Formulating the Moving Boundary Problem

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Formulating the Moving Boundary Problem (or
Conservation of Mass It isnt just a good idea,
its the law)
Andrew Yeckel
Derby research group meeting (add date)
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Solidification from the melt
Global mass conservation
Liquid
Time rate of change
Solid
Simplify to one-dimensional case
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Case 1 Equal densities
Liquid level
Rearrange
or
Volume of liquid lost equals volume of solid
gained
Fixed position in lab reference frame
Equal densities Liquid level remains constant
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Case 2 Unequal densities
Liquid level
Rearrange
Consider
Volume of liquid lost exceeds volume of solid
gained
Fixed position in lab reference frame
Unequal densities Liquid level falls (rises)
during growth
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Flow boundary conditions Equal densities
Introduce a coordinate relative to bottom of solid
Interface moves at velocity
What is correct velocity boundary condition at
interface?
Rigid boundary in motion

• Volume of liquid lost equals volume of solid
  gained
• Interface is not a rigid boundary
• Interface overtakes liquid
• Liquid is solidified in place
• No-slip is correct boundary condition

Interface motion does not induce any motion of
liquid
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Flow boundary conditions Unequal densities
Derivation 1
Mass of solid created
Mass of liquid overtaken
Mass shortage
Liquid must flow downward to make up shortfall
Rearrange,
By continuity, downward flow is uniform everywhere
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Flow boundary conditions Unequal densities
Derivation 2
Mass crossing a moving interface
Interface
Mass balance across interface
Rearrange, set
For equal density case
By continuity, downward flow is uniform everywhere
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Rate of translation of interface and top boundary
Required information
Interface location set at melting point isotherm
Mass balance at interface
Top rigid boundary in motion
We need one more relationship to complete the
formulation
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The global mass balance completes the picture
In this 1-D example, velocity is uniform
everywhere
Condition at interface and top boundary
Using mass balance at top
Trouble BC at top is function of interface
velocity
Conditions at top are non-local, destroys
sparseness of discretization
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Constraint-based approach to formulation
Use global mass balance directly
At interface
At top boundary
Global mass conservation
More general, does not assume that flow is
uniform everywhere
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Generalization to 2- and 3-Dimensions
Melting point isotherm
At interface
Normal velocity
Tangential velocity
Normal velocity
At top boundary
Tangential velocity
Everywhere
Global mass conservation
Completely general in any number of dimensions,
for non-uniform flow
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Implementation of global constraint
Global mass conservation
Discretized form element-by-element sum

• Consequences
• Single residual, non-local, Jacobian has one
  dense row
• Easily solved using Sherman-Morrison or Woodbury
  technique
• Requires one additional back-substitution

Arrowhead Solver
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Boundary conditions for mass transport
Segregation
Mass crossing a moving interface
Interface
Mass balance across interface
Diffusion and convection
Rearrange, use relations
Final result
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Boundary conditions for mass transport Top
boundary
Mass balance at top boundary
Flow at top boundary
Final result
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Boundary conditions for heat transport
Heat balance across interface
Interface
Conduction and convection
Rearrange, use relations
Final result
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Summary of boundary conditions/contraints
Melting point isotherm
Normal velocity
Interface
Tangential velocity
Segregation
Latent heat
Normal velocity
Tangential velocity
Top boundary
Mass flux
Heat flux
Everywhere
Global mass conservation