Nanofluid plays a crucial role in addressing the heat transmission challenges facing the industries including chemical processing systems, automobile radiators, spacecraft design, concrete heating, solar thermal conversion systems, etc. Consequently, this research is devoted to analyzing the solar radiation mechanism, thermophoresis and Brownian motion under unique conditions, specifically, temperature gradient within liquids with limiting viscosities or plastic dynamic viscosity at zero and infinite shear rate. To increase the model novelty, a model involving two phases of Casson–Carreau fluid conveying tiny particles is developed to analyze the flow of solar radiation mechanism over stationary and moving surfaces. Furthermore, the impacts of heat generation, chemical reaction and activation energy are also taken into account. The dimensionless equations are solved through numerical methods using the Galerkin-weighted residual technique and analytically employing the Homotopy analysis method with the assistance of MATHMATIACA 11.3 software. Our study reveals that the fluid temperature reduces for greater values of Weissenberg number, and Casson parameter. Additionally, the distribution of gyrotactic microorganisms decreased against the bio-convective Schmidt number and Peclet number.