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Searching fundamental information in ordinary differential equations. Nondimensionalization technique
J F Sánchez Pérez1, M Conesa1, I Alhama1
1Network Simulation Research Group, Universidad Politécnica de Cartagena, Cartagena, Spain.
Plos One
|October 4, 2017
Summary
Dimensional analysis and nondimensionalization simplify complex problems by creating dimensionless groups. This study presents a formal protocol for nondimensionalization, yielding more precise, physically interpretable results than traditional dimensional analysis.
Area of Science:
- Engineering
- Applied Mathematics
- Physics
Background:
- Dimensional analysis and nondimensionalization are established methods for simplifying complex physical problems.
- Both techniques aim to identify dimensionless groups that govern system behavior.
- Traditional dimensional analysis relies on physical understanding, while nondimensionalization uses governing equations.
Purpose of the Study:
- To propose a formal protocol for applying nondimensionalization to ordinary differential equations.
- To derive dimensionless normalized equations with physically interpretable dimensionless groups.
- To demonstrate the superior precision of nondimensionalization over dimensional analysis.
Main Methods:
- Development of a standardized protocol for nondimensionalization of ordinary differential equations.
- Mathematical manipulation of governing equations to achieve a dimensionless form.
- Analysis of resulting dimensionless groups for physical interpretation and order-of-magnitude unity.
Main Results:
- A formal protocol for nondimensionalization is presented.
- Dimensionless normalized equations are derived, featuring dimensionless groups with physical meaning.
- These dimensionless groups represent balances between opposing quantities and are of order unity.
- Nondimensionalization provides more precise solutions compared to dimensional analysis.
Conclusions:
- The proposed nondimensionalization protocol offers a rigorous approach to simplifying differential equations.
- The derived dimensionless groups offer enhanced physical insight and quantitative precision.
- Nondimensionalization is a more accurate and interpretable method for analyzing physical phenomena governed by differential equations.
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