Unsteady flow of hybrid nanofluid over a permeable shrinking inclined rotating disk with radiation and velocity slip effects

A nanofluid refers to a suspension of nanoparticles in a conventional fluid, which finds unique applications in diverse sectors, including engineering, technology, and medicine. When multiple nanoparticles are suspended, it creates a hybrid nanofluid. In this study, we aim to investigate an unsteady flow of hybrid nanofluid over a permeable shrinking inclined rotating disk subjected to heat radiation, magnetohydrodynamics and slip effects. The chosen nanoparticles for this study are alumina (Al2O3) and copper (Cu), incorporated into a base fluid of water (H2O) to create the hybrid nanofluid. An appropriate method of similarity transformation is executed along a set of partial differential equations that were reduced to a system of nonlinear ordinary differential equations, where numerical outcomes were then obtained via bvp4c in MATLAB software, with the influence of various parameters such as unsteadiness parameter, nanoparticle volume fraction, shrinking, radiation, magnetic and velocity slip parameters, shown in tables and figures. Multiple solutions (including dual, upper, and lower branch solutions) are identified for the governing similarity equations. Through the conducted stability analysis, it is determined that the upper branch solutions exhibit stability and physically realizable in practice, while the lower branch solutions are unstable. Our numerical findings showed that dual solutions exist when εc≤ε≤-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon_{c} \le \varepsilon \le - 1$$\end{document}, where εc<0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon_{c} < 0$$\end{document} is the critical value of ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon$$\end{document} for which the boundary value problem poses physical solutions applicable in practice. Yet, the boundary value problem lacks a similarity solution for ε≤εc≤0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \le \varepsilon_{c} \le 0$$\end{document}, and the complete set of partial differential equations needs to be solved numerically. Improvements in heat transfer rate are observed concerning the radiation parameter, nanoparticle fraction, and shrinking parameter. Furthermore, azimuthal velocity profiles show an increase influenced by velocity slip and magnetic parameters. The non-dimensional physical parameters, including stretching/shrinking, suction, slip, and unsteadiness, are also considered and their effects are presented in figures and tables.