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Researchers are paying close attention to Casson fluid due to their noteworthy applications in industries. Among the non-Newtonian substances, the Casson fluid is one of the most important types. In this paper, the flow of sodium alginate-based Casson hybrid nanofluid flow mixed with copper and alumina nanoparticles across a stretching sheet is investigated. The slips and convective boundary conditions are assumed in the problem. Moreover, the flow of the hybrid nanofluid is magnetically influenced. Additionally, the consequences of viscous dissipation are considered. Through similarity variables, the leading nonlinear PDEs are converted into ODEs, which are then simplified by the homotopy analysis method (HAM). The variations in velocities, temperature, skin friction, and Nusselt number are exhibited in the figures. With the help of figures, the convergence analysis of the present model has been shown. The result reveals that the Casson factor has a declining consequence on the velocity profile along the x − direction and y − directions. An improvement behavior is observed in the velocity plot in the y − direction through the stretching ratio parameter. The Casson parameter has a declining effect on the velocity profile along x-direction and ydirections. It is noticed that the temperature plot shows an escalating behavior through the Eckert number and magnetic parameter. From the numerical result, it is concluded that the Skin friction and Nusselt number of hybrid nanofluid is maximum as compared to nanofluid.

The current research focuses on harnessing the computational power of neural networks, notably the Levenberg-Marquardt backpropagation optimization algorithm (ANN-LMBOA), to simulate the behavior of magnetohydrodynamic (MHD) three-dimensional non-Newtonian Darcy-Forchheimer flow of Casson nanofluid over a stretching surface situated within a porous medium. The fluid flow is subject to the influence of Hall and ion slip forces, contributing to its behavior. Additionally, the Cattaneo–Christov heat flux model is incorporated to represent the temperature balance in the boundary layer accurately. Non-Newtonian fluids improve thermal transmission due to their shear-thinning propensity, increased turbulence, and more excellent convective heat transfer. The thermal conductivity of nanofluids is increased by the inclusion of nanoparticles, enabling more effective heat transmission. Buongiorno's model is used for the nanofluid phenomena that concentrate on thermophoresis and Brownian motion. To tackle the flow problem efficiently, the governing equations are transformed into non-linear differential equations using self-similar variables. The ANN-LMBOA algorithm is then employed to solve these equations, providing a practical and accurate approach to the simulation. To train and validate the ANN-LMBOA model, a dataset is generated using the bvp4c numerical method, which can easily handle highly non-linear coupled differential equations. The dataset is created for different scenarios of flow parameters, facilitating comprehensive training, testing, and authentication of the neural network. The accuracy of the ANN-LMBOA model is assessed through various statistical neural network tools, such as MSE, RP, CF graphs, and EH. The study thoroughly investigates flow model parameters related to momentum, energy, and concentration profiles, which are presented using visual representations to understand the system's behavior and characteristics comprehensively. The effects of various parameters of interest like Casson, ion slip, porosity, thermal relaxation, and magnetic parameters on velocity, temperature, and concentration distribution have been studied numerically via ANN-LMBOA networks. The absolute error of reference and target data is obtained in the range of 10−4 − 10−5which proves the best accuracy performance of ANN-LMBOA networks. By raising the Prandtl and Schmidt numbers, solutal, thermal, and diffusional variables, one can reduce the diffusion of mass and heat.