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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. Bubbles. Mixture properties:. Particles.

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Lecture 3 l.jpg
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


Mixture properties:


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Mixture properties (continue)

Continuity equations Mass fluxes

Momentum equations Momentum exchange

Energy equations Heat fluxes

Conduit flow during explosive eruption l.jpg

Schematic view of the system

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.

Conduit flow during explosive eruption



Modelling strategy l.jpg


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

Modelling strategy

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Models of fragmentation


FP - fragmentation at fixed porosity.


OP- critical












overpressure in a


growing bubble




























SR - critical

elongation strain-


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Chocked flows

High pressure

Low pressure


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Boundary conditions

  • Magma chamber:

    • pressure, temperature

    • initial concentration of dissolved gas - calculate volume fraction of bubbles


    • 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, m=const

  • 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 force=weight)

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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 > 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 forces:”sb” - between small and big particles

“gb” - between gas and big particles

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Model of vulcanian explosion generated by lava dome collapse

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  • Flow is 1D, transient

  • Velocity of gas and condensed phase are equal

  • Initial condition - V = 0, pressure at the top of the conduit > 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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Discharge rate and fragmentation depth

Results of simulations (no mt case)

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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