Scaling Theory of Hydrodynamic Singularities
Jens Eggers
University of Essen
The fundamental assumption underlying the equations of fluid
mechanics is that hydrodynamic fields vary over scales much larger than
those set by the size of molecules. Most of applied mathematical
research thus aims at showing that solutions remain smooth.
However a great variety of physically important problems such
as drop formation, drop coalescence, or the entrainment of air
bubbles below a fluid surface are characterized by finite-time
singularities of the hydrodynamic equations. This may seem like
an embarrassment of continuum modeling, but we show that on the
contrary the mathematical structure of singularities accurately
describes the important features of those flows. This includes the
surprising universality of singular solutions, and their simple
self-similar structure, which often allows one to arrive at explicit
solutions. The known structure of solutions is used to continue
the hydrodynamic equations across singularities.
We also stress experimental tests of our results and indicate
industrial applications.