Physical Volcanology Course Lecture 2 Dr. Oleg Melnik Institute of Mechanics Moscow State University

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Physical Volcanology CourseLecture 2Dr. Oleg
MelnikInstitute of MechanicsMoscow State
University
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Lecture 2 Fluid dynamics

• Basic principles of fluid dynamic and governing
  equations
• Laminar flow of the liquid between parallel
  plates and cylindrical pipe, resistance, Reynolds
  number
• Bubble growth in viscous liquid diffusion and
  overpressure development

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Conservation of Mass
Mass balance
Rate of mass flow out
Rate of mass flow in
Rate of accumulation
-
0

NET RATE OF MASS OUTFLOW
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Continuity Equation
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Momentum is Mass x Velocity
Each Plane has Velocity V
Each Plane has Mass m
Each Plane has Momentum ? P mV
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Change in Momentum
Consider a bouncing ball
Pafter momentum right after bouncing
Pafter - Pbefore F Dt
Note that pbefore lt 0 and pafter gt 0 ? F gt 0.
The ball is in contact with the floor for a
period of time Dt. The force exerted by the
floor causing the ball to bounce is F.
Pbefore momentum just before bouncing
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Momentum Balance in fluid
Sum of forces acting on system
Rate of momentum in
Rate of momentum out
Rate of accumulation of momentum
-


NET RATE OF MOMENTUM OUTFLOW
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Forces acting on surfaces
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Differential Momentum Balance
Estimation of forces acting on the element
(projection to x axis)
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Momentum Equations
x-component
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Stress-Deformation Relation
For closure of the problem, a constitutive
equation is needed to related stress to
deformation. For example the Newtonian
constitutive equation can be used for liquids
obeying Newtons law

• p - pressure, strain-rate tensor
• m - viscosity

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Components relation
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Navier-Stokes equations (rconst)
x-component
y-component
z-component
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Understanding pressureStatic case VxVyVz0

• Blaise Pascal
• 17th century

• Force acting on barrel walls

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Example 1

• Laminar flow of the liquid between parallel
  plates (dyke) and cylindrical pipe (conduit)
• Steady-state flow (nothing depends on time)
• Liquid is incompressible (densityconstant)
• VyVz0
• No gravity forces

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Mass conservation
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Momentum
x-component
y-component
z-component
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Momentum (continue)
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Final solution
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Example

• Cylindrical volcanic conduit D 50 m, L5000, Dp
  10 MPa, m 105 Pa s, Q-?

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Laminar and Turbulent Flows
Reynolds apparatus

• Reynolds number

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Laminar and Turbulent flows (continue)

• Magma - high viscosity, low velocity
• Re number is small
• LAMINAR FLOW

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Dynamics of bubble growth
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From individual bubble to multiphase flow

• Assumptions
• All bobbles has same radius and equally spaced
• Liquid around the bubble is Newtonian,
  incompressible
• Gas in the bubble is perfect (p rg RT )
• Bubble shall radii is bigger then r (small volume
  concentration of bubbles)
• No interaction!

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Mass conservation
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Momentum equation
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Boundary conditions at bubble-liquid interface
small
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Diffusion

• Due to random molecular motion concentration
  equilibrates

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Ficks law
Dm coefficient of molecular diffusion (m2/s) C
concentration (e.g. kg/m3) J mass flux
Does the gradient cause the diffusion? _____
NO!
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Ficks 2nd Law
dx
dA
Ficks 2nd Law
Jin
Jout
Chigh
Clow
The rate of change of the number of atoms in the
slice dV
dVdA?dx
The rate that atoms entering the slice the rate
that atoms leaving the slice

?
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Diffusive growth of bubble

• Large diffusion coefficient

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Bubble growth under sudden decompression (Navon,
Lensky,Lyahovsky)
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Initial and boundary conditions
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Solutions
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Conclusions

• Basic principles of fluid dynamics
• Flow in the pipes - resistance, laminar and
  turbulent flows
• Bubble growth dynamics - overpressure,
  fragmentation