Exact solutions of Navier-Stokes

Term exact solutions of Navier-Stokes refers to the special cases when the solution of the Navier-Stokes equation can be expressed in term of special functions. In principle, any function can be declared as "special", if its specific properties are described and the efficient algorithm of the evaluation is demonstrated.

Equation

In the simple case, the Navier-Stokes equation can be written in the following form:

\((1) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle u_{k,0} + u_j u_{k,j}+ b_k - \nu u_{k,j,j}=0 \)

where \(u_k\) refers to the \(k\)th component of its velocity, and number comma in the subscript indicates the derivative, and \(b_k\) is \(k\)th component of the pressure, and \(\nu\) is constant, that has sense of the kinematic viscosity]].

In principle, one can fake any smooth distribution of velocities, calculate the corresponding \(b\) and claim that the result is exact solution of (1). However, such a solution is of interest if and only if the resulting \(b\) may correspond to some conditions, realizable in the case of flow of some substance, at least asymptotically. In order to reduce variety of the non-realizable solution, the additional requirement on the divergence is useful; let

\((2) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle u_{k,k} + D_{,0}=0 \)

where \(D\) is density of the fluid. This density can be interpreted as zeroth component of the velocity. Then, the pressure \(p=P(D)\) may be interpreted as some given function of density, and the external force \(b\) may be expressed as function of the pressure gradient,

\((3) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle b_k=D_{,k}\)

However, \(b\) can be interpreted also in terms of the spatial derivative of the zeroth component of velosity, \((4) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle b_k=P(u_0)_{,k}\)

In the simplest case, one assume the compressivity zero; then, instead of (2), we have

\((5) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle u_{k,k}=0 \)

and the equation for pressure

\((6) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle P_{,j,j}=0 \)

In such a way, for the non-compressible fluid, the term exact solution of the Navier-Stokes refers to the non-trivial combonation of functions \(\vec u=\vec u(\vec x)\) and \(p=p(\vec x)\) such that

\((7) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle u_{k,0} + u_j u_{k,j}+ p_{,k} - \nu u_{k,j,j}=0 \)

and

\((8) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle p_{,j,j}=0 \)

Equation (8) is known also as the Laplace equation and also should be satisfied for the "exact solution".

The trivial "exact solution" of system (7),(8) is \(u\!=\!0\), \(p\!=\!\mathrm{const}\).

But there exist also another solutions considered in the following sections. Usually, they include the trivial solution as special case.

Laminar flow with gradient of velocity

The liquid between the pair of parallel moving plates refers to the solution

\((20) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \displaystyle p\!=\!0 ~\), \(~u_1\!=\!x_2~\), \(~u_2\!=\!u_3\!=0\)

Similar solutions can be obtained using the group of transformations of pressure and velocities that keep the action invariant, see article Navier-Stokes from variational principle.

The solution (20) is very important; locally, at the scale with small Reynolds number, every solution of the Navier-Stokes should behave as scaled and rotated image of equation (20).

Laminar flow in a tube with gradient of pressure

In analogy with (20), the solution for the laminar flow in a tube at the constant gradient of pressure can be written.

Taylor-Green vortex

There is special article Taylor–Green vortex. Some content from there may be copypasted below.

Wide wortex

For the rotating aperiodic solution see wide vortex. Some part of that article should be repeated below.

Transfer of momentum

The turbulent transfer of momentum is determined mainly by the correlator \(\langle u_1 u_2 \rangle\) and its rotations, where the mean value is assumed with respect to space. The solutions above do not provide such correlation. The simplest solution that would allow the correlation and the corresponding transfer of the momentum is highly required.