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Average shape of transport-limited aggregates
Benny Davidovitch1, Jaehyuk Choi, Martin Z Bazant
1Division of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA.
Physical Review Letters
|October 4, 2005
Summary
We explore stochastic and continuous growth models, deriving an equation for average aggregate shape. Stochastic growth is similar but not identical to continuous models, explaining discrepancies in DLA shapes and viscous fingers.
Area of Science:
- Physics
- Materials Science
- Complex Systems
Background:
- Stochastic and continuous models are used to describe transport-limited growth phenomena.
- Discrepancies exist between theoretical models like Diffusion Limited Aggregation (DLA) and experimental observations such as viscous fingering.
Purpose of the Study:
- To investigate the relationship between stochastic and continuous transport-limited growth models.
- To derive a governing equation for the average shape of stochastic aggregates.
- To explain observed differences between DLA and viscous fingers.
Main Methods:
- Derivation of a nonlinear integro-differential equation for the average shape of stochastic aggregates.
- Mean-field approximation to connect stochastic and continuous equations.
- Focus on the advection-diffusion-limited aggregation (ADLA) model.
Main Results:
- The derived integro-differential equation's mean-field approximation matches the continuous growth equation.
- The average shape of stochastic ADLA growth is similar, yet distinct from, continuous ADLA dynamics.
- This finding offers an explanation for discrepancies between DLA and viscous fingering.
Conclusions:
- Stochastic and continuous growth models are related but not equivalent.
- The derived framework provides insight into the limitations of continuous models for describing real-world growth phenomena.
- The study reconciles theoretical predictions with experimental observations in aggregation and fluid dynamics.