Advanced Scaling Techniques for the Modeling of Materials Processing

This paper presents a formalism to automate portions of the traditional scaling procedure. This formalism, called here Order of Magnitude Scaling (OMS), captures essential components of the governing equations into a set of matrices and arrays amenable to computer implementation using standard linear algebra tools. This approach is valid and specially useful for large systems of multicoupled equations. The matrix approach proposed enables one to automate operations such as normalization, dominant balance, obtention of characteristic values, and check for self-consistency. Also, automation permits an exhaustive exploration of all possible balances and the discovery of unsuspected scaling laws. As an example, OMS is applied to the analysis of the viscous boundary layer with no external pressure gradient. The obtained scaling laws for boundary layer thickness and transverse velocity are the same as those obtained with traditional scaling, and in addition, a meaningful estimation for pressure variations within the boundary layer is obtained; this last estimation is typically not addressed in the manual analysis.