  The hydrocyclone is a tool used in different fields but mostly in mining
industries to separate minerals. The minerals are put in a liquid or in a
stable suspension of a chosen density which must be less than the density of
the heavier minerals and greater than that of the lighter minerals. Because of
the effect of the walls the flow in a hydrocyclone is the result of a movement
of confined swirling (vortex) and recirculating flow, which are generated by
the difference of the means.
  We study turbulence and bifurcation in the motion of the Hydrocyclone by
using the Navier-Stokes for the conservation of the mass and of the momentum.
These equations are numerically solved with the finite element method (FEM) and
the $k-\epsilon$ model. Our FEM results are in quite good agreement with the
experimental data. Finally, we introduce a simple 3D family of maps which
models very well the behaviour of the trajectories of the light and heavy
particles in the Hydrocyclone.
