Flow through porous media: a momentum tracer approach

The flow of a fluid through a spatially periodic porous medium is studied using the momentum tracer approach, that is by monitoring the temporal spread of an instantaneous source of momentum (the “momentum tracer”), following its initial introduction into the interstitial fluid. The fluid flow is described macroscopically by a Brinkman-like effective equation, containing three effective tensorial quantities, namely the permeability dyadic, the generalized viscosity tetradic and a coupling triadic term. These phenomenological coefficients are shown to be expressible in terms of the solution of the characteristic eigenvalue problem associated with the (tensorial) Stokes operator, defined within the interstitial region of a single unit cell of the periodic porous medium. The phenomenological coefficients are shown to be independent of the initial position of the momentum tracer as well as of the choice of weight function used in averaging the microscale momentum density over the unit cell. Finally, we show that the analytical expressions for the phenomenological coefficients identically satisfy the symmetry requirements that can be obtained by applying the reciprocal theorem at the macro-scale level.