Navier-Stokes from variational principle

WARNING: this article has a serious (perhaps, unrecoverable) error in formulas. 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 .

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 , 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 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

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