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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Continuity equations Mass fluxes
Momentum equations Momentum exchange
Energy equations Heat fluxes
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.
Gasparticle dispersion from fragmentation till the vent.
Turbulent, high, nonequilibrium velocities.
subsonic in steady case, supersonic in transient.
Conduit flow during explosive eruption¶
¶
t
x
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 strategy4
FP  fragmentation at fixed porosity.
4
OP critical
&
R

=
p
p
m
+
4
g
m
overpressure in a
R
growing bubble
p
s
æ
ö
3
&
&
&
+
r
+
2
ç
÷
2
R
R
R
g
p
2
R
è
ø
m
small
4
SR  critical
elongation strain
rate
Atmosphere:
Need to calculate discharge rate
Main assumptions:
Mass conservation equations (bubbly zone)
a  volume concentration of gas (1a)  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
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
Additional relationships:
Equations in gasparticle dispersion
F  interaction forces:”sb”  between small and big particles
“gb”  between gas and big particles
Discharge rate and fragmentation depth
Results of simulations (no mt case)Plinian Collapsing
High  comes to stratosphere
Ash fallout, climate change
Acid rains, aviation hazards
Pyroclastic flow generation