Dual solutions on MHD radiative three-dimensional bidirectional nanofluid flow over a non-linearly permeable shrinking sheet

The bidirectional flow and thermal transfer of magnetohydrodynamics (MHD) and radiative nanofluid (magnetite-vacuum pump oil) due to a nonlinear shrinking surface in a three dimensional system is studied. The boundary layer model is first transformed into a set of ordinary differential equations using the similarity transformations, and then solved using the bvp4c solver. The accuracy of the present model is justified by comparing present data with the numerical values from the published findings. The effect of factors (magnetic parameter, radiation parameter and nanoparticles volumetric concentration) on the development of responses (skin friction coefficient and thermal rate) and critical value (separation value from laminar to turbulent flow) is observed through the graphical presentation. In addition, two solutions are attained where the first solution is affirmed as the reliable solution through stability analysis. Conclusively, the suction effect is nec-essary in generating the solutions under the phenomenon of opposing shrinking flow. The addition of magnetic parameter and nanoparticles concentration can enhance both responses as well as the critical value while the radiation parameter tends to reduce the heat transfer coefficient. The types of stretching/shrinking velocity (linear/nonlinear) also affect the heat transfer rate. The critical value can be extended by using the linear velocity, but, for thermal enhancement, the nonlinear form of velocity can significantly develop the thermal rate better than the linear shrinking surface.(c) 2023 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).