Double-stratified Marangoni boundary layer flow of Casson nanoliquid: probable error application

The present research is manifested to observe the impact of double stratification in the Marangoni convective flow of the Casson nanoliquid model toward an inclined surface. Due to permeable surface, Marangoni convective flow has been driven by concentration and temperature gradients. Investigation is compiled with the aid of Brownian diffusion, thermophoretic feature and double stratification in Marangoni convection of Casson nanoliquid. The analysis of heat and mass transfer is scrutinized through different physical flow parameters. The nonlinear structure of governing partial differential equations (PDEs) is reduced into nonlinear ordinary differential equations (ODEs), using some suitable classical transformations. Numerical technique is used for the solution of a modified set of ODEs by invoking Runge–Kutta fourth-order algorithm. Graphical illustration is focused to observe the effect of relevant flow parameters on the velocity, temperature and concentration profiles. To examine the variation in heat transfer and the significance in industrial zone, Nusselt number is organized in tabular form. Outcomes indicate that the impact of Marangoni ratio parameter on velocity field results in certain rise. The impact of inclined MHD results in a decline in fluid flow. A rise in heat flux is noted for increment in Prandtl number, while decline in the mass flux is noted. Furthermore, some values of correlation r are less than $$P.E.\left( r\right)$$ which shows the statistical insignificance of correlation coefficient. It is also noted that some correlation coefficients are remarkable, and the parameters are greatly interconnected to the physical attributes. The considered problem has multiple applications in physics, chemistry, industry, aerospace and biological fields.