Dual solutions of Darcy-Forchheimer radiative-convective magneto-nanofluid flow over an exponentially stretching/shrinking surface: Stability and entropy generation analysis

Abstract

A numerical investigation was conducted to examine the existence of dual solutions over exponentially stretching or shrinking permeable surfaces. The study aimed to analyze the impact of thermal and second-order velocity slip on magnetohydrodynamic (MHD) mixed convective radiative nanofluid flow within a Darcy-Forchheimer porous medium while considering heat generation. Two distinct types of nanofluids, namely Cu-water and TiO 2/water, were employed in the investigation. To address the nonlinear partial differential equations (PDEs), similarity transformations were utilized to transform them into a system of ordinary differential equations (ODEs). The resulting coupled higher-order nonlinear ODEs were then solved using a MATLAB-based boundary value algorithm known as bvp4c. Two distinct solutions are observed within a restricted range of the stretching/shrinking parameter ξ ( ξ > ξ c ) and suction parameter S ( S > S c ), terminating respectively at ξ = ξ c or S = S c . The critical values are identified as ξ c = − 1.1952 ( Cu − water ) , − 0.9304 ( Ti O 2 − water ) and S c = 1.6437 ( Cu − water ) , 1.8900 ( Ti O 2 − water ) , maintaining the fixed values of the remaining physical parameters. Notably, the critical value for Cu-water is lower than that of TiO 2-water. Moreover, in both solution branches, the range of skin-friction coefficient and local Nusselt number for Cu-water surpasses that of TiO 2-water. An evaluation of the temporal stability of these solutions reveals that only the first solution remains stable. This is supported by the positivity of the smallest eigenvalues ( γ 1 > 0 ), indicating stability. Conversely, for the second solution, the smallest eigenvalues ( γ 1 < 0 ) are negative, signifying its instability. Also, Cu-water nanofluid exhibits higher fluid velocity and lower fluid temperature than TiO 2-water nanofluid for stable solutions and the opposite behavior for unstable solution branches. Significantly, within the initial solution branch, there is an escalation in velocity profiles attributed to the magnetic parameter, Grashof number, porous parameter, and first- and second-order velocity slip parameter. Conversely, a decrease is observed for the Forchheimer number, while the opposite holds true for the second solution branch. For both solution branches, the temperature profiles experience a reduction due to thermal slip and stretching/shrinking parameters, whereas enhancements are noted for thermal radiation and heat generation parameters. Furthermore, an increase in the suction parameter leads to elevated skin friction coefficient and local Nusselt number in the first stable solution, thereby promptly improving heat transfer efficiency. Moreover, improvements in entropy generation arise from factors such as thermal radiation, magnetic field, and nanoparticle solid volume fraction, presenting opportunities to optimize heat transmission in cooling and heating systems.