Governing equations for multiphase flows. Continuum hypothesis.

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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.

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Dynamics of dispersed systems
Bubbles
Mixture properties
Particles
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Mixture properties (continue)
Continuity equations ?Mass fluxes Momentum
equations ?Momentum exchange Energy
equations ?Heat fluxes
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Conduit flow during explosive eruption

• Schematic view of the system

• Flow regimes and boundaries.
• Homogeneous from magma chamber until pressure gt
  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.

x
t
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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

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Models of fragmentation
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FP - fragmentation at fixed porosity.

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OP- critical

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p

m
4
g
m
overpressure in a
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growing bubble
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2
r

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R
2
g
p
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R
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m
small
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SR - critical
elongation strain-
rate
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Hydrostatic vs. Lithostatic pressure gradient
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Chocked flows
High pressure
Low pressure
Flow

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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

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Slezin (1982,1983,1992)

• Main assumptions
• Conduit has constant cross-section area
• Magma - Newtonian viscous liquid, mconst
• Bubbles do not rise in magma
• When a 0.7 - fragmentation, porous foam
• After fragmentation a 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 forceweight)
  
  
  

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Slezin (results)
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Woods, Koyaguchi (1994)

• Gas escape from ascending magma through the
  conduit walls.
• Fragmentation criteria a a.
• 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 gt rock permeability.
• Vertical pressure gradient to gas escape through
  the magma.

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Barmin, Melnik (2002)

• Magma - 3-phase system - melt, crystals and gas.
• Viscous liquid m (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.

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Mass conservation equations (bubbly zone)
a - volume concentration of gas (1-a) - of
condensed phase b - volume concentration of
crystals in condensed phase r - densities, m-
melt, c- crystals, g - gas c - mass fraction
of dissolved gas k pg1/2 V - velocities, Q -
discharge rates for m- magma, g - gas n -
number density of bubbles
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Momentum equations in bubbly zone
r - mixture density l - resistance coefficient
(32 - pipe, 12 -dyke) k(a) - permeability mg-
gas viscosity p- pressure s- mixture, m-
condensed phase, g-gas
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Rayleigh equation for bubble growth
Additional relationships
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Equations in gas-particle dispersion
F - interaction forcessb - between small and
big particles
gb - between gas and big particles
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Fragmentation wave
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Steady discharge vs. chamber pressure
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Pressure profiles in the conduit
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Model of vulcanian explosion generated by lava
dome collapse
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Assumptions

• Flow is 1D, transient
• Velocity of gas and condensed phase are equal
• Initial condition - V 0, pressure at the top of
  the conduit gt patm, drops down to patm 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

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Mechanical model
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Results of calculation (eq case)
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Discharge rate and fragmentation depth(eq case)
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Pulsing fragmentation
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Seismic record of eruption
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Results of simulations (no mt case)

• Discharge rate and fragmentation depth

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

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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

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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