Navier-Stokes equation

Navier-Stokes equation is system of equations for the vector of velosity of some isotropic fluid . In the simple case, it can be formulated as follows:
 * \( \rho \, \Big(\dot{\vec v}+ (\vec v \cdot \nabla) \vec v \Big)=

-\nabla p + \mu \nabla^2 \vec v + \vec f\)
 * \( \dot \rho + \nabla (\rho \vec v)=0\)
 * \( p=P(\rho)\)

Functions \(P\) and \(\vec f\!=\!\vec f(x)\) are supposed to be given; \(\nabla\) differentiates with respect to coordinates \(x\); \(\mu\) is real constant.

Cylindric coordinates
In the cylindric coordinates, the equation can be written as follows:

\rho \left(\frac{\partial u_r}{\partial t} + u_r \frac{\partial u_r}{\partial r} + \frac{u_{\phi}}{r} \frac{\partial u_r}{\partial \phi} + u_z \frac{\partial u_r}{\partial z} - \frac{u_{\phi}^2}{r}\right) = -\frac{\partial p}{\partial r} + \mu \left[\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_r}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 u_r}{\partial \phi^2} + \frac{\partial^2 u_r}{\partial z^2}-\frac{u_r}{r^2}-\frac{2}{r^2}\frac{\partial u_\phi}{\partial \phi} \right] + \rho g_r\)

\rho \left(\frac{\partial u_{\phi}}{\partial t} + u_r \frac{\partial u_{\phi}}{\partial r} + \frac{u_{\phi}}{r} \frac{\partial u_{\phi}}{\partial \phi} + u_z \frac{\partial u_{\phi}}{\partial z} + \frac{u_r u_{\phi}}{r}\right) = -\frac{1}{r}\frac{\partial p}{\partial \phi} + \mu \left[\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_{\phi}}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 u_{\phi}}{\partial \phi^2} + \frac{\partial^2 u_{\phi}}{\partial z^2} + \frac{2}{r^2}\frac{\partial u_r}{\partial \phi} - \frac{u_{\phi}}{r^2}\right] + \rho g_{\phi}\)

\rho \left(\frac{\partial u_z}{\partial t} + u_r \frac{\partial u_z}{\partial r} + \frac{u_{\phi}}{r} \frac{\partial u_z}{\partial \phi} + u_z \frac{\partial u_z}{\partial z}\right) = -\frac{\partial p}{\partial z} + \mu \left[\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_z}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 u_z}{\partial \phi^2} + \frac{\partial^2 u_z}{\partial z^2}\right] + \rho g_z\)

Polar coordinates
In the bi-dimensional case, the cylindric coordinates become the polar ones;

\rho \left(\frac{\partial u_r}{\partial t} + u_r \frac{\partial u_r}{\partial r} + \frac{u_{\phi}}{r} \frac{\partial u_r}{\partial \phi} - \frac{u_{\phi}^2}{r}\right) = -\frac{\partial p}{\partial r} + \mu \left[\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_r}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 u_r}{\partial \phi^2} -\frac{u_r}{r^2}-\frac{2}{r^2}\frac{\partial u_\phi}{\partial \phi} \right] + \rho g_r\)

\rho \left(\frac{\partial u_{\phi}}{\partial t} + u_r \frac{\partial u_{\phi}}{\partial r} + \frac{u_{\phi}}{r} \frac{\partial u_{\phi}}{\partial \phi} + \frac{u_r u_{\phi}}{r}\right) = -\frac{1}{r}\frac{\partial p}{\partial \phi} + \mu \left[\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_{\phi}}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 u_{\phi}}{\partial \phi^2} + \frac{2}{r^2}\frac{\partial u_r}{\partial \phi} - \frac{u_{\phi}}{r^2}\right] + \rho g_{\phi}\)

Case of symmetric solenoidal flow
For the symmetric solenoidal flow, \(v_r\) and all the derivatives with respect to \(\phi\) can be neglected; this leads to

\rho \frac{u_{\phi}^2}{r} = \frac{\partial p}{\partial r} \)

\rho \frac{\partial u_{\phi}}{\partial t} = \mu \frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_{\phi}}{\partial r}\right) - \mu \frac{u_\phi}{r^2}\)

These equations can be used to construct the quasi-stationary idealization of curl: \(\partial p/\partial r=\frac{\rho}{r} u_\phi^2\), and the relaxation rate is determined by the

\frac{\partial u_{\phi}}{\partial t} = \nu \frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_{\phi}}{\partial r}\right) - \nu \frac{u_\phi}{r^2}\) where \(~\nu=\mu/\rho~\) is kinematic viscosity. For the air, Landafshitz suggest the estimete \(\nu\approx 0.15~ \rm cm^2/sec\) while the density of air \(\rho \approx 1.2~ \rm kg/m^3\).

The last equation determines the decay of the curl in the approximation without turbulence. The angular momentum conserves; therefore, while the curl is alone, the only decay allowed is the spreading.

Exact solutions
The exact solutions for the Navier-Stokes equation are available for some symmetric cases. Such solutions will be collected in the special article Analytic solutions of the Navier-Stokes equation.

Numerical solutions
The hundreds programmers work on the efficient numerical approximation for the solutions of the Navier-Stokes equation during a half-century; Some commercial packets are available. For the beginning of century 21, the achievements are modest. For example, the efficient approximation of the Serega function that describe the shape of the breaking of the idealized oceanic wave (see the picture at the top) is not loaded.

There is special wiki dedicated to the numerical analysis of the equation of Navier-Stokes , that offers the collection of descriptions of the numerous software for the semi-authomatic approximation of its solutions.

Navier-Stokes from the variational principle
The Navier-Stokes equations follow from the principle of the stationary action for the Lagrangian
 * \( \displaystyle \mathcal L=u_k (u_{k,0} + u_{k,j} u_j+ b_k)- \frac{\nu}{2} (u_{i,j}+u_{j,i})^2 \)

under assumption, that variation of velocities does not affect variation of the accelerations .

Such a deduction is loaded as Navier-Stokes from variational principle. The deduction is, softly speaking, doubtful; so, the concept needs confirmation. In particular, the efficient approximations of the solutions could follow from such a strange variational principle.