Turbulent waves and oscillating amplitude of boundary-layers and radiative-heat transfer of nonlinear Williamson nanofluid over solar plate with periodic conditions

Turbulence and oscillating amplitude of boundary layers and nonlinear radiative heat and concentration motion of transient Williamson nanomaterial model alongside inclined solar surface with entropy optimization, magnetic oscillation, dynamic solar energy, and thermo diffusion is the novelty of this work. The objective of this problem is to scrutinize the speed streamlines, isothermal lines, energy/temperature and mass/concentration fields, steady heating ratio and instability of fluctuating energy/heat transfer alongside inclined plate at angle π/4. Governing mathematical formulation is developed to find the numerical outcomes of unknown physical properties. Fluctuating Stokes transformations and dimensionless variables are developed to reduce governing equations into oscillatory and steady equations with defined boundary values. Various pertinent parameters are generated for physical analysis of heat rate such as Eckert number ($E_c$), buoyancy number (${\lambda }_T$), oscillating magnetic parameter ($M_f$), Schmidt number ($S_c$), radiation number ($R_d$), thermophoresis number ($N_T$), Prandtl number (Pr), and Brownian motion number ($N_b$). Primitive variable transformation is applied to generate symmetry in all governing equations in FORTRAN programming tool for accurate asymptotic results. Implicit formula of finite difference scheme with central and backward relation is used to convert system of equations into algebraic equations. The obtained algebraic equations are solved through Gaussian elimination technique to explore fluid flow, heat and mass rate, and periodical fluctuations in thermal and concentration gradients. It is perceived that high largeness in fluid speed and energy/temperature field is depicted as thermal radiating constraint upsurges. The magnitude of streamlines ad isothermal lines is decreased as oscillating magnetic number and Eckert number increases but isothermal lines increased with maximum strong magnetic field. High oscillations in turbulent behavior of fluctuating heat and concentration/mass removal is noted for each value of magnetic field and Weissenberg number.