  In this paper, we deal with the controllability properties of a system of $m$
coupled Stokes systems or $m$ coupled Navier-Stokes systems. We show the
null-controllability of such systems in the case where the coupling is in a
cascade form and when the control acts only on one of the systems. Moreover, we
impose that this control has a vanishing component so that we control a
$m\times N$ state (corresponding to the velocities of the fluids) by $N-1$
distributed scalar controls. The proof of the controllability of the coupled
Stokes system is based on a Carleman estimate for the adjoint system. The local
null-controllability of the coupled Navier-Stokes systems is then obtained by
means of the source term method and a Banach fixed point.
