I have an equation for a non-viscous compressible fluid with density, pressure and velocity given by:

$$ \begin{align} \rho(x, y, z) &= \frac{3B}{a^2 + x^2 + y^2 + z^2} \\ p(x, y, z) &= \frac{-A^2B}{\left(a^2 + x^2 + y^2 + z^2\right)^3} \\ \mathbf{u}(x, y, z) &= \frac{A}{\left(a^2 + x^2 + y^2 + z^2\right)^2}\begin{pmatrix} 2(-ay + xz) \\ 2(ax + yz) \\ a^2 - x^2 - y^2 + z^2 \end{pmatrix} \\ g &= 0 \\ \mu &= 0 \end{align}$$

which satisfies the compressible Navier Stokes equations. (For some constants, $a , A, B$).

$$\rho (\mathbf{u}.\nabla\mathbf{u}) +\nabla p = 0$$

The flow lines form a Hopf fibration.

I'm looking for a similar solution for an incompressible fluid which follows the same flow lines. Are there any tricks to turn a compressible solution into an incompressible solution? (Which is the same except $\rho(x,y,z)$=constant.) Such as varying the velocity on the flow lines to even out the density?Or to prove that such a solution can't exist?

I would imagine that if the fluid moved faster in places where the flow lines come together that would decrease the density. But at the same time, the density here goes to zero at infinity. Whereas we would need the density to go to a finite constant at infinity?


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