On the effective viscosity for the Darcy–Brinkman equation

Up-scaling of the Stokes equations with non-slip boundary condition describing the flow in a porous medium, leads to the Darcy–Brinkman equationɛβμβvD,β=-Kβ·(∇Pm,β-ρβg)+Kβ·μβ∇2vD,β.ɛβμβvD,β=-Kβ·(∇Pm,β-ρβg)+Kβ·μβ∇2vD,β.The second-order term -μβ∇2vD,β-μβ∇2vD,β recovers the viscous drag effects and uses the fluid viscosity coefficient. However, experimental measurements and computer simulation results have suggested that the Darcy–Brinkman equation should incorporate an effective viscosity:ɛβμβvD,β=-Kβ·(∇Pm,β-ρβg)+Kβ·∇(μβ,eff∇vD,β).ɛβμβvD,β=-Kβ·(∇Pm,β-ρβg)+Kβ·∇(μβ,eff∇vD,β).To the best of our knowledge, a theoretical back-up for the existence of an effective viscosity for the Stokes flow within a porous medium, has not been provided yet. This work focuses in this issue and shows that the use of a slip boundary condition is required to obtain an effective viscosity different from the one corresponding to the fluid phase. This is done by means of an up-scaling procedure based on volume averaging methods [S. Whitaker, The Method of Volume Averaging, Kluwer Academic Publisher, Amsterdam, 1999], which provides a boundary-value problem to compute the underlying effective viscosity. By imposing certain values of a slip coefficient γγ, the effect of the slip boundary condition on the superficial average velocity is provided as a function of porosity. Our calculations show a non-monotonous dependence for γ⩽1γ⩽1, while for γ⪢1γ⪢1 the average velocity tends to the one obtained by imposing non-slip conditions.