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Navier-Stokes equations - Revision history
2024-03-29T13:43:56Z
Revision history for this page on the wiki
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Bluebase: Reference correction
2012-08-28T18:17:33Z
<p>Reference correction</p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{reference-paper | author=Pieter Wesseling | year=2001 | title=Principles of computational fluid dynamics | rest=Springer-Verlag Berlin Heidelberg }}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{reference-paper | author=Pieter Wesseling | year=2001 | title=Principles of computational fluid dynamics | rest=Springer-Verlag Berlin Heidelberg }}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">Please correct </del>-<del class="diffchange diffchange-inline">-[[User:Michail</del>|<del class="diffchange diffchange-inline">Michail]] 01:52</del>, <del class="diffchange diffchange-inline">26 August 2012 (MDT) (I mean here Dr</del>. <del class="diffchange diffchange-inline">Schlichting Boundary layer theory</del>, <del class="diffchange diffchange-inline">but I have only russian edition</del>)</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">{{reference</ins>-<ins class="diffchange diffchange-inline">paper </ins>| <ins class="diffchange diffchange-inline">author=Schlichting</ins>, <ins class="diffchange diffchange-inline">H</ins>., <ins class="diffchange diffchange-inline">Gersten, K. | year=2006 | title=Grenzschicht Theorie | rest=10th edn. Springer-Verlag Berlin Heidelberg. (german</ins>)<ins class="diffchange diffchange-inline">}}</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>{{reference-paper | author=<del class="diffchange diffchange-inline">Hermann </del>Schlichting | year=<del class="diffchange diffchange-inline">???? </del>| title=<del class="diffchange diffchange-inline">Grenzschicht Theorie </del>| rest=<del class="diffchange diffchange-inline">????, ????? </del>}}</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>{{reference-paper | author=Schlichting<ins class="diffchange diffchange-inline">, H., Gersten, K. </ins>| year=<ins class="diffchange diffchange-inline">2000 </ins>| title=<ins class="diffchange diffchange-inline">Boundary Layer Theory </ins>| rest=<ins class="diffchange diffchange-inline">8th edn. Springer-Verlag Berlin Heidelberg </ins>}}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{reference-paper | author=Герман Шлихтинг | year=1974 | title=Теория пограничного слоя | rest=translation from 5th German edition corrected by 6th american edition edited by L.G. Loiciansky, Nauka, Moscow}}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{reference-paper | author=Герман Шлихтинг | year=1974 | title=Теория пограничного слоя | rest=translation from 5th German edition corrected by 6th american edition edited by L.G. Loiciansky, Nauka, Moscow}}</div></td></tr>
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Bluebase
https://cfd-online.com/W/index.php?title=Navier-Stokes_equations&diff=15098&oldid=prev
Michail: /* References */
2012-08-26T12:53:25Z
<p><span class="autocomment">References</span></p>
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<td colspan='2' style="background-color: white; color:black;">Revision as of 12:53, 26 August 2012</td>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{reference-paper | author=Hermann Schlichting | year=???? | title=Grenzschicht Theorie | rest=????, ????? }}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{reference-paper | author=Hermann Schlichting | year=???? | title=Grenzschicht Theorie | rest=????, ????? }}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>{{reference-paper | author=Герман Шлихтинг | year=1974 | title=Теория пограничного слоя | rest=translation from 5th German edition corrected by 6th american edition by <del class="diffchange diffchange-inline">edition of </del>L.G. Loiciansky, Nauka, Moscow}}</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>{{reference-paper | author=Герман Шлихтинг | year=1974 | title=Теория пограничного слоя | rest=translation from 5th German edition corrected by 6th american edition <ins class="diffchange diffchange-inline">edited </ins>by L.G. Loiciansky, Nauka, Moscow}}</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==External links==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==External links==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>*[http://scienceworld.wolfram.com/physics/Navier-StokesEquations.html Navier-Stokes equations at mathworld.com]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>*[http://scienceworld.wolfram.com/physics/Navier-StokesEquations.html Navier-Stokes equations at mathworld.com]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>*[http://www.claymath.org/millennium/Navier-Stokes_Equations/ Millenium Problem]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>*[http://www.claymath.org/millennium/Navier-Stokes_Equations/ Millenium Problem]</div></td></tr>
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Michail
https://cfd-online.com/W/index.php?title=Navier-Stokes_equations&diff=15077&oldid=prev
Michail: /* References */ please help with original name of Dr. Hermann Schlichting boundary layer theory book
2012-08-26T07:59:24Z
<p><span class="autocomment">References: </span> please help with original name of Dr. Hermann Schlichting boundary layer theory book</p>
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<td colspan='2' style="background-color: white; color:black;">Revision as of 07:59, 26 August 2012</td>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{reference-paper | author=Hermann Schlichting | year=???? | title=Grenzschicht Theorie | rest=????, ????? }}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{reference-paper | author=Hermann Schlichting | year=???? | title=Grenzschicht Theorie | rest=????, ????? }}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>{{reference-paper | author=Герман Шлихтинг | year=1974 | title=Теория пограничного слоя | rest=translation from 5th German edition corrected by 6th american edition by edition of L.G. Loiciansky }}</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>{{reference-paper | author=Герман Шлихтинг | year=1974 | title=Теория пограничного слоя | rest=translation from 5th German edition corrected by 6th american edition by edition of L.G. Loiciansky<ins class="diffchange diffchange-inline">, Nauka, Moscow</ins>}}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==External links==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==External links==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>*[http://scienceworld.wolfram.com/physics/Navier-StokesEquations.html Navier-Stokes equations at mathworld.com]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>*[http://scienceworld.wolfram.com/physics/Navier-StokesEquations.html Navier-Stokes equations at mathworld.com]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>*[http://www.claymath.org/millennium/Navier-Stokes_Equations/ Millenium Problem]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>*[http://www.claymath.org/millennium/Navier-Stokes_Equations/ Millenium Problem]</div></td></tr>
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Michail
https://cfd-online.com/W/index.php?title=Navier-Stokes_equations&diff=15076&oldid=prev
Michail: /* References */
2012-08-26T07:56:55Z
<p><span class="autocomment">References</span></p>
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<td colspan='2' style="background-color: white; color:black;">Revision as of 07:56, 26 August 2012</td>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{reference-paper | author=Hermann Schlichting | year=???? | title=Grenzschicht Theorie | rest=????, ????? }}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{reference-paper | author=Hermann Schlichting | year=???? | title=Grenzschicht Theorie | rest=????, ????? }}</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">{{reference-paper | author=Герман Шлихтинг | year=1974 | title=Теория пограничного слоя | rest=translation from 5th German edition corrected by 6th american edition by edition of L.G. Loiciansky }}</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==External links==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==External links==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>*[http://scienceworld.wolfram.com/physics/Navier-StokesEquations.html Navier-Stokes equations at mathworld.com]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>*[http://scienceworld.wolfram.com/physics/Navier-StokesEquations.html Navier-Stokes equations at mathworld.com]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>*[http://www.claymath.org/millennium/Navier-Stokes_Equations/ Millenium Problem]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>*[http://www.claymath.org/millennium/Navier-Stokes_Equations/ Millenium Problem]</div></td></tr>
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Michail
https://cfd-online.com/W/index.php?title=Navier-Stokes_equations&diff=15075&oldid=prev
Michail: /* References */
2012-08-26T07:52:53Z
<p><span class="autocomment">References</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{reference-paper | author=Pieter Wesseling | year=2001 | title=Principles of computational fluid dynamics | rest=Springer-Verlag Berlin Heidelberg }}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{reference-paper | author=Pieter Wesseling | year=2001 | title=Principles of computational fluid dynamics | rest=Springer-Verlag Berlin Heidelberg }}</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Please correct --[[User:Michail|Michail]] 01:52, 26 August 2012 (MDT) (I mean here Dr. Schlichting Boundary layer theory, but I have only russian edition)</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">{{reference-paper | author=Hermann Schlichting | year=???? | title=Grenzschicht Theorie | rest=????, ????? }}</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==External links==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==External links==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>*[http://scienceworld.wolfram.com/physics/Navier-StokesEquations.html Navier-Stokes equations at mathworld.com]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>*[http://scienceworld.wolfram.com/physics/Navier-StokesEquations.html Navier-Stokes equations at mathworld.com]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>*[http://www.claymath.org/millennium/Navier-Stokes_Equations/ Millenium Problem]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>*[http://www.claymath.org/millennium/Navier-Stokes_Equations/ Millenium Problem]</div></td></tr>
</table>
Michail
https://cfd-online.com/W/index.php?title=Navier-Stokes_equations&diff=15074&oldid=prev
Michail: /* Derivation of momentum equations */
2012-08-26T07:36:30Z
<p><span class="autocomment">Derivation of momentum equations</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===Derivation of momentum equations===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===Derivation of momentum equations===</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">All is started from Newton's law:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">FORCE is equal MASS time ACCELERATION</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><table width="70%"><tr><td></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math> </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">F=ma </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></td><td width="5%">(2)</td></tr></table></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">and which is applied to the fluid parsel</ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Image:Motion.JPG]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Image:Motion.JPG]]</div></td></tr>
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Michail
https://cfd-online.com/W/index.php?title=Navier-Stokes_equations&diff=14172&oldid=prev
Michail: /* Derivation of the energy equation */
2012-04-16T08:55:07Z
<p><span class="autocomment">Derivation of the energy equation</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(2)</td></tr></table></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(2)</td></tr></table></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Assuming that heat is added to each material particle at a rate <math>q</math> per unit of mass, and that there is a heat flux per unit of area through <math>S(t)</math> we find</ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><table width="70%"><tr><td></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><table width="70%"><tr><td></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(2)</td></tr></table></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(2)</td></tr></table></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Let heat diffusion be governed by Fourier's law:</ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(2)</td></tr></table></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(2)</td></tr></table></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">with <math>k</math> the thermal conductivity, and <math>T</math> the tempereture, which is another state variable. Using the divergence theorem we find</ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(2)</td></tr></table></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(2)</td></tr></table></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">By application of the transport theorem and substitution we obtain</ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(2)</td></tr></table></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(2)</td></tr></table></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Since this holds for every <math>V(t)</math> , we have</ins></div></td></tr>
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</table>
Michail
https://cfd-online.com/W/index.php?title=Navier-Stokes_equations&diff=14165&oldid=prev
Michail: /* References */
2012-04-14T14:56:16Z
<p><span class="autocomment">References</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{reference-paper | author=C. L. M. H. Navier | year=1822 | title=Memoire sur les lois du mouvement des fluides | rest=Mem. Acad. Sci. Inst. France, 6, 389-440}}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{reference-paper | author=C. L. M. H. Navier | year=1822 | title=Memoire sur les lois du mouvement des fluides | rest=Mem. Acad. Sci. Inst. France, 6, 389-440}}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>{{reference-paper | author=Loiciansky, L.G. | year=1978 | title=Mechanics of Fluid and Gas | rest=5 edn., <del class="diffchange diffchange-inline">p. 736. </del>Nauka, Moscow }}</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>{{reference-paper | author=Loiciansky, L.G. | year=1978 | title=Mechanics of Fluid and Gas | rest=5 edn., Nauka, Moscow }}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{reference-paper | author=Pieter Wesseling | year=2001 | title=Principles of computational fluid dynamics | rest=Springer-Verlag Berlin Heidelberg }}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{reference-paper | author=Pieter Wesseling | year=2001 | title=Principles of computational fluid dynamics | rest=Springer-Verlag Berlin Heidelberg }}</div></td></tr>
</table>
Michail
https://cfd-online.com/W/index.php?title=Navier-Stokes_equations&diff=14164&oldid=prev
Michail: /* Derivation of the energy equation */
2012-04-14T14:42:37Z
<p><span class="autocomment">Derivation of the energy equation</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===Derivation of the energy equation===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===Derivation of the energy equation===</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>we follow Wesseling</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>we follow Wesseling <ins class="diffchange diffchange-inline">(2001)</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">By applying the first law of thermodynamics to a material volume <math>V(t)</math> we find</ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><table width="70%"><tr><td></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><table width="70%"><tr><td></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(2)</td></tr></table></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(2)</td></tr></table></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">with <math>E</math> the ''total energy'' per unit mass</ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(2)</td></tr></table></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(2)</td></tr></table></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Furthermore, <math>W</math> is the rate of work expended by the surroundings on the fluid in <math>V(t)</math>, and <math>Q</math> is the rate of heat addition </ins></div></td></tr>
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Michail
https://cfd-online.com/W/index.php?title=Navier-Stokes_equations&diff=14163&oldid=prev
Michail: /* References */
2012-04-14T14:37:25Z
<p><span class="autocomment">References</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{reference-paper | author=C. L. M. H. Navier | year=1822 | title=Memoire sur les lois du mouvement des fluides | rest=Mem. Acad. Sci. Inst. France, 6, 389-440}}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{reference-paper | author=C. L. M. H. Navier | year=1822 | title=Memoire sur les lois du mouvement des fluides | rest=Mem. Acad. Sci. Inst. France, 6, 389-440}}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>{{reference-paper | author=Loiciansky, L.G. | year=1978 | title=Mechanics of Fluid and Gas | rest=5 edn., p. 736. Nauka, Moscow <del class="diffchange diffchange-inline">(1978)</del>}}</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>{{reference-paper | author=Loiciansky, L.G. | year=1978 | title=Mechanics of Fluid and Gas | rest=5 edn., p. 736. Nauka, Moscow <ins class="diffchange diffchange-inline">}}</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">{{reference-paper | author=Pieter Wesseling | year=2001 | title=Principles of computational fluid dynamics | rest=Springer-Verlag Berlin Heidelberg </ins>}}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==External links==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==External links==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>*[http://scienceworld.wolfram.com/physics/Navier-StokesEquations.html Navier-Stokes equations at mathworld.com]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>*[http://scienceworld.wolfram.com/physics/Navier-StokesEquations.html Navier-Stokes equations at mathworld.com]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>*[http://www.claymath.org/millennium/Navier-Stokes_Equations/ Millenium Problem]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>*[http://www.claymath.org/millennium/Navier-Stokes_Equations/ Millenium Problem]</div></td></tr>
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Michail