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      denotes the gradient operator with respect to the dimensionless spatial variable .

Exercise 2.9 Substitute these operators into the Navier–Stokes equation (2.15) and simplify to get the dimensionless Navier–Stokes equation:

(2.16)

      where .

The dimensionless parameter in Eq. (2.16) is the Reynolds number, named after Irish‐born fluid mechanician Osborne Reynolds [128]. This number serves as a unit‐free gauge of the ratio of inertial effects to viscous effects and, heuristically, as an index of mathematical intractability. We associate the regime with slow flows in which viscous effects dominate those associated with inertia. When is much smaller than 1, it is common to neglect the inertial terms.

2.4 Two Classic Problems in Fluid Mechanics

As mentioned in Section 2.3, the Navier–Stokes equation (2.15) poses formidable mathematical challenges. Exact solutions are known only in special geometries and only under highly restrictive assumptions, many of which allow us to neglect the nonlinear inertial term . We now examine two such problems in fluid mechanics that bear on the analysis of flows in porous media. Each serves as a highly simplified model of fluid flow in the interstices of a porous medium, and each involves significant reductions in complexity compared with the full Navier–Stokes equation. The Hagen–Poiseuille problem is a simple model of fluid flows in a straight, cylindrical tube, which one can envision as an idealized pore channel. The Stokes problem models slow, viscous flows around a solid sphere, which one can imagine as an idealized solid grain.

      2.4.1 Hagen–Poiseuille Flow

      One of the earliest known exact solutions to the Navier–Stokes equation arose from a simple but important model examined by Gotthilf Hagen, a German fluid mechanician, and French physicist J.L.M. Poiseuille, mentioned in Section 2.3. Citing Hagen's 1839 work [67], in 1840, Poiseuille [122] developed a classic solution for flow through a pipe. The derivation presented here follows that given by British mathematician G.K. Batchelor [16, Section 4.2].

      Consider steady flow in a thin, horizontal, cylindrical tube having circular cross‐section and radius . Let the fluid's density and viscosity be constant. Orient the Cartesian coordinate system so that the ‐axis coincides with the axis of the tube.

      The problem simplifies if we temporarily convert to cylindrical coordinates, defined by the coordinate transformation

      (B.5)

      reviewed in Appendix B. Here represents position along the axis of the tube, represents distance from the axis, and

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