Exemplification of Dimensional Analysis via MATLAB® Using Eigen Values

Abstract

In this research article, a set of dimensional physical quantities is transformed into a dimensionless group (or ratio). For a given set of dimensional variables, the physical variables represent the rows and their dimensions represent the columns of a dimensions-matrix. The dimensions-matrix is rearranged both column- and row-wise. The columns are sorted in ascending order based on the column sum and then on the largest negative entry (i.e., cell value). On the other hand, the rows are sorted in descending order based on the number of non-zero entries found in each row and then on the higher first entry. With the aid of MATLAB®, it was found that the proposed method leads to a permutation matrix that has an Eigen vector whose elements represent the exponent for each physical dimensional quantity such that, at the end, a dimensionless group (or ratio) can be formulated, like Schmidt, Nusselt, Reynolds, and Peclet number. The method, however, was found to work well with a set of physical quantities where each is raised to an exponent of ±1.