Symmetric flows and Darcy's law in Curved Spaces

TL;DR

This paper investigates symmetric solutions of the Stokes equation on curved three-dimensional manifolds, deriving conditions for their existence and generalizing Darcy's law for fluid flow in curved environments using differential forms.

Contribution

It provides necessary and sufficient conditions for symmetric unidirectional flows on curved spaces and extends Darcy's law to spaces with constant curvature.

Findings

Conditions for existence of symmetric solutions identified

Generalized Darcy's law for curved spaces derived

Differential forms used for elegant formulation

Abstract

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.