Stability analysis of dual solutions for mixed convection and thermal radiation with hybrid nanofluid flow past shrinking/stretching curved surface

Abstract

This study portrays the stability analysis and dual solutions of mixed convection and thermal radiation of hybrid nano-fluid flow past stretching/shrinking a curved surface in the presence of injection/suction conditions. A hybrid nano-fluid, in which water is used as the base fluid, copper and alumina are used as nano-particles, and the magnetic field is taken into account. The present study’s findings will provide fruitful implications for future research in the field of fluid dynamics. The bvp4c method using Matlab software is implemented to get the numerical solution of the nonlinear partial differential equation transformed into the ordinary differential equation. The behavior of the first and second solutions under the governing parameters on the curved surface of dimensionless velocity g′(ξ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g'(\\xi )$$\\end{document}, shear stress profile g′′(ξ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g''(\\xi )$$\\end{document}, temperature profile θ(ξ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta (\\xi )$$\\end{document}, skin friction coefficient Cfs, and local Nusselt’s number Nus were visualized in figurative and tabular form. From this the following are investigated: as the values of ϕj\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi j$$\\end{document} increase, the velocity profile for the second solution decreases, and the opposite trend is observed for the first solution. For the values of K and λ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda 1$$\\end{document}, the shear stress profile increased for the first solution, and the opposite trend was observed for the second solution, though after some interval points, the inverse of this statement was observed.For the values of S,Pr,Rd,M,andλ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S,Pr,Rd,M, \\;\ ext{and}\\; \\lambda$$\\end{document}, the upwind thermal boundary layer of the first solution is larger than the second solution.For the value of M uphill, the estimation of the absolute value of λci\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda ci$$\\end{document} increases for both the skin friction coefficient and the local Nusselt number. In the second solution, increasing the values of β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta$$\\end{document}, Pr, S, Ec, and M has a similar effect on g′′(0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g''(0)$$\\end{document}, Cfs, -θ′(0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta '(0)$$\\end{document}, and Nus. In the first solution, increasing the values of Ec, S, and Pr on g′′(0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g''(0)$$\\end{document} and Cfs results in a decrease. The first solution has a positive eigenvalue, whereas the second solution has a negative eigenvalue. Agreement between the present analysis and literature is acceptable.