(
2010-10-23)
Holmes-Cerfon, Miranda C.; Whitehead, John A.
Previous works have shown that when liquid flows in a pipe whose boundary
temperature is below freezing, a tubular drainage conduit forms surrounded by solidified
material that freezes shut under the appropriate combination of forcing conditions. We
conduct laboratory experiments with wax in which the tube freezes shut below a certain
value of flux from a pump. As the flux is gradually decreased to this value, the total
pressure drop across the length of the tube first decreases to a minimum value and then
rises before freezing. Previous theoretical models of a tube driven by a constant pressure
drop suggest that once the pressure minimum is reached, the states for a lower flux
should be unstable and the tube should therefore freeze up. In our experiments, flux and
pressure drop were coupled, and this motivates us to extend the theory for low-Reynolds
number flow through a tube with solidification to incorporate a simple pressure drop-flux
relationship. Our model predicts a steady-state relationship between flux and pressure
drop that has a minimum of the pressure as the flux is varied. The stability properties of
these steady states depend on the boundary conditions: for a fixed flux, they are all stable,
whereas for fixed pressure drop, only those with a flux larger than that at the pressure
drop minimum are stable. For a mixed pressure-flux condition, the stability threshold of
the steady states lies between these two end members. This provides a possible
mechanism for the experimental observations.
