Navier–Stokes Equations - Exact Solutions of The Navier–Stokes Equations

Exact Solutions of The Navier–Stokes Equations

Some exact solutions to the Navier–Stokes equations exist. Examples of degenerate cases — with the non-linear terms in the Navier–Stokes equations equal to zero — are Poiseuille flow, Couette flow and the oscillatory Stokes boundary layer. But also more interesting examples, solutions to the full non-linear equations, exist; for example the Taylor–Green vortex. Note that the existence of these exact solutions does not imply they are stable: turbulence may develop at higher Reynolds numbers.

A two dimensional example

For example, in the case of an unbounded planar domain with two-dimensional — incompressible and stationary — flow in polar coordinates the velocity components and pressure p are:

u_r=\frac{A}{r}, \qquad u_\phi=B\left(\frac{1}{r}-r^\left(A/\nu+1\right)\right), \qquad p = -\frac{A^2+B^2}{2r^2} - \frac{2B^2 \nu r^\left(A/\nu\right)}{A} + \frac{B^2 r^\left(\frac{2A}{\nu}+2\right)}{\frac{2A}{\nu}+2},

where A and B are arbitrary constants. This solution is valid in the domain r ≥ 1 and for

In Cartesian coordinates, when the viscosity is zero, this is:

\mathbf{v}(x,y) = \frac{1}{x^2+y^2}\begin{pmatrix} Ax+By \\ Ay-Bx \end{pmatrix} , \qquad p(x,y) = -\frac{A^2+B^2}{2(x^2+y^2)}


A three dimensional example

For example, in the case of an unbounded Euclidean domain with three-dimensional — incompressible, stationary and with zero viscosity — radial flow in Cartesian coordinates the velocity vector and pressure p are:(Is there a reference?)

\mathbf{v}(x,y,z) = \frac{A}{x^2+y^2+z^2}\begin{pmatrix} x \\ y\\ z \end{pmatrix} , \qquad p(x,y,z) = -\frac{A^2}{2(x^2+y^2+z^2)}

There is a singularity at .


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