Validation method for the systematization of results based on a similarity concept

Abstract

The development of a functional methodological approach is presented here, to clarify a globally valid way of evaluating the precision of mathematical modeling of physical and/or chemical processes. Starting from the description of the system, a phenomenon accompanied by a disclaiming hypothesis is investigated, against which the knowledge is accumulated with time. Moreover, the possibility of the evolution of any phenomenon being interrupted when a parameter overpasses a critical threshold, after which the hypothesis is not any more valid, is introduced. This possibility should be obtained through the dependence of a selected macroscopic quantity (marker) on a specific parameter. To apply this methodology, the problem of Stokes flow through a granular medium of spheroidal grains has been selected as an indicative case study. The prolate spheroidal configuration is considered, since the results for the oblate spheroid can be recovered via a simple transformation. Therein, the three‐dimensional flow fields are initially constructed analytically via the Papkovich‐Neuber differential representation, which provides the velocity and pressure fields in terms of harmonic spheroidal eigenfunctions. Next, under the Kuwabara‐type spheroidal 2D unit cell concept, the above expressions degenerate to the axisymmetric case, and the full solution is then obtained, keeping the leading terms of the series, which are adequate for most engineering applications for specific aspect ratio of the spheroids. In the sequel, the aforementioned problem is solved numerically for a 3D extension of the same model, where this numerical solution has been achieved by using the finite volumes method, while the resulting linear systems were approximated by applying the well‐known successful over‐relaxation concept. Finally, outcomes by both models have been compared via the above methodology, resulting to objective and reliable accuracy criteria.