Symmetric flows and Darcy's law in Curved Spaces

We consider the problem of existence of certain symmetrical solutions of Stokes equation on a three-dimensional manifold $M$ with a general metric possessing symmetry. These solutions correspond to unidirectional flows. We have been able to determine necessary and sufficient conditions for their existence. Symmetric unidirectional flows are fundamental for deducing the so-called Darcy's law, which is the law governing fluid flow in a Hele-Shaw cell embedded in the environment $M$. Our main interest is to depart from the usual, flat background environment, and consider the possibility of an environment of arbitrary constant curvature $K$ in which a cell is embedded. We generalize Darcy's law for particular models of such spaces obtained from $\real^3$ with a conformal metric. We employ the calculus of differential forms for a simpler and more elegant approach to the problems herein discussed.