Magnetohydrodynamic flow of Cu–Fe3O4/H2O hybrid nanofluid with effect of viscous dissipation: dual similarity solutions

This study shows multiple solutions, heat transfer characteristics, and stability analysis of the magnetohydrodynamic (MHD) flow of hybrid nanofluid caused by the nonlinear shrinking/stretching surface. To investigate the effects of high temperature on the porous surface, the energy dissipation function and porous term are considered in the momentum and energy equations. We used Tiwari and Das’s model for nanofluid in which water is considered as a base fluid. A new kind of fluid is made in which two kinds of nanoparticles, namely copper (Cu) and iron oxide (Fe3O4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{Fe}}_{3} {\text{O}}_{4}$$\end{document}), are considered. The system of ordinary differential equations (ODEs) is obtained by applying similarity transformations on the modeled of partial differential equations. Both shooting and Runge–Kutta fourth-order methods are employed to solve the resultant ODEs. The equations for stability analysis have been derived and then solved by using a three-stage Lobatto IIIa formula for the smallest eigenvalue. It is noticed that the obtained value is in a good agreement with the previously published literature, hence validating the results of the shooting method. Furthermore, parametric studies also have been conducted and found that dual solutions only exist on the shrinking surface. In addition, it is also observed from the profile that dual solutions exist only for the case of suction where bc1=-3.0582\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_{{{\text{c}}1}} = - 3.0582$$\end{document}, bc2=-3.0788,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_{{{\text{c}}2}} = - 3.0788,$$\end{document} and bc3=-3.1249\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_{{{\text{c}}3}} = - 3.1249$$\end{document} are the critical values for the respective values of ϕFe3O4=0.5%,5%,1%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi_{{{\text{Fe}}_{3} {\text{O}}_{4} }} = 0.5\% ,5\% , 1\%$$\end{document}. Moreover, the velocity of hybrid nanofluid decreases (increases) in the first (second) solution when both magnetic and permeability coefficient parameters are increased.