# Governing equations and models for multiphase flows

This lecture covers the use of continuum hypothesis, fragmentation mechanisms, and conduit flow models to describe multiphase flows.

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## About Governing equations and models for multiphase flows

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1. Lecture 3 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.

2. Dynamics of dispersed systems Dynamics of dispersed systems Mixture properties: Bubbles Particles

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

4. Conduit flow during explosive eruption Conduit flow during explosive eruption Schematic view of the system x t 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.

5. Modelling strategy 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

6. Models of fragmentation Models of fragmentation FP - fragmentation at fixed porosity . SR - critical elongation strain- rate OP- critical overpressure in a growing bubble p g p m p p m g R R 4 3 R R R R 2 2 2 small

7. Hydrostatic vs. Lithostatic pressure gradient Hydrostatic vs. Lithostatic pressure gradient

8. Chocked flows Chocked flows Flow High pressure Low pressure

9. Boundary conditions 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

10. Slezin (1982,1983,1992) 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)

11. Slezin (results) Slezin (results)

12. Woods , Koyaguchi (1994) 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.

13. Barmin, Melnik (2002) 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.

14. Mass conservation equations (bubbly zone) 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

15. Momentum equations in bubbly zone 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

16. Rayleigh equation for bubble growth Rayleigh equation for bubble growth Additional relationships:

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

18. Fragmentation wave Fragmentation wave

19. Steady discharge vs. chamber pressure Steady discharge vs. chamber pressure

20. Pressure profiles in the conduit Pressure profiles in the conduit

21. Model of vulcanian explosion generated by lava dome collapse Model of vulcanian explosion generated by lava dome collapse

22. Assumptions 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

23. Mechanical model Mechanical model

24. Results of calculation (eq case) Results of calculation (eq case)

25. Discharge rate and fragmentation depth (eq case) Discharge rate and fragmentation depth (eq case)

26. Pulsing fragmentation Pulsing fragmentation

27. Seismic record of eruption Seismic record of eruption

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

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

32. Unsolved problems 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

33. Unsolved problems (cont) 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

34. Unsolved problems (cont.)! 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