Brinkman’s and Generalized Brinkman’s Equations and Associated Viscosities

Under Poiseuelle flow, the effective viscosity associated with Brinkman’s equation is investigated and quantified for a high-porosity porous layer whose geometric factors are given by Ergun’s equation and the Kozney-Carmen equation. An expression for the relative viscosity is derived and expressed in terms of the geometric factors. Viscous and Darcy flow limits are quantified in terms of the relative viscosity and the geometric factors. This work also initiates investigations of the effective viscosity when the Brinkman flow is through a variable permeability porous layer and when the governing equation is the generalized Brinkman equation. When permeability is variable, it is shown that the relative viscosity is always less than unity when the Brinkman pressure gradient is the same as the corresponding Navier-Stokes pressure gradient. A condition on the pressure gradients under which the relative viscosity is greater than unity is derived. When the flow is governed by the generalized Brinkman’s equation, the problem of quantifying the effective viscosity is replaced by that of determining variations in viscosity due to pressure, and variations in pressure due to changes in the porous medium parameters.