Myron B. Allen, III - The Mathematics of Fluid Flow Through Porous Media

By the chain rule, for any sufficiently differentiable function ,

Figure 2.9 Geometry of the Stokes problem for slow fluid flow around a solid sphere.

Here,

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.

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 problemis a simple model of fluid flows in a straight, cylindrical tube, which one can envision as an idealized pore channel. The Stokes problemmodels slow, viscous flows around a solid sphere, which one can imagine as an idealized solid grain.

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 the angle represents the azimuth about the axis. In this coordinate system, the Laplace operator has the form

(B.7)

Appendix Breviews the derivation of this expression.

In view of the symmetry of the problem about the axis of the tube, we seek solutions of the form

(2.17)

that is, the axial component depends only on distance from the axis of the tube, and the radial and azimuthal coordinates of the velocity vanish, as drawn in Figure 2.10. We allow the pressure to vary with .

Exercise 2.10 Show that, under these conditions, the nonlinear term vanishes. Work in Cartesian coordinates .

Figure 2.10 Profile of flow through a thin circular cylinder having radius .

Since the flow is steady, the Navier–Stokes equation therefore reduces to

(Here we use the dimensionless form.)