# Lecture 3 Governing equations for multiphase flows. Continuum hypothesis. Fragmentation mechanisms. Models of conduit flows during explosive eruptions.

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Lecture 3 Governing equations for multiphase flows. Continuum hypothesis. Fragmentation mechanisms. Models of conduit flows during explosive eruptions and results. Volcanic plume dynamics in the atmosphere.

Dynamics of dispersed systems Mixture properties: Bubbles Particles

Mixture properties (continue) Continuity equations  Mass fluxes Momentum equations  Momentum exchange Energy equations  Heat fluxes

Conduit flow during explosive eruption Schematic view of the system  xt Flow regimes and boundaries.  Homogeneous from magma chamber until pressure > saturation pressure.  Constant density, viscosity and velocity, laminar.  Vesiculated magma from homogeneous till magma fragmentation.  Bubbles grow due to exsolution of the gas and decompression.  Velocity and viscosity increases.  Flow is laminar with sharp gradients before fragmentation due to viscous friction.  Fragmentation zone or surface (?).  Fragmentation criteria.  Gas-particle dispersion from fragmentation till the vent.  Turbulent, high, nonequilibrium velocities.  subsonic in steady case, supersonic in transient.

Modelling strategy Equations Mass conservation for liquid and gas phases –intensity of mass transfer, bubble nucleation and diffusive growth Momentum equations –gravity forces, conduit resistance, inertia –momentum transfer between phases Energy equations –energy transfer between phases –dissipation of energy by viscous forces Bubble growth equation - nonequilibrium pressure distribution Physical properties of magma - density, gas solubility, viscosity Fragmentation mechanism Boundary conditions - chamber, atmosphere, between flow zones

Models of fragmentation  FP - fragmentation at fixed porosity.  SR - critical elongation strain- rate  OP- critical overpressure in a growing bubble p g p m    pp m g  R R 4   3      RRR R  2 2 2   small

Chocked flows Flow High pressure Low pressure

Boundary conditions Magma chamber:  pressure, temperature  initial concentration of dissolved gas - calculate volume fraction of bubbles Atmosphere:  Pressure is equal to atmospheric if flow is subsonic  Chocked flow conditions - velocity equal to velocity of sound Need to calculate discharge rate

Slezin (1982,1983,1992) Main assumptions: Conduit has constant cross-section area Magma - Newtonian viscous liquid,  =const Bubbles do not rise in magma When  = 0.7 - fragmentation, porous foam After fragmentation  = 0.7, all extra gas goes to interconnected voids. When concentration of gas in voids = 0.4 - transition to gas particle dispersion. Particles are suspended (drag force=weight)

Slezin (results)

Woods, Koyaguchi (1994) Gas escape from ascending magma through the conduit walls. Fragmentation criteria  =  *.  Magma ascends slowly - looses its gas - no fragmentation - lava dome extrusion.  Magma ascends rapidly - no gas loss - fragmentation - explosive eruption. Contra arguments:  Magma permeability should be > rock permeability.  Vertical pressure gradient to gas escape through the magma.

Barmin, Melnik (2002) Magma - 3-phase system - melt, crystals and gas. Viscous liquid  (concentrations of dissolved gas and crystals). Account for pressure disequilibria between melt and bubbles. Permeable flow through the magma. Fragmentation in “fragmentation wave.” 2 particle sizes - “small” and “big.”

Mass conservation equations (bubbly zone)  - volume concentration of gas (1-  ) - of condensed phase  - volume concentration of crystals in condensed phase  - densities, “m”- melt, “c”- crystals, “g” - gas c - mass fraction of dissolved gas = k p g 1/2 V - velocities, Q - discharge rates for “m”- magma, “g” - gas n - number density of bubbles

Momentum equations in bubbly zone  - mixture density - resistance coefficient (32 - pipe, 12 -dyke) k(  ) - permeability  g - gas viscosity p- pressure “s”- mixture, “m”- condensed phase, “g”- gas

Rayleigh equation for bubble growth Additional relationships:

Equations in gas-particle dispersion F - interaction forces:”sb” - between small and big particles “gb” - between gas and big particles

Fragmentation wave

Pressure profiles in the conduit

Model of vulcanian explosion generated by lava dome collapse

Assumptions Flow is 1D, transient Velocity of gas and condensed phase are equal Initial condition - V = 0, pressure at the top of the conduit > p atm, drops down to p atm at t =0 Two cases of mass transfer: equilibrium (fast diffusion), no mass transfer (slow diffusion) Pressure disequilibria between bubbles and magma No bubble additional nucleation

Mechanical model

Results of calculation (eq case)

Discharge rate and fragmentation depth (eq case)

Pulsing fragmentation

Seismic record of eruption

Results of simulations (no mt case) Discharge rate and fragmentation depth

Volcanic plumes PlinianCollapsing High - comes to stratosphere Ash fallout, climate change Acid rains, aviation hazards Pyroclastic flow generation

Unsolved problems Physical properties of magma –Magma rheology for high strain-rates and high bubble and crystal content Bubbly flow regime –Incorporation of bubble growth model into the conduit model –Understanding bubble interaction for high bubble concentrations –Understanding of bubble coalescence dynamics, permeability development –Thermal effects during magma ascent - viscous dissipation, gas exsolution

Unsolved problems (cont) Fragmentation –Fragmentation in the system of partly interconnected bubbles –Partial fragmentation, structure of fragmentation zone, particle size distribution Gas-particle dispersion –Momentum and thermal interaction in highly concentrated gas- particle dispersions

Unsolved problems (cont.)! General –Coupling of conduit flow model with a model of magma chamber and atmospheric dispersal model –Deformation of the conduit walls during explosive eruption Visco-elastic deformation Erosion –Interaction of magma conduit flow with permeable water saturated layers - phreato-magmatic eruptions

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