Navier-Stokes from variational principle - Revision history

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'''WARNING''': this article has a serious (perhaps, unrecoverable) error in formulas. [[User:Kouznetsov|Kouznetsov]] 09:59, 5 July 2011 (JST)

Consideration of the [[equation of Navier-Stokes]] as result of the formal (id est, correct) application of the [[principle of stationary action]] might be a key to the breakthrough in the efficient approximation of the solutions. The text below uses the ideas by [[Enrico Sciubba]]
<ref name="enrico">
http://www.icatweb.org/vol7/7.3/06-scubba.pdf Enrico Sciubba. Flow Exergy as a Lagrangian for the Navier-Stokes Equations for Incompressible Flow. Int. J. Thermodynamics, ISSN 1301-9724 Vol. 7, (No. 3), pp.115-122, September-2004
</ref>.


Consider the Lagrangian
:$ (1) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle
\mathcal L=u_k \left(q u_{k,0} + u_{k,j} u_j+ b_k\right)- \frac{\nu}{2} (u_{i,j}+u_{j,i})^2
$

where $u_k$ are assumed to be functions of $x_1, x_2, x_3$ ; $k$ and $j$ and $i$ take integer values from unity to $3$; $q$ is constant.
The additional index after comma indicates the derivative with respect to specified coordinate, time is assumed to be coordinate number zero. Summation is assumed over the repeating indices. The power two in ($1$) is also treated as repetition of indices.

The principle of stationary action is written as follows:
:$ (2) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle
\frac{\partial \mathcal L}{\partial u_k}
- \frac{\partial}{\partial x_j}\left( \frac{\partial \mathcal L}{\partial u_{k,j} } \right)
- \frac{\partial}{\partial t}\left( \frac{\partial \mathcal L}{\partial u_{k,0} } \right) =0
$

From expression ($1$), the derivatives of the Lagrangian can be estimated as follows:
:$ (3) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle
\frac{\partial \mathcal L}{\partial u_k}=q u_{k,0} + u_{k,j} u_j+ b_k ~+~ u_\ell u_{\ell,k}=
%u_{k,0} + u_{k,j} u_j+ b_k ~+~ u_j u_{j,k}=
q u_{k,0} + (u_{k,j}+ u_{j,k}) u_j+ b_k
$

:$ (3.5) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle
\frac{\partial \mathcal L}{\partial u_{k,0}}= q u_{k}
$

:$(4) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle
\frac{\partial \mathcal L}{\partial u_{k,j}} = u_k u_j - \nu (u_{k,j}+u_{j,k})
$

Then, the derivation of ($3.5$) with respect to $t$ gives
:$(4.5) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle
\partial_0
\frac{\partial \mathcal L}{\partial u_{k,0}} = q u_{k,0}$
and the derivation of
of ($4$) with respect to $x_j$ gives
:$(5) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle
\partial_j
\frac{\partial \mathcal L}{\partial u_{k,j}} =
u_{k,j} u_j +
u_k u_{j,j} +
\nu (u_{k,j,j}+u_{j,k,j})
$

Assuming low
compressivity
of the fluid, the terms with $ u_{j,j}$ and $ u_{j,k,j} = u_{j,j,k}$ are neglected; then, equation ($2$) gives
:$(6) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle
q u_{k,0} + (u_{k,j}+ u_{j,k}) u_j+ b_k - q u_{k,0} - \big (u_{k,j} u_j + \nu u_{k,j,j}\big)=0
$

In (6), terms with time derivative cancel, giving the equation in the following form:
:$(7) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle
u_{j,k} u_j+ b_k - \nu u_{k,j,j} =0
$
Such an equation does not corresponds to the [[Equation of Navier-Stokes]]
<ref name="landafshitzFluidMechanics">
http://www.scribd.com/doc/8314363/Fluid-Mechanics-L-D-Landau-E-M-Lifschitz L.D.Landau. E.M.Lifshitz. Fluid Mechanics. Volume 6 of the course of Theoretical Physics, Translation from Russian by J.B.Sykes and W.H.Reid. par.15, page 46, eq.(15.7)</ref>, and such an equation cannot be used to predict the time evolution of the system.

The question how the Navier-Stokes were obtained from variational principle in
<ref name="enrico">
http://www.icatweb.org/vol7/7.3/06-scubba.pdf Enrico Sciubba. Flow Exergy as a Lagrangian for the Navier-Stokes Equations for Incompressible Flow. Int. J. Thermodynamics, ISSN 1301-9724 Vol. 7, (No. 3), pp.115-122, September-2004
</ref> arises.

Perhaps, the deviation of the measurements in the atmospheric science from the predictions by the models
may be attributed to the error of the deduction of the Navier-Stokes from variational principle; the laws of conservation (there should be of order of 10 of conserving quantities), that follow from the [[Noeter theorem]], should be revised.
==References==

<references/>

[[Copyleft]] 2011 by Dmitrii Kouznetsov. This deduction may be used for free; attribute the source.

[[Category:Physics]]
[[Category:Fluid Dynamics]]
[[Category:Articles in English]]
[[Category:stationary action]]

@@ Line 11: / Line 11: @@
 Consider the Lagrangian
-:$ (1) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle
+:\( (1) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle
 \mathcal L=u_k \left(q u_{k,0} + u_{k,j} u_j+ b_k\right)-  \frac{\nu}{2} (u_{i,j}+u_{j,i})^2
-where $u_k$ are assumed to be functions of $x_1,  x_2, x_3$ ; $k$ and $j$ and $i$ take integer values from unity to $3$; $q$ is constant.
+where \(u_k\) are assumed to be functions of \(x_1,  x_2, x_3\) ; \(k\) and \(j\) and \(i\) take integer values from unity to \(3\); \(q\) is constant.
-The additional index after comma indicates the derivative with respect to specified coordinate, time is assumed to be coordinate number zero. Summation is assumed over the repeating indices. The power two in ($1$) is also treated as repetition of indices.
+The additional index after comma indicates the derivative with respect to specified coordinate, time is assumed to be coordinate number zero. Summation is assumed over the repeating indices. The power two in (\(1\)) is also treated as repetition of indices.
 The principle of stationary action is written as follows:
-:$ (2) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle
+:\( (2) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle
 \frac{\partial \mathcal L}{\partial u_k}
 - \frac{\partial}{\partial x_j}\left( \frac{\partial \mathcal L}{\partial u_{k,j} } \right)
 - \frac{\partial}{\partial t}\left( \frac{\partial \mathcal L}{\partial u_{k,0} } \right) =0
-From expression ($1$), the derivatives of the Lagrangian can be estimated as follows:
+From expression (\(1\)), the derivatives of the Lagrangian can be estimated as follows:
-:$ (3) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle
+:\( (3) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle
 \frac{\partial \mathcal L}{\partial u_k}=q u_{k,0} + u_{k,j} u_j+ b_k ~+~ u_\ell u_{\ell,k}=
 %u_{k,0} + u_{k,j} u_j+ b_k ~+~ u_j u_{j,k}=
 q u_{k,0} + (u_{k,j}+ u_{j,k}) u_j+ b_k
-:$ (3.5) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle
+:\( (3.5) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle
 \frac{\partial \mathcal L}{\partial u_{k,0}}= q u_{k}
-:$(4) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle
+:\((4) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle
 \frac{\partial \mathcal L}{\partial u_{k,j}} = u_k u_j -  \nu (u_{k,j}+u_{j,k})
-Then, the derivation of ($3.5$) with respect to $t$ gives
+Then, the derivation of (\(3.5\)) with respect to \(t\) gives
-:$(4.5) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle
+:\((4.5) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle
 \partial_0
-\frac{\partial \mathcal L}{\partial u_{k,0}} = q u_{k,0}$
+\frac{\partial \mathcal L}{\partial u_{k,0}} = q u_{k,0}\)
 and the derivation of
-of ($4$) with respect to $x_j$ gives
+of (\(4\)) with respect to \(x_j\) gives
-:$(5) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle
+:\((5) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle
 \partial_j
 \frac{\partial \mathcal L}{\partial u_{k,j}} =
@@ Line 52: / Line 52: @@
 u_k u_{j,j} +
   \nu (u_{k,j,j}+u_{j,k,j})
 Assuming low
 compressivity
-of the fluid, the terms with $ u_{j,j}$ and $ u_{j,k,j} = u_{j,j,k}$ are neglected; then, equation ($2$) gives
+of the fluid, the terms with \( u_{j,j}\) and \( u_{j,k,j} = u_{j,j,k}\) are neglected; then, equation (\(2\)) gives
-:$(6) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle
+:\((6) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle
 q u_{k,0} + (u_{k,j}+ u_{j,k}) u_j+ b_k - q u_{k,0} - \big (u_{k,j} u_j + \nu u_{k,j,j}\big)=0
 In (6), terms with time derivative cancel, giving the equation in the following form:
-:$(7) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle
+:\((7) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle
  u_{j,k} u_j+ b_k - \nu u_{k,j,j} =0
 Such an equation does not corresponds to the [[Equation of Navier-Stokes]]
 <ref name="landafshitzFluidMechanics">

Navier-Stokes from variational principle - Revision history

T: Text replacement - "\$([^\$]+)\$" to "\\(\1\\)"

Maintenance script at 05:30, 20 June 2013