[prolyxis]

A simple example of a function with a singularity is f(t)=1/t. Note that at t=0, f(t) is undefined due to division by zero. On either side of zero, the absolute value of f(t) approaches infinity.

In this case, we are tracking the flow of an incompressible fluid over time. This flow is represented by a velocity field evolving over time, under the constraint of no net inflow/outflow of material into any region of space. Thus, the singularity corresponds to a portion of fluid speeding up and approaching an infinite speed as you approach some finite time.

Because the fluid cannot be compressed, the only way the singularity can be produced is for a portion of the liquid to swirl, increasingly rapidly, about some point: hence the discussion in the article about vorticity.

As isoprophlex pointed out, this undefined value of the velocity field prevents you from (or at least complicates) computing the further evolution of the fluid.

[nico]

Thank you for the great explanation.

Do these swirls shed energy? Is it considered in these equations that for example friction within the swirls would slow them down (and hence not reach a singularity)?

[pas]

In real fluids yes, absolutely, they basically transform/branch/divide/split into smaller and smaller scale vortices and then those dissipate the energy into the fluid (heating it up a bit).