{*}^{3}}}{{{{\delta }^{2}}{{\varepsilon }^{2}}\left( {\omega _{*}^{2} - 1} \right)}} + \frac{{W\left( {\sqrt 2 \left( {i - 1} \right)\omega _{*}^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2}}} + 6{{\varepsilon }^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2}}}}{{\omega }_{*}}\left( {\omega _{*}^{2} - 1} \right)} \right)}}{{2{{\delta }^{3}}{{\varepsilon }^{2}}\sqrt {\omega _{*}^{2} - 1} }},$$

$$\begin{gathered} {{\alpha }_{W}} = \frac{2}{{{{\delta }^{6}}{{\varepsilon }^{3}}}} + \frac{{\left( {1 + i} \right)\sqrt 2 W\left( { - 9 + 2{{W}^{2}}} \right)\omega _{*}^{{{9 \mathord{\left/ {\vphantom {9 2}} \right. \kern-0em} 2}}} + 54iW{{\varepsilon }^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2}}}}\omega _{*}^{4}\left( {\omega _{*}^{2} - 1} \right)}}{{2{{\delta }^{3}}{{\varepsilon }^{3}}{{{\left( {\omega _{*}^{2} - 1} \right)}}^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2}}}}}} \\ + \,\,\frac{{3\left( {1 + i} \right)\omega _{*}^{2}}}{{2{{\delta }^{4}}}}\left( {\frac{{3i\sqrt 2 \left( { - 3 + 2{{W}^{2}}} \right)\sqrt {{{\omega }_{*}}} }}{{{{\varepsilon }^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2}}}}}} + \frac{{\left( {1 + i} \right)\left( { - 6 + {{W}^{2}}} \right){{\omega }_{*}}}}{{{{\varepsilon }^{3}}\left( {\omega _{*}^{2} - 1} \right)}} + 9\left( {1 - i} \right)\left( {\omega _{*}^{2} - 1} \right)} \right) \\ + \,\,\frac{{27\left( {1 + i} \right){{\omega }^{{{{11} \mathord{\left/ {\vphantom {{11} 2}} \right. \kern-0em} 2}}}}}}{{\sqrt 2 {{\delta }^{2}}{{\varepsilon }^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2}}}}\left( {\omega _{*}^{2} - 1} \right)}} + \frac{{W\left( {3\left( {1 - i} \right)\sqrt 2 \omega _{*}^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2}}} - 18{{\varepsilon }^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2}}}}{{\omega }_{*}}\left( {\omega _{*}^{2} - 1} \right)} \right)}}{{2{{\delta }^{5}}{{\varepsilon }^{3}}\sqrt {\omega _{*}^{2} - 1} }}. \\ \end{gathered} $$

Appendix B

Equation (36) breaks down into three independent equations:

$${{k}_{{*x}}} = 0,$$

(B.1)

$$\frac{{1 - i}}{{\sqrt {2\varepsilon } }}\sqrt {{{\omega }_{*}}} - {{k}_{{*x}}}\frac{{\sqrt {\omega _{*}^{2} - 1} }}{{{{\omega }_{*}}}} = 0,$$

(B.2)

$$\begin{gathered} \frac{{1 - i}}{{\sqrt {2\varepsilon } }}\sqrt {{{\omega }_{*}}} {{k}_{{*x}}} + \frac{{\sqrt {\omega _{*}^{2} - 1} }}{{{{\omega }_{*}}}}k_{{*x}}^{2} + \left( {\frac{{1 - i}}{{\sqrt 2 }}{{\delta }^{2}}\sqrt {{{\omega }_{*}}} k_{{*x}}^{3} - \frac{{1 - i}}{{\sqrt 2 }}\omega _{*}^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2}}}\sqrt {\omega _{*}^{2} - 1} } \right)\sqrt \varepsilon \\ + \,\,\left( {{{k}_{{*x}}}\left( {1 - \omega _{*}^{2}} \right) + \frac{{{{\delta }^{2}}\sqrt {\omega _{*}^{2} - 1} }}{{{{\omega }_{*}}}}k_{{*x}}^{4}} \right)\varepsilon = 0. \\ \end{gathered} $$

(B.3)

The solutions to Eqs. (B.1)–(B.3) will be written as follows:

$${{k}_{{*x}}} = 0,$$

(B.4)

$${{k}_{{*x}}} = \frac{{\left( {1 - i} \right)\omega _{*}^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2}}}}}{{\sqrt {2\varepsilon \left( {\omega _{*}^{2} - 1} \right)} }},$$

(B.5)

$${{k}_{{*x}}} = - \frac{1}{2}\sqrt \eta - \frac{{\mu {{\omega }_{*}}}}{{4{{\delta }^{2}}\varepsilon \sqrt {\omega _{*}^{2} - 1} }} - \frac{1}{2}\sqrt { - \eta + \chi - \frac{\varpi }{{4\sqrt \eta }}} ,$$

(B.6)

$${{k}_{{*x}}} = - \frac{1}{2}\sqrt \eta - \frac{{\mu {{\omega }_{*}}}}{{4{{\delta }^{2}}\varepsilon \sqrt {\omega _{*}^{2} - 1} }} + \frac{1}{2}\sqrt { - \eta + \chi - \frac{\varpi }{{4\sqrt \eta }}} ,$$

(B.7)

$${{k}_{{*x}}} = \frac{1}{2}\sqrt \eta - \frac{{\mu {{\omega }_{*}}}}{{4{{\delta }^{2}}\varepsilon \sqrt {\omega _{*}^{2} - 1} }} - \frac{1}{2}\sqrt { - \eta + \chi + \frac{\varpi }{{4\sqrt \eta }}} ,$$

(B.8)

$${{k}_{{*x}}} = \frac{1}{2}\sqrt \eta - \frac{{\mu {{\omega }_{*}}}}{{4{{\delta }^{2}}\varepsilon \sqrt {\omega _{*}^{2} - 1} }} + \frac{1}{2}\sqrt { - \eta + \chi + \frac{\varpi }{{4\sqrt \eta }}} ,$$

(B.9)

$$\begin{gathered} \eta = \frac{1}{{12{{{\left( {\alpha + \sqrt {{{\alpha }^{2}} + 4{{\beta }^{3}}} } \right)}}^{{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0em} 3}}}}{{\delta }^{4}}{{\varepsilon }^{2}}{{{\left( {\omega _{*}^{2} - 1} \right)}}^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2}}}}}}\left[ {{{{\sqrt {\omega _{*}^{2} - 1} }}^{{}}}} \right. \\ \times \,\,\left( {{{2}^{{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0em} 3}}}} \times 4\left( {\omega _{*}^{2} - 1} \right) + {{{\left( {\alpha + \sqrt {{{\alpha }^{2}} + 4{{\beta }^{3}}} } \right)}}^{{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0em} 3}}}}{{\delta }^{4}}\varepsilon \left( { - 3i + {{2}^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0em} 3}}}} \times 2\varepsilon {{{\left( {\alpha + \sqrt {{{\alpha }^{2}} + 4{{\beta }^{3}}} } \right)}}^{{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0em} 3}}}}\left( {\omega _{*}^{2} - 1} \right)} \right)} \right. \\ \left. {\left. { + \,\,{{\delta }^{2}}\left( {{{2}^{{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0em} 3}}}} \times 12i\omega _{*}^{3} - 8\varepsilon {{{\left( {\alpha + \sqrt {{{\alpha }^{2}} + 4{{\beta }^{3}}} } \right)}}^{{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0em} 3}}}}\left( {\omega _{*}^{2} - 1} \right) - 18\left( {1 - i} \right){{2}^{{{5 \mathord{\left/ {\vphantom {5 6}} \right. \kern-0em} 6}}}}{{\varepsilon }^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2}}}}{{\omega }^{{{5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-0em} 2}}}}\left( {\omega _{*}^{2} - 1} \right)} \right)} \right)} \right], \\ \mu = \frac{{\left( {1 - i} \right)}}{{\sqrt 2 }}{{\delta }^{2}}\sqrt \varepsilon {{\omega }_{*}}, \\ \end{gathered} $$

$$\begin{gathered} \chi = \frac{{3{{\mu }^{2}}\omega _{*}^{2}}}{{4{{\delta }^{4}}{{\varepsilon }^{2}}\left( {\omega _{*}^{2} - 1} \right)}} - \frac{2}{{{{\delta }^{2}}\varepsilon }}, \\ \varpi = - \frac{{{{\mu }^{3}}\omega _{*}^{3}}}{{{{\delta }^{6}}{{\varepsilon }^{3}}{{{\left( {\omega _{*}^{2} - 1} \right)}}^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2}}}}}} + \frac{{4\mu {{\omega }_{*}}}}{{{{\delta }^{4}}{{\varepsilon }^{2}}\sqrt {\omega _{*}^{2} - 1} }} - \frac{{8{{\omega }_{*}}}}{{{{\delta }^{2}}\varepsilon \sqrt {\omega _{*}^{2} - 1} }}\left( {\varepsilon \left( {1 - \omega _{*}^{2}} \right) + \frac{{\left( {1 - i} \right)\sqrt {{{\omega }_{*}}} }}{{\sqrt 2 \sqrt \varepsilon }}} \right), \\ \end{gathered} $$

$$\beta = - \frac{1}{{{{\delta }^{4}}{{\varepsilon }^{2}}}} + \frac{{9\left( {1 - i} \right)\omega _{*}^{{{5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-0em} 2}}}}}{{\sqrt 2 {{\delta }^{2}}\sqrt \varepsilon }} - \frac{{3i\omega _{*}^{3}}}{{{{\delta }^{2}}{{\varepsilon }^{2}}\left( {\omega _{*}^{2} - 1} \right)}},$$

$$\alpha = \frac{2}{{{{\delta }^{6}}{{\varepsilon }^{3}}}} + \frac{{3\left( {1 + i} \right)\omega _{*}^{2}}}{{2{{\delta }^{4}}}}\left( { + 9\left( {1 - i} \right)\left( {\omega _{*}^{2} - 1} \right)} \right) + \frac{{27\left( {1 + i} \right){{\omega }^{{{{11} \mathord{\left/ {\vphantom {{11} 2}} \right. \kern-0em} 2}}}}}}{{\sqrt 2 {{\delta }^{2}}{{\varepsilon }^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2}}}}\left( {\omega _{*}^{2} - 1} \right)}}.$$