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Magma flow in volcanic conduits involves complicated physico-chemical transformations during ascent, including gas exsolution, bubble nucleation and growth, gas escape from the magma, and magma fragmentation (in the case of explosiveeruptions). These changes are accompanied by major changes in the rheological properties of magma. The structure of the flow can change from homogeneous liquid flow at depth to gas-particle dispersion flow in the upper part of a conduit.There are two distinct zones of the flow: the zone where the liquid is a continuous phase and flow is mainly controlled by viscous resistance, and the zone with continuous gas phase where the flow is dominated by inertia. These zones are separated by a fragmentation front, whose position must be determined during the solution of the flow dynamics. This makes modelling of conduit flows a difficult problem and requires strict constraints on the accuracy and stability of the numerical method.

The literature on the modelling of conduit flow processes during explosive eruptions contains many tens of papers.Good overviews include those by Woods (1995), Sparks et al. (1997), Papale (1998), Melnik (2000), Slezin (2003), Sahagian (2005), Mader (2006). Most of the models presented in the literature describe conduit flow during volcanic eruptions as a 1D steady-state process based on the assumptions that the length of the conduit is much largerthan its radii and the time scale of parameter variations is much longer than the residence time of an individual

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