Potential Driven Flows Through Bifurcating Networks Aerospace and Mechanical Engineering Graduate Student Conference October, 2006 Jason Mayes Advisor: Dr. Mihir Sen

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

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."

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?

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

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 q q1q1 q2q2 q = q 1 +q 2

A self-similar model potential-driven transfer Assumptions Transfer governed by a linear operator i.e., for each branch:

A self-similar model simplification Goal is to reduce or simplify the system to the single equation Generation two (N=2) Network

A self-similar model simplification Apply recursively to eliminate q 1,1 and q 2,2

A self-similar model a simple result Result of simplification for N=2 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

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

Examples tree of resistors Symmetric with generation independent operators

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

Examples fractional order visco-elasticity models Asymmetric with generation independent operators

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

Examples laminar pipe flows in branching networks Symmetric with generation dependent operators

Examples laminar pipe flows in branching networks Each pipe governed by In Laplace domain In the time domain

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

Questions?