  We construct supergravity backgrounds representing non-homogeneous
compactifications of d=10,11 supergravities to four dimensions, which cannot be
written as a direct product. The geometries are regular and approach
$AdS_7\times S^4$ or $AdS_5\times S^5$ at infinity; they are generically
non-supersymmetric, except in a certain "extremal" limit, where a Bogomol'nyi
bound is saturated and a naked singularity appears. By using these spaces, one
can construct a model of QCD that generalizes by one (or two) extra parameters
a recently proposed model of QCD based on the non-extremal D4 brane. This
allows for some extra freedom to circumvent some (but not all) limitations of
the simplest version.
