Many liquids are hard to compress (i.e., their volumes or densities don't
change much when you squeeze them), so that the density 
is
essentially a constant. For such an incompressible fluid the equation of continuity simplifies to
so that the velocity field 
is a solenoidal vector. This
condition is analogous to the condition 
which we
encounter in electromagnetism--the magnetic field has zero divergence.
Incompressible flows are much easier to treat theoretically than
compressible flows, so this is a very useful simplifying assumption. You
might worry about applying it to gases, but it turns out that even there it
often works well. In fact, if U is some characteristic flow velocity, and c is the speed of sound in the fluid, then the fluid can be treated as
incompressible whenever 
. This is often expressed in terms of the
Mach number Ma, defined as
then if 
the fluid can be treated as incompressible. For
instance, the speed of sound in air is 330 m/s (740 mi/h); therefore, fluid
flow around people, cars, trains, as well as fluid flow in hurricanes and
tornadoes, all satisfy the incompressibility condition. Air flow around
commercial airliners ( 
) violates this condition, as does
supersonic flight.
