  In this paper, we analyze a scheme for the time-dependent variable density
Navier-Stokes equations. The algorithm is implicit in time, and the space
approximation is based on a low-order staggered non-conforming finite element,
the so-called Rannacher-Turek element. The convection term in the momentum
balance equation is discretized by a finite volume technique, in such a way
that a solution obeys a discrete kinetic energy balance, and the mass balance
is approximated by an upwind finite volume method. We first show that the
scheme preserves the stability properties of the continuous problem (L
$\infty$-estimate for the density, L $\infty$ (L 2)-and L 2 (H 1)-estimates for
the velocity), which yields, by a topological degree technique, the existence
of a solution. Then, invoking compactness arguments and passing to the limit in
the scheme, we prove that any sequence of solutions (obtained with a sequence
of discretizations the space and time step of which tend to zero) converges up
to the extraction of a subsequence to a weak solution of the continuous
problem.
