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Solidification of pressure-driven flow in
a finite rigid channel with application to volcanic eruptions

J. R. Lister and P. J. Dellar (1996) Solidification of pressure-driven flow in
a finite rigid channel with application to volcanic eruptions J.
Fluid Mech. 323 267-283

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#### Abstract

Competition between conductive cooling and advective heating occurs
whenever hot fluid invades a cold environment. Here the solidification
of hot viscous flow driven by a fixed pressure drop through an
initially planar or cylindrical channel embedded in a cold rigid solid
is analysed. At early times, or far from the channel entrance, the flow
starts to solidify and block the channel, thus reducing the flow rate.
Close to the channel entrance, and at later times, the supply of new
hot fluid starts to melt back the initial chill. Eventually, either
solidification or meltback becomes dominant throughout the channel, and
flow either ceases or continues until the source is exhausted. The
evolution of the dimensionless system, which is characterized by the
initial Peclet number Pe, the Stefan number S and the dimensionless
solidification temperature Theta, is calculated numerically and by a
variety of asymptotic schemes. The results show the importance of
variations along the channel and caution against models based on a
single 'representative' width. The critical Peclet number Pe(c), which
marks the boundary between eventual solidification and eventual
meltback, is determined for a wide range of parameters and found to be
much larger for cylindrical channels than for planar channels, owing to
the slower rate of decay of the heat flux into the solid in a
cylindrical geometry. For a planar channel Pe(c) is given by the simple
algebraic result Pe(c) similar to 0.46[Theta(2)/(1 - Theta)(S +
2/pi)](3) when (1 - Theta)(-1) much greater than S much greater than 1,
but in general it requires numerical solution. Similar analyses, in
which there is a spatially varying and time-dependent interaction
between the rates of solidification and flow, have a range of
applications to geological and industrial processes.

@Article{ListerDellar96,

author = {John R.
Lister and Paul J. Dellar},

title =
{Solidification of pressure-driven flow in a finite rigid channel with
application to volcanic eruptions},

journal = {J. Fluid
Mech.},

year =
{1996},

volume = {232},

pages =
{267--384},

DOI =
{doi:10.1017/S0022112096000912}

}