  A description of the dynamics of a collisionless, self-gravitating fluid is
developed and applied to follow the development of Large Scale Structures in
the Universe. Such description takes on one of the assumptions of the Adhesion
Approximation (AA) model, i.e., the introduction of an artificial viscosity
$\nu$ in the Euler equation, but extends it to deeper non-linear stages, where
the extrapolation of the linear relation $\psi=-\phi$ between the velocity and
the gravitational potentials -- at the basis of both the Zel'dovich and the
Adhesion models -- is no longer valid. This is achieved by expanding the
relation bewteen the gravitational potential $\phi$ and the velocity potential
$\psi$, in powers of the small viscosity $\nu$. When the results are compared
to the outcomes of an N-body simulation, the proposed extension of the Adhesion
approach provides a better description of small-scale, deeply non-linear
regions, as it is quantitatively shown by the computation of some statistical
indicators. At larger scales, the satisfactory description of the large scale
texture and of the voids given by the canonical Adhesion Approximation is
preserved in the extended model.
