## 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.