Nonlinear equation governing flow in a saturated porous medium

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Abstract
It is argued that the appropriate generalization of Darcy's law when inertia effects are included takes the form ∇p = −(μ/k) V − (ρc/k½)|V|V, div V = 0, where k is the permeability of the medium and the ‘form drag constant’ c is a coefficient which is independent of the pressure p, the seepage velocity V, and the density ρ and viscosity μ of the fluid but which is dependent on the geometry of the medium. We formulate a nonlinear extension of Brinkman's self‐consistent theory for the flow of a viscous fluid through a swarm of spherical particles. We equate the drag per unit volume given by the right hand side of the first of the above equations to the total drag ND on the N particles contained within that unit volume, in an infinite region Ω, where D is the drag on a single particle placed in a velocity field v subject to ρ(v · ∇)v + grad p = μ∇2v−μ/k v − (cρ/k½)|v|v, div v = 0, v|∂Ω is a prescribed constant, where μ is the viscosity. Without solving these equations, we obtain an estimate for c from the known experimental drag law for a solid sphere placed in a uniform stream.