Chapter 14

Convection in a porous medium

Before analysing porous medium convection, which represents perhaps the simplest paradigm of a spatially distributed system with instability, there is a discursive prologue which considers first linear stability, and then nonlinear stability (via multiple scales), for a general nonlinear evolution equation.

The Boussinesq approximation for porous medium convection, and linear stability analysis (for impermeable boundary conditions) and nonlinear stability analysis are both done explicitly.

Lastly, a boundary layer analysis is given for high Rayleigh number convection. This is an interesting problem, and relates to other studies involving high Rayleigh number convection which I have studied in the past, particularly when the viscosity is a strongly variable function of temperature and possibly pressure; the application is to convection in the Earth's mantle. See papers 30 and 65 in my c.v. What is interesting about the porous medium problem is that no definitive theory seems to exist. Experiments, both laboratory and numerical, are unclear, and there are different asymptotic theories. I haven't pursued whether anybody has cleaned this up in the last 10 years, and if anybody out there knows more than I do, I'd be pleased to hear about it.

Question 6 of the exercises is about the Welander loop, which is a simple model for the convective circulation in a back boiler. When I was installing a back boiler in my own home about 10 years ago, I plumbed in one of the upward sections with a downward (reverse) slope. Sure enough, when I lit the fire, geysering occurred: the water in the pipes remained stationary until boiling occurred at the boiler, and the resulting creation of suddenly much lighter steam/water mixture caused a rapid convective surge, accompanied by much gurgling, at the end of which cold fluid came to the boiler, the flow ceased, and the cycle resumed. I replumbed the offending section of pipe, and the boiler has been working happily since. Geysering is discussed briefly on page 232, and again (in the chapter on two-phase flow) on page 284. I didn't seem to give any external reference to it, but it is discussed in Turcotte and Schubert's book on page 417.