http://alt-sci.ru/_en/wiki/index.php?title=Vortex&feed=atom&action=historyVortex - Revision history2024-03-28T21:50:01ZRevision history for this page on the wikiMediaWiki 1.25.1http://alt-sci.ru/_en/wiki/index.php?title=Vortex&diff=707&oldid=prevAdmin at 08:15, 29 August 20162016-08-29T08:15:18Z<p></p>
<table class='diff diff-contentalign-left'>
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<td colspan='2' style="background-color: white; color:black; text-align: center;">← Older revision</td>
<td colspan='2' style="background-color: white; color:black; text-align: center;">Revision as of 08:15, 29 August 2016</td>
</tr><tr><td colspan="2" class="diff-lineno" id="L58" >Line 58:</td>
<td colspan="2" class="diff-lineno">Line 58:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Thus, the velocity curl flux through any tube section is constant over its length, and according to the [[wikipedia:Kelvin–Stokes theorem|Stokes theorem]], it’s equal to a circulation along the section contour:</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Thus, the velocity curl flux through any tube section is constant over its length, and according to the [[wikipedia:Kelvin–Stokes theorem|Stokes theorem]], it’s equal to a circulation along the section contour:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>\int{curl\,\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{S}}}=\oint{\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{l}}}=const\label{Helmholtz2}</math></div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>\int{curl\,\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{S}}}=\oint{\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{l}}}=const\label{Helmholtz2}</math></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The equation \ref{Helmholtz2} is known as the [[wikipedia:Helmholtz's theorems|<del class="diffchange diffchange-inline">3nd </del>Helmholtz's theorem]], which <del class="diffchange diffchange-inline">implies</del>:</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The equation \ref{Helmholtz2} is known as the [[wikipedia:Helmholtz's theorems|<ins class="diffchange diffchange-inline">3rd </ins>Helmholtz's theorem]], <ins class="diffchange diffchange-inline">from </ins>which <ins class="diffchange diffchange-inline">follow</ins>:</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* The flow velocity <del class="diffchange diffchange-inline">over the </del>tube <del class="diffchange diffchange-inline">surface </del>is inversely proportional to the length (radius) of the tube section contour.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* The flow velocity <ins class="diffchange diffchange-inline">within a </ins>tube is inversely proportional to the length (radius) of the tube section contour.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* The [[Helix and spiral|helical]] flow is possible along the tube of any shape without <del class="diffchange diffchange-inline">breaking </del>the <del class="diffchange diffchange-inline">stream continuity</del>.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* The [[Helix and spiral|helical]] flow is possible along the tube of any shape without the <ins class="diffchange diffchange-inline">flow discontinuity</ins>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* The tube is continuous between the continuum interfaces. A stable vortex is either located between the interfaces or closed into the ring.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* The tube is continuous between the continuum interfaces. A stable vortex is either located between the interfaces or closed into the ring.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
</table>Adminhttp://alt-sci.ru/_en/wiki/index.php?title=Vortex&diff=705&oldid=prevAdmin at 07:15, 29 August 20162016-08-29T07:15:39Z<p></p>
<table class='diff diff-contentalign-left'>
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
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<td colspan='2' style="background-color: white; color:black; text-align: center;">← Older revision</td>
<td colspan='2' style="background-color: white; color:black; text-align: center;">Revision as of 07:15, 29 August 2016</td>
</tr><tr><td colspan="2" class="diff-lineno" id="L22" >Line 22:</td>
<td colspan="2" class="diff-lineno">Line 22:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>\frac{\partial}{\partial t}curl\,\mathbf{\overrightarrow{v}}-curl(\mathbf{\overrightarrow{v}}\times curl\mathbf{\overrightarrow{v}})=curl\,\mathbf{\overrightarrow{f}}\label{Helmholtz1}</math></div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>\frac{\partial}{\partial t}curl\,\mathbf{\overrightarrow{v}}-curl(\mathbf{\overrightarrow{v}}\times curl\mathbf{\overrightarrow{v}})=curl\,\mathbf{\overrightarrow{f}}\label{Helmholtz1}</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">Thus, the </del>vortical force field (<math>curl\,\mathbf{\overrightarrow{f}}\neq 0</math>) causes an accelerated circular motion, so <del class="diffchange diffchange-inline">in the steady-state equation \ref{Helmholtz1}, </del><math>curl\,\mathbf{\overrightarrow{f}}=0</math>:</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">The </ins>vortical force field (<math>curl\,\mathbf{\overrightarrow{f}}\neq 0</math>) <ins class="diffchange diffchange-inline">obviously </ins>causes an accelerated circular motion, so <math>curl\,\mathbf{\overrightarrow{f}}=0</math> <ins class="diffchange diffchange-inline">in the steady-state equation \ref{Helmholtz1}</ins>:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>curl(\mathbf{\overrightarrow{v}}\times curl\,\mathbf{\overrightarrow{v}})=0\label{Helmholtz1_steady}</math></div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>curl(\mathbf{\overrightarrow{v}}\times curl\,\mathbf{\overrightarrow{v}})=0\label{Helmholtz1_steady}</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td colspan="2" class="diff-lineno" id="L54" >Line 54:</td>
<td colspan="2" class="diff-lineno">Line 54:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The velocity curl flux through a tube surface is:</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The velocity curl flux through a tube surface is:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>\oint{curl\,\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{S}}}=\int_{S_1}{curl\,\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{S}}}-\int_{S_2}{curl\,\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{S}}}\label{Helmholtz2_1}</math></div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>\oint{curl\,\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{S}}}=\int_{S_1}{curl\,\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{S}}}-\int_{S_2}{curl\,\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{S}}}\label{Helmholtz2_1}</math></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>where <math>S_1</math> and <math>S_2</math> are the cross sections of the tube. <del class="diffchange diffchange-inline">At </del>the <del class="diffchange diffchange-inline">same time</del>, according to the [[wikipedia:Divergence theorem|divergence theorem]] and the identity <math>div\;curl\,\mathbf{\overrightarrow{v}}=0</math>:</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>where <math>S_1</math> and <math>S_2</math> are the cross sections of the tube. <ins class="diffchange diffchange-inline">On </ins>the <ins class="diffchange diffchange-inline">other hand</ins>, according to the [[wikipedia:Divergence theorem|divergence theorem]] and the identity <math>div\;curl\,\mathbf{\overrightarrow{v}}=0</math>:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>\oint{curl\,\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{S}}}=\oint{div\;curl\,\mathbf{\overrightarrow{v}}\mathrm{d}V}=0\label{Helmholtz2_2}</math></div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>\oint{curl\,\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{S}}}=\oint{div\;curl\,\mathbf{\overrightarrow{v}}\mathrm{d}V}=0\label{Helmholtz2_2}</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Thus, the velocity curl flux through any tube section is constant over its length, and according to the [[wikipedia:Kelvin–Stokes theorem|Stokes theorem]], it’s equal to a circulation along the section contour:</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Thus, the velocity curl flux through any tube section is constant over its length, and according to the [[wikipedia:Kelvin–Stokes theorem|Stokes theorem]], it’s equal to a circulation along the section contour:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>\int{curl\,\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{S}}}=\oint{\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{l}}}=const\label{Helmholtz2}</math></div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>\int{curl\,\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{S}}}=\oint{\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{l}}}=const\label{Helmholtz2}</math></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The equation \ref{Helmholtz2} is known as the <del class="diffchange diffchange-inline">2nd </del>Helmholtz theorem, which implies:</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The equation \ref{Helmholtz2} is known as the <ins class="diffchange diffchange-inline">[[wikipedia:</ins>Helmholtz<ins class="diffchange diffchange-inline">'s theorems|3nd Helmholtz's </ins>theorem<ins class="diffchange diffchange-inline">]]</ins>, which implies:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* The flow velocity over the tube surface is inversely proportional to the length (radius) of the tube section contour.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* The flow velocity over the tube surface is inversely proportional to the length (radius) of the tube section contour.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* The [[Helix and spiral|helical]] flow is possible along the tube of any shape without breaking the stream continuity.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* The [[Helix and spiral|helical]] flow is possible along the tube of any shape without breaking the stream continuity.</div></td></tr>
</table>Adminhttp://alt-sci.ru/_en/wiki/index.php?title=Vortex&diff=704&oldid=prevAdmin at 05:01, 25 August 20162016-08-25T05:01:47Z<p></p>
<a href="http://alt-sci.ru/_en/wiki/index.php?title=Vortex&diff=704&oldid=502">Show changes</a>Adminhttp://alt-sci.ru/_en/wiki/index.php?title=Vortex&diff=502&oldid=prevAdmin at 10:43, 18 October 20152015-10-18T10:43:59Z<p></p>
<table class='diff diff-contentalign-left'>
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
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<tr style='vertical-align: top;'>
<td colspan='2' style="background-color: white; color:black; text-align: center;">← Older revision</td>
<td colspan='2' style="background-color: white; color:black; text-align: center;">Revision as of 10:43, 18 October 2015</td>
</tr><tr><td colspan="2" class="diff-lineno" id="L13" >Line 13:</td>
<td colspan="2" class="diff-lineno">Line 13:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>\rho\frac{\partial\mathbf{\overrightarrow{v}}}{\partial t}+\frac{\rho}{2}grad\;v^2-\rho\mathbf{\overrightarrow{v}}\times curl\mathbf{\overrightarrow{v}}+grad(P+U)=\rho\mathbf{\overrightarrow{f}}\label{gromeka_lamb1}</math></div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>\rho\frac{\partial\mathbf{\overrightarrow{v}}}{\partial t}+\frac{\rho}{2}grad\;v^2-\rho\mathbf{\overrightarrow{v}}\times curl\mathbf{\overrightarrow{v}}+grad(P+U)=\rho\mathbf{\overrightarrow{f}}\label{gromeka_lamb1}</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>\frac{\partial\mathbf{\overrightarrow{v}}}{\partial t}+\frac{1}{2}grad\;v^2-\mathbf{\overrightarrow{v}}\times curl\mathbf{\overrightarrow{v}}+\frac{1}{\rho}grad(P+U)=\mathbf{\overrightarrow{f}}\label{gromeka_lamb2}</math></div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>\frac{\partial\mathbf{\overrightarrow{v}}}{\partial t}+\frac{1}{2}grad\;v^2-\mathbf{\overrightarrow{v}}\times curl\mathbf{\overrightarrow{v}}+\frac{1}{\rho}grad(P+U)=\mathbf{\overrightarrow{f}}\label{gromeka_lamb2}</math></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">The strict theory of vortices is applicable either for </del>the <del class="diffchange diffchange-inline">ideal incompressible flows (</del><math>\<del class="diffchange diffchange-inline">rho</del>=<del class="diffchange diffchange-inline">const</del></math><del class="diffchange diffchange-inline">)</del>, <del class="diffchange diffchange-inline">or for the ideal compressible [[Continuum#Barotropic flow|quadratic barotropic flows]]. For both cases, the Gromeka-Lamb equation can be written (regardless the gradient factor) as</del>:</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">Applying </ins>the <ins class="diffchange diffchange-inline">curl operator to \ref{gromeka_lamb2}, and using the identity </ins><math><ins class="diffchange diffchange-inline">curl</ins>\<ins class="diffchange diffchange-inline">;grad\;X</ins>=<ins class="diffchange diffchange-inline">0</ins></math>, <ins class="diffchange diffchange-inline">produces</ins>:  </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:<math>\frac{\partial<del class="diffchange diffchange-inline">\mathbf{\overrightarrow{v}}</del>}{\partial t}<del class="diffchange diffchange-inline">+</del>\<del class="diffchange diffchange-inline">frac</del>{<del class="diffchange diffchange-inline">1}</del>{<del class="diffchange diffchange-inline">2}grad\;</del>v<del class="diffchange diffchange-inline">^2</del>-\mathbf{\overrightarrow{v}}\times curl\mathbf{\overrightarrow{v}}+grad\frac{<del class="diffchange diffchange-inline">P+U</del>}{\rho}=\mathbf{\overrightarrow{f}}\label{<del class="diffchange diffchange-inline">gromeka_lamb3</del>}</math></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:<math>\frac{\partial}{\partial t}<ins class="diffchange diffchange-inline">curl</ins>\<ins class="diffchange diffchange-inline">mathbf</ins>{<ins class="diffchange diffchange-inline">\overrightarrow</ins>{v<ins class="diffchange diffchange-inline">}}</ins>-<ins class="diffchange diffchange-inline">curl(</ins>\mathbf{\overrightarrow{v}}\times curl\mathbf{\overrightarrow{v}}<ins class="diffchange diffchange-inline">)</ins>+grad\frac{<ins class="diffchange diffchange-inline">1</ins>}{\rho}<ins class="diffchange diffchange-inline">\times grad(P+U)</ins>=<ins class="diffchange diffchange-inline">curl</ins>\mathbf{\overrightarrow{f}}\label{<ins class="diffchange diffchange-inline">pre_Helmholtz1</ins>}</math></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">Applying the curl operator </del>to <del class="diffchange diffchange-inline">\ref{gromeka_lamb3}, and using </del>the <del class="diffchange diffchange-inline">identity </del><math><del class="diffchange diffchange-inline">curl</del>\<del class="diffchange diffchange-inline">;grad\;X</del>=<del class="diffchange diffchange-inline">0</del></math>, <del class="diffchange diffchange-inline">produces </del>the Helmholtz equation, which is independent of density and pressure:</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">The strict theory of vortices is applicable either </ins>to the <ins class="diffchange diffchange-inline">incompressible flows (</ins><math>\<ins class="diffchange diffchange-inline">rho</ins>=<ins class="diffchange diffchange-inline">const</ins></math><ins class="diffchange diffchange-inline">) or to the compressible [[Continuum#Barotropic flow|barotropic flows]]. In both cases</ins>, <ins class="diffchange diffchange-inline">the equation \ref{pre_Helmholtz1} is converted into </ins>the Helmholtz equation, which is independent of density and pressure:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>\frac{\partial}{\partial t}curl\mathbf{\overrightarrow{v}}-curl(\mathbf{\overrightarrow{v}}\times curl\mathbf{\overrightarrow{v}})=curl\mathbf{\overrightarrow{f}}\label{Helmholtz1}</math></div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>\frac{\partial}{\partial t}curl\mathbf{\overrightarrow{v}}-curl(\mathbf{\overrightarrow{v}}\times curl\mathbf{\overrightarrow{v}})=curl\mathbf{\overrightarrow{f}}\label{Helmholtz1}</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Thus, the vortical force field (<math>curl\mathbf{\overrightarrow{f}}\neq 0</math>) causes a circular motion of the continuum particles. Certainly, it causes an accelerated motion, so in the steady-state Helmholtz equation <math>curl\mathbf{\overrightarrow{f}}=0</math>:</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Thus, the vortical force field (<math>curl\mathbf{\overrightarrow{f}}\neq 0</math>) causes a circular motion of the continuum particles. Certainly, it causes an accelerated motion, so in the steady-state Helmholtz equation <math>curl\mathbf{\overrightarrow{f}}=0</math>:</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="L21" >Line 21:</td>
<td colspan="2" class="diff-lineno">Line 21:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The steady-state continuity equation ([[Continuum#Eq-5|"Continuum", 5]]) is:</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The steady-state continuity equation ([[Continuum#Eq-5|"Continuum", 5]]) is:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>div\;\rho\mathbf{\overrightarrow{v}}=\mathbf{\overrightarrow{v}}grad\;\rho+\rho\;div\mathbf{\overrightarrow{v}}=0\label{continuity_steady2}</math></div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>div\;\rho\mathbf{\overrightarrow{v}}=\mathbf{\overrightarrow{v}}grad\;\rho+\rho\;div\mathbf{\overrightarrow{v}}=0\label{continuity_steady2}</math></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>For an incompressible flow (<math>grad\;\rho=0</math>), the velocity is a vortical field (<math>div\mathbf{\overrightarrow{v}}=0</math>). For a compressible flow, the velocity is a vortex field too, if the <del class="diffchange diffchange-inline">force of a pressure and a potential energy are perpendicular to the velocity vector, producing only a normal acceleration in a steady-state vortex</del>:</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>For an incompressible flow (<math>grad\;\rho=0</math>), the velocity is a vortical field (<math>div\mathbf{\overrightarrow{v}}=0</math>). For a compressible flow, the velocity is a vortex field too, if <ins class="diffchange diffchange-inline">its density is constant along </ins>the <ins class="diffchange diffchange-inline">flow</ins>:</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:<math>\mathbf{\overrightarrow{v}}grad<del class="diffchange diffchange-inline">(P+U)</del>=0\label{<del class="diffchange diffchange-inline">barotropic_steady_condition</del>}</math></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:<math>\mathbf{\overrightarrow{v}}grad<ins class="diffchange diffchange-inline">\;\rho</ins>=0\label{<ins class="diffchange diffchange-inline">continuity_steady3</ins>}</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>As a result, a stable (steady-state) vortex is generally described by the system of equations:</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>As a result, a stable (steady-state) vortex is generally described by the system of equations:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* Kinematic Helmholtz equation \ref{Helmholtz1_steady}.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* Kinematic Helmholtz equation \ref{Helmholtz1_steady}.</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="L30" >Line 30:</td>
<td colspan="2" class="diff-lineno">Line 30:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The conservative and pressure forces are shaping a vortex, and cause it to move over a continuum. This follows from the [[wikipedia:Kelvin's circulation theorem|Kelvin’s circulation theorem]], whereby any conservative force does not change a vortex circulation, and consequently the energy of this force can change a velocity of the entire vortex:</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The conservative and pressure forces are shaping a vortex, and cause it to move over a continuum. This follows from the [[wikipedia:Kelvin's circulation theorem|Kelvin’s circulation theorem]], whereby any conservative force does not change a vortex circulation, and consequently the energy of this force can change a velocity of the entire vortex:</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:<math>\frac{\mathrm{d}}{\mathrm{d}t}\oint{\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{l}}}=\oint{\frac{\mathrm{d}\mathbf{\overrightarrow{v}}}{\mathrm{d}t}\mathrm{d}\mathbf{\overrightarrow{l}}}=-\oint{<del class="diffchange diffchange-inline">grad</del>\frac{<del class="diffchange diffchange-inline">P+U</del>}{\rho}\mathrm{d}\mathbf{\overrightarrow{l}}}=0\label{Kelvin_basic}</math></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:<math>\frac{\mathrm{d}}{\mathrm{d}t}\oint{\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{l}}}=\oint{\frac{\mathrm{d}\mathbf{\overrightarrow{v}}}{\mathrm{d}t}\mathrm{d}\mathbf{\overrightarrow{l}}}=-\oint{\frac{<ins class="diffchange diffchange-inline">1</ins>}{\rho}<ins class="diffchange diffchange-inline">grad(P+U)</ins>\mathrm{d}\mathbf{\overrightarrow{l}}}=0\label{Kelvin_basic}</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The proof of \ref{Kelvin_basic} uses the kinematic Kelvin's theorem:</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The proof of \ref{Kelvin_basic} uses the kinematic Kelvin's theorem:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>\frac{\mathrm{d}}{\mathrm{d}t}\oint{\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{l}}}=\oint{\frac{\mathrm{d}\mathbf{\overrightarrow{v}}}{\mathrm{d}t}\mathrm{d}\mathbf{\overrightarrow{l}}}\label{Kelvin_kinematic}</math></div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>\frac{\mathrm{d}}{\mathrm{d}t}\oint{\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{l}}}=\oint{\frac{\mathrm{d}\mathbf{\overrightarrow{v}}}{\mathrm{d}t}\mathrm{d}\mathbf{\overrightarrow{l}}}\label{Kelvin_kinematic}</math></div></td></tr>
</table>Adminhttp://alt-sci.ru/_en/wiki/index.php?title=Vortex&diff=497&oldid=prevAdmin at 10:05, 8 October 20152015-10-08T10:05:19Z<p></p>
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<td colspan='2' style="background-color: white; color:black; text-align: center;">← Older revision</td>
<td colspan='2' style="background-color: white; color:black; text-align: center;">Revision as of 10:05, 8 October 2015</td>
</tr><tr><td colspan="2" class="diff-lineno" id="L13" >Line 13:</td>
<td colspan="2" class="diff-lineno">Line 13:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>\rho\frac{\partial\mathbf{\overrightarrow{v}}}{\partial t}+\frac{\rho}{2}grad\;v^2-\rho\mathbf{\overrightarrow{v}}\times curl\mathbf{\overrightarrow{v}}+grad(P+U)=\rho\mathbf{\overrightarrow{f}}\label{gromeka_lamb1}</math></div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>\rho\frac{\partial\mathbf{\overrightarrow{v}}}{\partial t}+\frac{\rho}{2}grad\;v^2-\rho\mathbf{\overrightarrow{v}}\times curl\mathbf{\overrightarrow{v}}+grad(P+U)=\rho\mathbf{\overrightarrow{f}}\label{gromeka_lamb1}</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>\frac{\partial\mathbf{\overrightarrow{v}}}{\partial t}+\frac{1}{2}grad\;v^2-\mathbf{\overrightarrow{v}}\times curl\mathbf{\overrightarrow{v}}+\frac{1}{\rho}grad(P+U)=\mathbf{\overrightarrow{f}}\label{gromeka_lamb2}</math></div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>\frac{\partial\mathbf{\overrightarrow{v}}}{\partial t}+\frac{1}{2}grad\;v^2-\mathbf{\overrightarrow{v}}\times curl\mathbf{\overrightarrow{v}}+\frac{1}{\rho}grad(P+U)=\mathbf{\overrightarrow{f}}\label{gromeka_lamb2}</math></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The strict theory of vortices is applicable either for the ideal incompressible flows (<math><del class="diffchange diffchange-inline">grad\;</del>\rho=<del class="diffchange diffchange-inline">0</del></math>), or for the ideal [[Continuum#Barotropic flow|barotropic flows]]. For both cases, the Gromeka-Lamb equation can be written as:</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The strict theory of vortices is applicable either for the ideal incompressible flows (<math>\rho=<ins class="diffchange diffchange-inline">const</ins></math>), or for the ideal <ins class="diffchange diffchange-inline">compressible </ins>[[Continuum#Barotropic flow|<ins class="diffchange diffchange-inline">quadratic </ins>barotropic flows]]. For both cases, the Gromeka-Lamb equation can be written <ins class="diffchange diffchange-inline">(regardless the gradient factor) </ins>as:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>\frac{\partial\mathbf{\overrightarrow{v}}}{\partial t}+\frac{1}{2}grad\;v^2-\mathbf{\overrightarrow{v}}\times curl\mathbf{\overrightarrow{v}}+grad\frac{P+U}{\rho}=\mathbf{\overrightarrow{f}}\label{gromeka_lamb3}</math></div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>\frac{\partial\mathbf{\overrightarrow{v}}}{\partial t}+\frac{1}{2}grad\;v^2-\mathbf{\overrightarrow{v}}\times curl\mathbf{\overrightarrow{v}}+grad\frac{P+U}{\rho}=\mathbf{\overrightarrow{f}}\label{gromeka_lamb3}</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Applying the curl operator to \ref{gromeka_lamb3}, and using the identity <math>curl\;grad\;X=0</math>, produces the Helmholtz equation, which is independent of density and pressure:</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Applying the curl operator to \ref{gromeka_lamb3}, and using the identity <math>curl\;grad\;X=0</math>, produces the Helmholtz equation, which is independent of density and pressure:</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="L21" >Line 21:</td>
<td colspan="2" class="diff-lineno">Line 21:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The steady-state continuity equation ([[Continuum#Eq-5|"Continuum", 5]]) is:</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The steady-state continuity equation ([[Continuum#Eq-5|"Continuum", 5]]) is:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>div\;\rho\mathbf{\overrightarrow{v}}=\mathbf{\overrightarrow{v}}grad\;\rho+\rho\;div\mathbf{\overrightarrow{v}}=0\label{continuity_steady2}</math></div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>div\;\rho\mathbf{\overrightarrow{v}}=\mathbf{\overrightarrow{v}}grad\;\rho+\rho\;div\mathbf{\overrightarrow{v}}=0\label{continuity_steady2}</math></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>For an incompressible flow (<math>grad\;\rho=0</math>), the velocity is a vortical field (<math>div\mathbf{\overrightarrow{v}}=0</math>). <del class="diffchange diffchange-inline">And for </del>a <del class="diffchange diffchange-inline">barotropic </del>flow, the velocity is a vortex field too, if the force of a pressure and a potential energy are perpendicular to the velocity vector, producing only a normal acceleration in a steady-state vortex:</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>For an incompressible flow (<math>grad\;\rho=0</math>), the velocity is a vortical field (<math>div\mathbf{\overrightarrow{v}}=0</math>). <ins class="diffchange diffchange-inline">For </ins>a <ins class="diffchange diffchange-inline">compressible </ins>flow, the velocity is a vortex field too, if the force of a pressure and a potential energy are perpendicular to the velocity vector, producing only a normal acceleration in a steady-state vortex:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>\mathbf{\overrightarrow{v}}grad(P+U)=0\label{barotropic_steady_condition}</math></div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>\mathbf{\overrightarrow{v}}grad(P+U)=0\label{barotropic_steady_condition}</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>As a result, a stable (steady-state) vortex is generally described by the system of equations:</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>As a result, a stable (steady-state) vortex is generally described by the system of equations:</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="L27" >Line 27:</td>
<td colspan="2" class="diff-lineno">Line 27:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* Balance equation for the pressure, the [[wikipedia:Centrifugal force|centrifugal]] and the [[wikipedia:Conservative force|conservative]] forces.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* Balance equation for the pressure, the [[wikipedia:Centrifugal force|centrifugal]] and the [[wikipedia:Conservative force|conservative]] forces.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">In </del>the vortices, the centrifugal force is <del class="diffchange diffchange-inline">balanced against </del>the force of the static pressure gradient, which is decreasing towards the center of a vortex due to <del class="diffchange diffchange-inline">the growth of </del>a velocity and a dynamic pressure.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">Within </ins>the vortices, the centrifugal force is <ins class="diffchange diffchange-inline">compensated by </ins>the force of the static pressure gradient, which is decreasing towards the center of a vortex due to <ins class="diffchange diffchange-inline">increase in </ins>a velocity and a dynamic pressure.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The conservative and pressure forces are shaping a vortex, and cause it to move over a continuum. This follows from the [[wikipedia:Kelvin's circulation theorem|Kelvin’s circulation theorem]], whereby any conservative force does not change a vortex circulation, and consequently the energy of this force can change a velocity of the entire vortex:</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The conservative and pressure forces are shaping a vortex, and cause it to move over a continuum. This follows from the [[wikipedia:Kelvin's circulation theorem|Kelvin’s circulation theorem]], whereby any conservative force does not change a vortex circulation, and consequently the energy of this force can change a velocity of the entire vortex:</div></td></tr>
</table>Adminhttp://alt-sci.ru/_en/wiki/index.php?title=Vortex&diff=61&oldid=prevAdmin: Protected "Vortex" ([Edit=Allow only administrators] (indefinite) [Move=Allow only administrators] (indefinite))2015-07-25T14:37:13Z<p>Protected "<a href="/en/wiki/Vortex" title="Vortex">Vortex</a>" ([Edit=Allow only administrators] (indefinite) [Move=Allow only administrators] (indefinite))</p>
<table class='diff diff-contentalign-left'>
<tr style='vertical-align: top;'>
<td colspan='1' style="background-color: white; color:black; text-align: center;">← Older revision</td>
<td colspan='1' style="background-color: white; color:black; text-align: center;">Revision as of 14:37, 25 July 2015</td>
</tr><tr><td colspan='2' style='text-align: center;'><div class="mw-diff-empty">(No difference)</div>
</td></tr></table>Adminhttp://alt-sci.ru/_en/wiki/index.php?title=Vortex&diff=60&oldid=prevAdmin at 14:35, 25 July 20152015-07-25T14:35:43Z<p></p>
<table class='diff diff-contentalign-left'>
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
<tr style='vertical-align: top;'>
<td colspan='2' style="background-color: white; color:black; text-align: center;">← Older revision</td>
<td colspan='2' style="background-color: white; color:black; text-align: center;">Revision as of 14:35, 25 July 2015</td>
</tr><tr><td colspan="2" class="diff-lineno" id="L4" >Line 4:</td>
<td colspan="2" class="diff-lineno">Line 4:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Corresponding Wikipedia article: [[wikipedia:Vortex|Vortex]]</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Corresponding Wikipedia article: [[wikipedia:Vortex|Vortex]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>-----</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>-----</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The vortex is a [[wikipedia:Circular motion|circular motion]] <del class="diffchange diffchange-inline">in </del>a [[Continuum|continuum]]. The known theory of vortices is applicable only for the 3-dimensional space, because it uses the [[Physical field#Vector analysis formulas|differentiation formulas for products of the vector fields]].</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The vortex is a [[wikipedia:Circular motion|circular motion]] <ins class="diffchange diffchange-inline">within </ins>a [[Continuum|continuum]]. The known theory of vortices is applicable only for the 3-dimensional space, because it uses the [[Physical field#Vector analysis formulas|differentiation formulas for products of the vector fields]].</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The [[wikipedia:Vorticity|vorticity]] measure is the [[wikipedia:Curl (mathematics)|curl]] of a velocity field. According to one of the curl definitions by the [[wikipedia:Circulation (fluid dynamics)|circulation]], choosing a circle of radius <math>r</math> as the contour, the curl becomes:</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The [[wikipedia:Vorticity|vorticity]] measure is the [[wikipedia:Curl (mathematics)|curl]] of a velocity field. According to one of the curl definitions by the [[wikipedia:Circulation (fluid dynamics)|circulation]], choosing a circle of radius <math>r</math> as the contour, the curl becomes:</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="L10" >Line 10:</td>
<td colspan="2" class="diff-lineno">Line 10:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Then, using the kinematic formula <math>\mathbf{\overrightarrow{\omega}}=(\mathbf{\overrightarrow{r}}\times\mathbf{\overrightarrow{v}})/r^2</math>, the curl of a velocity field is the twice [[wikipedia:Angular velocity|angular velocity]] of an infinitesimally small continuum element:</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Then, using the kinematic formula <math>\mathbf{\overrightarrow{\omega}}=(\mathbf{\overrightarrow{r}}\times\mathbf{\overrightarrow{v}})/r^2</math>, the curl of a velocity field is the twice [[wikipedia:Angular velocity|angular velocity]] of an infinitesimally small continuum element:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>curl\mathbf{\overrightarrow{v}}=2\mathbf{\overrightarrow{\omega}}\label{curl_2omega}</math></div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>curl\mathbf{\overrightarrow{v}}=2\mathbf{\overrightarrow{\omega}}\label{curl_2omega}</math></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The Euler’s equation ([[Continuum#Eq-4|"Continuum", 4]]) using the known identity <math>(\mathbf{\overrightarrow{v}}\cdot\nabla)\mathbf{\overrightarrow{v}}=\frac{1}{2}grad\;v^2-\mathbf{\overrightarrow{v}}\times curl\mathbf{\overrightarrow{v}}</math>, is converted to the Gromeka-[[wikipedia:Horace Lamb|Lamb]] equation:</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The Euler’s equation ([[Continuum#Eq-4|"Continuum", 4]])<ins class="diffchange diffchange-inline">, </ins>using the known identity <math>(\mathbf{\overrightarrow{v}}\cdot\nabla)\mathbf{\overrightarrow{v}}=\frac{1}{2}grad\;v^2-\mathbf{\overrightarrow{v}}\times curl\mathbf{\overrightarrow{v}}</math>, is converted to the Gromeka-[[wikipedia:Horace Lamb|Lamb]] equation:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>\rho\frac{\partial\mathbf{\overrightarrow{v}}}{\partial t}+\frac{\rho}{2}grad\;v^2-\rho\mathbf{\overrightarrow{v}}\times curl\mathbf{\overrightarrow{v}}+grad(P+U)=\rho\mathbf{\overrightarrow{f}}\label{gromeka_lamb1}</math></div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>\rho\frac{\partial\mathbf{\overrightarrow{v}}}{\partial t}+\frac{\rho}{2}grad\;v^2-\rho\mathbf{\overrightarrow{v}}\times curl\mathbf{\overrightarrow{v}}+grad(P+U)=\rho\mathbf{\overrightarrow{f}}\label{gromeka_lamb1}</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>\frac{\partial\mathbf{\overrightarrow{v}}}{\partial t}+\frac{1}{2}grad\;v^2-\mathbf{\overrightarrow{v}}\times curl\mathbf{\overrightarrow{v}}+\frac{1}{\rho}grad(P+U)=\mathbf{\overrightarrow{f}}\label{gromeka_lamb2}</math></div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>:<math>\frac{\partial\mathbf{\overrightarrow{v}}}{\partial t}+\frac{1}{2}grad\;v^2-\mathbf{\overrightarrow{v}}\times curl\mathbf{\overrightarrow{v}}+\frac{1}{\rho}grad(P+U)=\mathbf{\overrightarrow{f}}\label{gromeka_lamb2}</math></div></td></tr>
</table>Adminhttp://alt-sci.ru/_en/wiki/index.php?title=Vortex&diff=59&oldid=prevAdmin: Created page with "ru:Вихрь {{Book_page|Waves|Physics|Helix and spiral}} Corresponding Wikipedia article: Vortex ----- The vortex is a wikipedia:Circular motion|..."2015-07-25T14:33:03Z<p>Created page with "<a href="http://alt-sci.ru/wiki/%D0%92%D0%B8%D1%85%D1%80%D1%8C" class="extiw" title="ru:Вихрь">ru:Вихрь</a> {{Book_page|Waves|Physics|Helix and spiral}} Corresponding Wikipedia article: <a href="http://en.wikipedia.org/wiki/Vortex" class="extiw" title="wikipedia:Vortex">Vortex</a> ----- The vortex is a wikipedia:Circular motion|..."</p>
<p><b>New page</b></p><div>[[ru:Вихрь]]<br />
{{Book_page|Waves|Physics|Helix and spiral}}<br />
<br />
Corresponding Wikipedia article: [[wikipedia:Vortex|Vortex]]<br />
-----<br />
The vortex is a [[wikipedia:Circular motion|circular motion]] in a [[Continuum|continuum]]. The known theory of vortices is applicable only for the 3-dimensional space, because it uses the [[Physical field#Vector analysis formulas|differentiation formulas for products of the vector fields]].<br />
<br />
The [[wikipedia:Vorticity|vorticity]] measure is the [[wikipedia:Curl (mathematics)|curl]] of a velocity field. According to one of the curl definitions by the [[wikipedia:Circulation (fluid dynamics)|circulation]], choosing a circle of radius <math>r</math> as the contour, the curl becomes:<br />
:<math>curl\mathbf{\overrightarrow{v}}=\lim_{S\to 0}\frac{1}{S}\oint{\mathbf{\overrightarrow{r}}\times\mathbf{\overrightarrow{v}}\mathrm{d}\alpha}=\frac{1}{\pi}\lim_{r\to 0}\oint{\frac{\mathbf{\overrightarrow{r}}\times\mathbf{\overrightarrow{v}}}{r^2}\mathrm{d}\alpha}\label{vorticity}</math><br />
Then, using the kinematic formula <math>\mathbf{\overrightarrow{\omega}}=(\mathbf{\overrightarrow{r}}\times\mathbf{\overrightarrow{v}})/r^2</math>, the curl of a velocity field is the twice [[wikipedia:Angular velocity|angular velocity]] of an infinitesimally small continuum element:<br />
:<math>curl\mathbf{\overrightarrow{v}}=2\mathbf{\overrightarrow{\omega}}\label{curl_2omega}</math><br />
The Euler’s equation ([[Continuum#Eq-4|"Continuum", 4]]) using the known identity <math>(\mathbf{\overrightarrow{v}}\cdot\nabla)\mathbf{\overrightarrow{v}}=\frac{1}{2}grad\;v^2-\mathbf{\overrightarrow{v}}\times curl\mathbf{\overrightarrow{v}}</math>, is converted to the Gromeka-[[wikipedia:Horace Lamb|Lamb]] equation:<br />
:<math>\rho\frac{\partial\mathbf{\overrightarrow{v}}}{\partial t}+\frac{\rho}{2}grad\;v^2-\rho\mathbf{\overrightarrow{v}}\times curl\mathbf{\overrightarrow{v}}+grad(P+U)=\rho\mathbf{\overrightarrow{f}}\label{gromeka_lamb1}</math><br />
:<math>\frac{\partial\mathbf{\overrightarrow{v}}}{\partial t}+\frac{1}{2}grad\;v^2-\mathbf{\overrightarrow{v}}\times curl\mathbf{\overrightarrow{v}}+\frac{1}{\rho}grad(P+U)=\mathbf{\overrightarrow{f}}\label{gromeka_lamb2}</math><br />
The strict theory of vortices is applicable either for the ideal incompressible flows (<math>grad\;\rho=0</math>), or for the ideal [[Continuum#Barotropic flow|barotropic flows]]. For both cases, the Gromeka-Lamb equation can be written as:<br />
:<math>\frac{\partial\mathbf{\overrightarrow{v}}}{\partial t}+\frac{1}{2}grad\;v^2-\mathbf{\overrightarrow{v}}\times curl\mathbf{\overrightarrow{v}}+grad\frac{P+U}{\rho}=\mathbf{\overrightarrow{f}}\label{gromeka_lamb3}</math><br />
Applying the curl operator to \ref{gromeka_lamb3}, and using the identity <math>curl\;grad\;X=0</math>, produces the Helmholtz equation, which is independent of density and pressure:<br />
:<math>\frac{\partial}{\partial t}curl\mathbf{\overrightarrow{v}}-curl(\mathbf{\overrightarrow{v}}\times curl\mathbf{\overrightarrow{v}})=curl\mathbf{\overrightarrow{f}}\label{Helmholtz1}</math><br />
Thus, the vortical force field (<math>curl\mathbf{\overrightarrow{f}}\neq 0</math>) causes a circular motion of the continuum particles. Certainly, it causes an accelerated motion, so in the steady-state Helmholtz equation <math>curl\mathbf{\overrightarrow{f}}=0</math>:<br />
:<math>curl(\mathbf{\overrightarrow{v}}\times curl\mathbf{\overrightarrow{v}})=0\label{Helmholtz1_steady}</math><br />
The steady-state continuity equation ([[Continuum#Eq-5|"Continuum", 5]]) is:<br />
:<math>div\;\rho\mathbf{\overrightarrow{v}}=\mathbf{\overrightarrow{v}}grad\;\rho+\rho\;div\mathbf{\overrightarrow{v}}=0\label{continuity_steady2}</math><br />
For an incompressible flow (<math>grad\;\rho=0</math>), the velocity is a vortical field (<math>div\mathbf{\overrightarrow{v}}=0</math>). And for a barotropic flow, the velocity is a vortex field too, if the force of a pressure and a potential energy are perpendicular to the velocity vector, producing only a normal acceleration in a steady-state vortex:<br />
:<math>\mathbf{\overrightarrow{v}}grad(P+U)=0\label{barotropic_steady_condition}</math><br />
As a result, a stable (steady-state) vortex is generally described by the system of equations:<br />
* Kinematic Helmholtz equation \ref{Helmholtz1_steady}.<br />
* Balance equation for the pressure, the [[wikipedia:Centrifugal force|centrifugal]] and the [[wikipedia:Conservative force|conservative]] forces.<br />
<br />
In the vortices, the centrifugal force is balanced against the force of the static pressure gradient, which is decreasing towards the center of a vortex due to the growth of a velocity and a dynamic pressure.<br />
<br />
The conservative and pressure forces are shaping a vortex, and cause it to move over a continuum. This follows from the [[wikipedia:Kelvin's circulation theorem|Kelvin’s circulation theorem]], whereby any conservative force does not change a vortex circulation, and consequently the energy of this force can change a velocity of the entire vortex:<br />
:<math>\frac{\mathrm{d}}{\mathrm{d}t}\oint{\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{l}}}=\oint{\frac{\mathrm{d}\mathbf{\overrightarrow{v}}}{\mathrm{d}t}\mathrm{d}\mathbf{\overrightarrow{l}}}=-\oint{grad\frac{P+U}{\rho}\mathrm{d}\mathbf{\overrightarrow{l}}}=0\label{Kelvin_basic}</math><br />
The proof of \ref{Kelvin_basic} uses the kinematic Kelvin's theorem:<br />
:<math>\frac{\mathrm{d}}{\mathrm{d}t}\oint{\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{l}}}=\oint{\frac{\mathrm{d}\mathbf{\overrightarrow{v}}}{\mathrm{d}t}\mathrm{d}\mathbf{\overrightarrow{l}}}\label{Kelvin_kinematic}</math><br />
The [[wikipedia:Streamlines, streaklines, and pathlines|streamline]] is a line, the tangent of which is parallel to the velocity vector at a given point. The [[Continuum#Bernoulli's law|Bernoulli's law]] is valid for any streamline, because any parameter along it is a function of a single coordinate, and the force vector <math>\mathbf{\overrightarrow{v}}\times curl\mathbf{\overrightarrow{v}}</math> is always perpendicular to the flow, and does not produce a tangent acceleration.<br />
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The [[wikipedia:Vorticity#Vortex_lines_and_vortex_tubes|vortex line]] is a line, the tangent of which is parallel to the velocity curl vector at a given point. The Bernoulli's law is valid for any vortex line.<br />
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The [[wikipedia:Vorticity#Vortex_lines_and_vortex_tubes|vortex tube]] is a closed surface, which is produced by the vortex lines. An example of a vortex tube is the inner surface of the vortex in a liquid or a gas with the maximum speed on it and the resting continuum inside.<br />
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The velocity curl flux through a tube surface:<br />
:<math>\oint{curl\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{S}}}=\int_{S_1}{curl\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{S}}}-\int_{S_2}{curl\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{S}}}\label{Helmholtz2_1}</math><br />
where <math>S_1</math> and <math>S_2</math> are the cross sections of the tube. At the same time, according to the[[wikipedia:Divergence theorem|divergence theorem]] and the identity <math>div\;curl\;\mathbf{\overrightarrow{v}}=0</math>:<br />
:<math>\oint{curl\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{S}}}=\oint{div\;curl\mathbf{\overrightarrow{v}}\mathrm{d}V}=0\label{Helmholtz2_2}</math><br />
Thus, the velocity curl flux through any tube section is constant over its length, and according to the [[wikipedia:Kelvin–Stokes theorem|Stokes theorem]], it’s equal to a circulation along the section contour:<br />
:<math>\int{curl\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{S}}}=\oint{\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{l}}}=const\label{Helmholtz2}</math><br />
The equation \ref{Helmholtz2} is known as the 2nd Helmholtz theorem, which implies:<br />
* The flow velocity over the tube surface is inversely proportional to the length (radius) of the tube section contour.<br />
* The [[Helix and spiral|helical]] flow is possible along the tube of any shape without breaking the stream continuity.<br />
* The tube is continuous between the continuum interfaces. The stable vortices are either between the interfaces, or are closed into the ring.<br />
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A funnel is a typical vortex, which occurs, for example, when draining the fluid from a reservoir through a small hole at its bottom. The flow velocity increases as it get closer to the hole, and a weak fluid rotation around it can raise a powerful whirl.<br />
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The [[Ring|vortex ring]] is a vortex, which is closed to the annular [[wikipedia:Solid of revolution|solid of revolution]]. Generally, the generatrix and the guide of this solid have an arbitrary shape. The vortex ring is the only possible form of a stable vortex, which may not consume energy from the environment. In the real fluids and gases, the viscous friction dissipates the energy of a vortex, and it decays. The streamlines of the vortex ring can be:<br />
* Circular, along a generatrix of the solid. Such vortices arise in the fluids and gases by the action of a jet. For example, the stable [[wikipedia:Smoke ring|smoke rings]] arise when the jet of smoke passes through a narrow opening.<br />
* Helical, which consists of a motion both along a generatrix and along a guide of the solid.<br />
<font style="font-size:85%"><br />
'''Exmple. «A storm in a teacup».'''<br />
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The rotation of a liquid in a cup causes the centrifugal force and boosts a pressure at the walls of the cup. An annular cylindrical vortex appears at a certain speed and lifts from the bottom a stuff, which is heavier than the liquid (tea leaves, for example). <br />
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The flow direction along the generatrix of a vortex ring is determined by the pressure gradient. The greatest pressure is always at the edge of the cup bottom, and it’s a sum of the liquid column pressure and the cup walls pressure. If the walls pressure is high enough, the stream may run from the bottom edge to the bottom center, ascending along the rotation axis and descending along the cup walls.<br />
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The decaying vortex causes the accumulation of the heavy objects (tea leaves) at the bottom center, because the flow force is sufficient to move the objects over the bottom, but not enough to lift them off the bottom.<br />
</font><br />
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==See also==<br />
* [[Aetheric vortex]]<br />
* [[Liquid vortices]]<br />
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{{Book_page|Waves|Physics|Helix and spiral}}</div>Admin