This is a special case of
multiphase flow, in which one or more fluid phase flows through a
permeable matrix. In many cases, the matrix itself is deformable, for
example when water is squeezed out of a sponge. This leads to
mathematically interesting problems, in which the motions of the
and solid phases are coupled through the mutual drag force that each
exerts on the other. We are involved with many applications of these
ideas, including the following.
The rock that constitutes the Earth's crust is essentially a porous
medium that deforms over geological timescales.
The flow through, and erosion of, this medium by magma leads to such
phenomena as layered magma chambers and volcanic eruptions.
The flow of groundwater through soil and/or rock has important
applications in agriculture and in pollution control.
Other topics of interest include compaction of sedimentary basins and
the phenomenon of frost heave, which occurs when groundwater freezes.
As well as damaging roads and pavements, frost heave is responsible
for geological formations like the stone garland shown opposite.
One of the original motivations for studying porous-medium flow was
the extraction of oil from porous rock.
It was found that suction of viscous fluid from a porous medium
is often unstable, tending to leave behind a sizeable
proportion of the oil in small isolated packets, and increasing the
extracted fraction is a constant challenge. Sometimes solvents are used to enhance recovery.
Another important issue for the oil industry is
from locally measured properties (e.g. permeability) and
laboratory experiments, so that useful predictions can be made over
the much longer scales (hundreds of metres) relevant to an oil well.
Most of the tissues in the body (e.g. bone, cartilage, muscle)
are deformable porous media. The proper functioning of such materials
depends crucially on the flow of blood, nutrients and so forth
Porous-medium models are used to understand various medical
and treatments (such as injections).
H. Ockendon & E. L. Terrill,
1993 A mathematical model for the wet-spinning process,
Euro. J. Appl. Math. 4, 341-360.
A. C. Fowler & W. B. Krantz,
1994 A generalized secondary frost heave model,
SIAM J. Appl. Math. 54 1650-1675.
A. C. Fowler & X. S. Yang,
1998 Fast and slow compaction in sedimentary basins,
SIAM J. Appl. Math. 59, 365-385.
A. D. Fitt, P. D. Howell, J. R. King, C. P. Please &
D. W. Schwendeman,
2002 Multiphase flow in a roll press nip,
Euro. Jnl Appl. Maths 13, 225-259.
C. J. W. Breward, H. M. Byrne & C. E. Lewis,
2002 The role of cell-cell interactions in a two-phase model for
avascular tumour growth,
J. Math. Biol. 45(2), 125-152.
People working in this area within OCIAM