Potential Driven Flows Through Bifurcating Networks

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Potential Driven Flows Through Bifurcating
Networks

• Aerospace and Mechanical Engineering Graduate
  Student Conference 2006
• 19 October, 2006
• Jason Mayes
• Advisor Dr. Mihir Sen

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Outline

• Background/Motivation
• Objectives
• A self-similar model
• Simplifying the model
• Forms of similarity
• Examples
• Conclusions / Future work

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Background / Motivation self-similarity

• Natural physical examples
• Broccoli
• Artificially occurring examples
• Artificial terrain
• Non-physical examples
• Data series
• Music

"When each piece of a shape is geometrically
similar to the whole, both the shape and the
cascade that generate it are called
self-similar." - Mandelbrot
From Hofstadter's classic, Godel, Escher, Bach
"A fugue is like a canon, in that it is usually
based on one theme which gets played in different
voices and different keys, and occasionally at
different speeds or upside down or backwards."
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Background / Motivation self-similar systems

• Self-similarity in engineering
• many systems can be considered self-similar over
  several scales
• Large scale problems
• as systems grow, solutions become more difficult
• can we simplify?

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Objectives simplification and reduction

• Take advantage of similarity to simplify analysis
• Use known structure / behavior to simplify
• Using structure
• Extending behavior from one scale to another
• Reduce large equation sets

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A self-similar model a model for analysis

• The bifurcating tree geometry
• Geometry seen in a wide variety of applications
• Potential-driven flow or transfer
• ex. heat, fluid, energy, ect.
• Conservation at bifurcation points

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A self-similar model potential-driven transfer

• Assumptions
• Transfer governed by a linear operator
• i.e., for each branch

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A self-similar model simplification

• Goal is to reduce or simplify the system to the
  single equation

Generation two (N2) Network
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A self-similar model simplification
Apply recursively to eliminate q1,1 and q2,2
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A self-similar model a simple result

• Result of simplification for N2 network
• For the more general N-generation network
• For integro-differential operators
• Process repeated in the Laplace domain
• Inverses become simple algebraic inverses
• Can be written as a continued fraction

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Forms of similarity within and between

• Similarity can be used to further simplify
• Two forms of similarity
• Similarity within a generation
• Symmetric networks the operators within a
  generation are identical
• Asymmetric networks the operators within a
  generation are not identical
• Similarity between generations
• Generation dependent operators depend on the
  generation in which the operator occurs and
  change between successive generations
• Generation independent operators do not change
  between generations
• Four possible combinations
• Symmetric with generation independent operators
• Asymmetric with generation independent operators
• Symmetric with generation dependent operators
• Asymmetric with generation independent operators

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Examples tree of resistors

• Symmetric with generation independent operators

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Examples tree of resistors

• Tree constructed of identical resistors
• Current i(t) through each branch is driven by the
  potential difference v(t) across the branch
• Each branch governed by
• For an N generational tree of resistors
• For an infinite tree of resistors

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Examples fractional order visco-elasticity models

• Asymmetric with generation independent operators

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Examples fractional order visco-elasticity models

• Tree is composed of springs and dashpots
• Branches governed by linear operators
• Result is a fractional-order visco-elasticity
  model

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Examples laminar pipe flows in branching networks

• Symmetric with generation dependent operators

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Examples laminar pipe flows in branching networks

• Each pipe governed by
• In Laplace domain
• In the time domain

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Conclusions / Future Work

• Known behavior on one scale can be extended to
  better understand behavior on another
• Similarity or structure can be used to help
  simplify
• For equation sets patterns or structures can be
  used to simplify
• Future work
• Analyze other self-similar geometries
• Study probabilistically self-similar geometries

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Questions?