next up previous
Next: Streamlines and streamtubes Up: The equation of continuity Previous: The equation of continuity

Incompressible flow

Many liquids are hard to compress (i.e., their volumes or densities don't change much when you squeeze them), so that the density tex2html_wrap_inline913 is essentially a constant. For such an incompressible fluid the equation of continuity simplifies to


so that the velocity field tex2html_wrap_inline887 is a solenoidal vector. This condition is analogous to the condition tex2html_wrap_inline917 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 tex2html_wrap_inline923 . This is often expressed in terms of the Mach number Ma, defined as


then if tex2html_wrap_inline925 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 ( tex2html_wrap_inline927 ) violates this condition, as does supersonic flight.

Vittorio Celli
Mon Aug 11 22:46:35 EDT 1997