Computational study with neural networks to double diffusion in Prandtl thermal nanofluid flow adjacent to a stretching surface design with numerical treatment

Abstract

The study focuses on the implementation of neural networks in analyzing the behavior of Prandtl nanofluid near a extending surface in the existence of dual diffusion. The research investigates the merged consequence of thermal and concentration gradients on the fluid flow and heat transfer characteristics using advanced computational techniques. The investigation delves into the phenomenon of double diffusion in the flow of Prandtl nanofluid near a stretching surface (DD-PNSS), employing the Levenberg-Marquardt scheme with Artificial Neural Networks (LMS-ANNs). The application of similarity variables converts the non-linear partial differential equations into non-linear ordinary differential equations. Through the application of the Lobatto IIIa formula in a three-stage process, various data sets are generated for the LMS-ANNs by varying parameters such as the Prandtl fluid parameter ( $\alpha$ ), Prandtl number (Pr), Brownian motion parameter (Nb), thermophoresis parameter (Nt), and Dufour-solutal Lewis number (Ld). The range of parameter α is from 2 to 3.5, parameter β from 0.1 to 0.9, Nb is from 0.1 to 0.4, Nt is from 0.1 to 0.7, Le is from 1 to 3 and Ld is from 0.1 to 0.5. The intrinsic behavior of embedded parameters on dimensionless velocity, temperature, solutal concentration, and nanoparticle concentration profiles is graphically evaluated. The proposed LMS-ANNs model is meticulously tested, validated, and trained using a multi-stage approach, and its performance is compared to established references to ensure its reliability. The effectiveness of the suggested LMS-ANNs model is further affirmed through regression analysis, Mean Squared Error (MSE) evaluation, and histogram studies, showcasing an exceptional accuracy level ranging from 10⁻⁰⁸ to 10⁻¹⁰, setting it apart from alternative approaches and reference models.