Entropy optimization of MHD second-grade nanofluid thermal transmission along stretched sheet with variable density and thermal-concentration slip effects

The goal of present investigation is to explore the influence of exponential variable density and entropy optimization on second-grade nanofluid heating efficiency and mass-concentration transmission along extended surface using external magnetic-field and temperature-concentration slip effects. To enhance the motion of nanoparticles and thermal efficiency, the influence of exponential form of temperature-based density on magnetically charged second-grade nanomaterial is main novelty of this research. For higher temperature difference, the entropy optimization is used. The defined formulation of stream functions and similarities are used to convert leading second-grade nanofluid model into ordinary differential form. The efficient Keller box method and Newton Raphson technique are applied to compute numerical results. The final algebraic equations are solved through global matrix for unknown physical quantities. The consequence of all physical constraints on velocity/U profile, temperature/θ field, concentration/ϕ shapes, skin friction coefficient, Nusselt and Sherwood number are analyzed pictorially and numerically. The following range of parameters 0.1 ≤ $\xi$ ≤ 2.0, 0.0 ≤ $n$ ≤ 1.2, 0.1 ≤ $E_c$ ≤ 2.0, 0.07 ≤ $P_r$ ≤ 7.0, 0.01 ≤ $N_t$ ≤ 0.8, 0.01 ≤ $N_b$ ≤ 0.9 is used. It is found that velocity field increases with maximum amplitude as variable density, magnetic force and temperature-slip constraint. It is noted that the slip behavior in temperature field and concentration field are increased with convective boundary conditions. It is depicted that local Nusselt quantity and local Sherwood quantity increases as buoyancy force and Prandtl coefficient increases.