Catalogue of existence of the multiple physical solutions of hydromagnetic flow over a stretching/shrinking sheet for viscoelastic second-grade and Walter's B fluids

In this work, the author restudied a fourth-order nonlinear ordinary differential equation that commonly appears in investigating the hydromagnetic flow of two types of viscoelastic fluids over a stretching/shrinking sheet. In this type of problem, vital question arises: what are the parameters values needed to obtain regions and numbers of the physical solutions? To answer this question, the researchers worked on suggesting/estimating the initial values, or even chose them randomly. This usually leads to wasting a lot of time, effort and maybe to untrustworthy results. Towards avoiding this rugged road, the present work answered this question decisively and definitively. In particular, an explicit and direct catalogue of existence multiple real positive solutions was in detail introduced. MATHEMATICA code was programmed and, therefore, many tables and figures of the critical values and regions were produced at different values of the controlled parameters. In addition, and maybe for the first time, the mathematical expressions of these critical values were secured. Hence, remarkable differences were detected between the existing results and those obtained in the literature, therefore, spurious physical sights were gained. In conclusion, this research determined the whole values of the parameters that are required to control the number of solutions or even not to obtain solutions at all. Therefore, the current catalogue aims to help the researchers to avoid guessing the parameters' values on solving the current type of problem.