  We study the geometry of foliated non-Lorentzian spacetimes in terms of the
Godbillon-Vey class of the foliation. We relate the intrinsic torsion of a
foliated Aristotelian manifold to its Godbillon-Vey class, and interpret it as
a measure of the local spin of the spatial leaves in the time direction. With
this characterisation, the Godbillon-Vey class is an obstruction to
integrability of the $\mathsf{G}$-structure defining the Aristotelian
spacetime. We use these notions to formulate a new geometric approach to
hydrodynamics of fluid flows by endowing them with Aristotelian structures. We
establish conditions under which the Godbillon-Vey class represents an
obstruction to steady flow of the fluid and prove new conservation laws.
