Viscosity renormalization in the Brinkman equation

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Abstract
The Brinkman equation purports to describe low‐Reynolds‐number flow in porous media in situations where velocity gradients are non‐negligible. The equation involves modifying the usual Darcy law by the addition of a standard viscosity term whose coefficient is usually identified with the pure‐fluid viscosity. It is argued instead that the porous medium induces a renormalization of viscosity, which is calculated in the dilute limit and separately in a self‐consistent approximation. The effective Brinkman viscosity is found to decrease from the pore‐fluid value. The calculation fails at low porosity but agrees at least in part with experiment. In addition, the relationship between the Brinkman equation and the phenomenological boundary condition of Beavers and Joseph is discussed and it is pointed out that their experimental configuration provides a simple means of measuring viscosity renormalization.