Modelling the flow of a second order fluid through and over a porous medium using the volume averages. I. The generalized Brinkman's equation

In this paper, the generalized Brinkman's equation for a viscoelastic fluid is derived using the volume averages. Darcy's generalised equation is consequently obtained neglecting the first and the second Brinkman's correction with respect to the drag term. The latter differs from the Newtonian drag because of an additional term quadratic in the velocity and inversely proportional to a "viscoelastic" permeability defined in the paper. The viscoelastic permeability tensor can be calculated by solving a boundary value problem, but it must be in fact experimentally measured. To isolate the elastic contribution, the constitutive equation of the second order fluid of Coleman and Noll is chosen because, in simple shear at steady state, second order fluids show a constant viscosity and first and second normal stress differences quadratic in the shear rate. The model predictions are compared with data of the literature obtained in a Darcy's experiment and the agreement is good.