The matrix of coefficients in order of magnitude scaling

Abstract

This paper introduces the matrix of coefficients, which summarizes the physical insight into a problem. This matrix is part of a larger methodology, Order of Magnitude Scaling, which provides closed form estimations of the unknowns, their range of validity, and a set of dimensionless groups that indicate the true ratio of driving forces. These results are obtained even when the problems are described by non-linear partial differential equations. Order of Magnitude Scaling focuses on problems with many driving forces and relatively simple geometries. The matrix of coefficients is the starting point for obtaining these results in a systematic way through matrix operations. This methodology can be computationally much faster than methods that numerically integrate the governing equations, and does not present convergence problems.