  We formulate the thin-film hydrodynamics of a suspension of polar self-driven
particles and show that it is prone to several instabilities through the
interplay of activity, polarity and the existence of a free surface. Our
approach extends, to self-propelling systems, the work of Ben Amar and Cummings
[Phys Fluids 13 (2001) 1160] on thin-film nematics. Based on our estimates the
instabilities should be seen in bacterial suspensions and the lamellipodium,
and are potentially relevant to the morphology of biofilms. We suggest several
experimental tests of our theory.
