  In the first part of the paper we provide a new classification of
incompressible fluids characterized by a continuous monotone relation between
the velocity gradient and the Cauchy stress. The considered class includes
Euler fluids, Navier-Stokes fluids, classical power-law fluids as well as
stress power-law fluids, and their various generalizations including the fluids
that we refer to as activated fluids, namely fluids that behave as an Euler
fluid prior activation and behave as a viscous fluid once activation takes
place. We also present a classification concerning boundary conditions that are
viewed as the constitutive relations on the boundary. In the second part of the
paper, we develop a robust mathematical theory for activated Euler fluids
associated with different types of the boundary conditions ranging from no-slip
to freeslip and include Navier's slip as well as stick-slip. Both steady and
unsteady flows of such fluids in three-dimensional domains are analyzed.
