Convective heat and zero-mass flux conditions in the time-dependent second-grade nanofluid flow by unsteady bidirectional surface movement

In this paper, a theoretical model is developed for the deliberation of time-dependent 3D magnetohydrodynamics second-grade (SG) nanofluid with radiation effects past an unsteady stretched surface. The work is carried out by involving the physical effects of thermophoresis and Brownian movement. To find the desired solutions, more practical boundary conditions are introduced familiar as the Robin and zero-mass diffusion conditions. The derived model of partial differential form is reframed into the structure of ordinary differential expressions by implementing the dimensionless variables which are reported numerically through the Runge-KuttaFehlberg (RKF) scheme with shooting procedure. Numerically determined skin-friction data and energy transportation rate are also addressed with appropriate analysis. Solutions are performed for nanofluid velocity, concentration of nanoparticles and nanofluid temperature. The analysis of dissimilar parametric values on liquid velocity, concentration of nanoparticles and temperature is reported using plots and tabulated data. The presence of Hartman and Biot numbers demonstrated the higher temperature curves. The nanoparticle concentration and temperature distributions reduced for larger unsteady parameter values. A comparison is also provided for limiting cases to authenticate our obtained results.