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Research Papers

Order of Magnitude Scaling: A Systematic Approach to Approximation and Asymptotic Scaling of Equations in Engineering

[+] Author and Article Information
Patricio F. Mendez

Department of Chemical and Materials Engineering,
University of Alberta,
9107 116th St.,
Edmonton, Alberta T6G 2V4, Canada
e-mail: pmendez@ualberta.ca

Thomas W. Eagar

Department of Materials Science and Engineering,
Massachusetts Institute of Technology,
77 Massachusetts Ave.,
Cambridge, MA 02139, USA
e-mail: tweagar@mit.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the Journal of Applied Mechanics. Manuscript received November 24, 2011; final manuscript received February 21, 2012; accepted manuscript posted May 17, 2012; published online October 29, 2012. Assoc. Editor: Martin Ostoja-Starzewski.

J. Appl. Mech 80(1), 011009 (Oct 29, 2012) (9 pages) Paper No: JAM-11-1448; doi: 10.1115/1.4006839 History: Received November 24, 2011; Revised February 21, 2012; Accepted May 17, 2012

This work introduces the “order of magnitude scaling” (OMS) technique, which permits for the first time a simple computer implementation of the scaling (or “ordering”) procedure extensively used in engineering. The methodology presented aims at overcoming the limitations of the current scaling approach, in which dominant terms are manually selected and tested for consistency. The manual approach cannot explore all combinations of potential dominant terms in problems represented by many coupled differential equations, thus requiring much judgment and experience and occasionally being unreliable. The research presented here introduces a linear algebra approach that enables unassisted exhaustive searches for scaling laws and checks for their self-consistency. The approach introduced is valid even if the governing equations are nonlinear, and is applicable to continuum mechanics problems in areas such as transport phenomena, dynamics, and solid mechanics. The outcome of OMS is a set of power laws that estimates the characteristic values of the unknowns in a problem (e.g., maximum velocity or maximum temperature variation). The significance of this contribution is that it extends the range of applicability of scaling techniques to large systems of coupled equations and brings objectivity to the selection of small terms, leading to simplifications. The methodology proposed is demonstrated using a linear oscillator and thermocapillary flows in welding.

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Figures

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Fig. 1

Structure of the matrix of coefficients [C]. Line Cj,k represents coefficient k in Eq. i, and each column represents a numerical constant (K1,K2,…), a problem parameter (P1,P2,…), or an unknown characteristic value (S1,S2,…).

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Fig. 2

Flow chart of the OMS procedure

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Fig. 3

System coordinates and problem configuration for thermocapillary flows in welding (modified from Ref. [16]). Only half the enclosure is shown.

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