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.
