Fuzzy Analysis for Thin-Film Flow of a Third-Grade Fluid Down an Inclined Plane

Abstract

We examined the thin-film flow problem of a third-grade fluid on an inclined plane under a fuzzy environment. The highly nonlinear flow governing differential equations (DEs) with the boundary conditions are fuzzified using the triangular fuzzy numbers (TFNs) developed by $\alpha$ -cut $\left(\alpha \in \left[\mathrm{0,1}\right]\right).$ The fuzzy perturbation (FPM) method is adopted to calculate the fuzzified form of the governing equations as well as the fuzzified boundary conditions. For the validation, the present work is in good agreement as compared to existing work in the literature under the crisp form. For various values of the fluid parameter $\lambda ,$ inclined parameter $\gamma$ and fuzzy parameter $\alpha$ -cut is presented in graphical form. The $\alpha$ -cut controls TFNs, and the variability of uncertainty is investigated using a triangular membership function (MF). Using TFNs, the middle (crisp), left, and right values of the fuzzy velocity profile are used for fuzzy linear regression analysis. The outcome of this study and the fuzzy velocity profile have the maximum rate of flow as compared to the crisp velocity profile (mid values).