Artificial neural network–levenberg marquardt modeling of TiO₂–water nanofluid boundary layer on howarth's wavy cylinder with sustainable engineering perspectives

Abstract Recent advancements in the field have demonstrated that nanofluids possess a remarkable capacity to significantly enhance both hydrodynamic and thermal performance. The physical mechanisms governing mass and heat transfer are closely tied to fluid motion across a wide range of geometric configurations. One of the most practical approaches to modernizing phase change systems involves introducing nanoparticles into conventional base fluids, thereby elevating their thermal conductivity. This technique not only accelerates thermal processes but also improves structural efficiency and measurement precision. Among the emerging geometries being explored for heat transfer optimization, the sinusoidal circular cylinder has shown considerable promise across various sustainable energy applications. However, a notable gap exists in the current literature: no prior study simultaneously examined thermophoretic particle deposition along a sinusoidal circular cylinder under steady three-dimensional flow conditions, particularly when incorporating stagnation-point nanofluid flow alongside Boger fluid effects and thermal radiation. The present work addresses this gap through a comprehensive numerical investigation based on the aforementioned assumptions, using water as the primary base fluid. The central focus of this study is the flow dynamics and thermal behavior of an H₂O–TiO₂ nanofluid in the vicinity of a sinusoidally contoured cylinder. The governing partial differential equations are systematically reduced and solved using the bvp4c solver within the MATLAB computational environment, which enables accurate and efficient numerical treatment of the resulting ordinary differential equations. The outcomes are thoroughly analyzed and illustrated through graphical representations. The results consistently indicate that nanoparticle shape plays a decisive role in determining the overall heat transfer performance of the system. The improved values of solid volume fraction from 0.01 to 0.05, the surface drag force Cfx increases from 0.668% to 3.11% and Cfy increases from 0.766% to 3.36%. The dimensionless boundary-layer system, reduced via similarity transformation to a set of coupled nonlinear ordinary differential equations (ODEs), is solved numerically and the solution dataset is subsequently used to train a feedforward artificial neural network (ANN) via the Levenberg–Marquardt backpropagation (LMB) algorithm. This article details, with equation-level specificity, how each architectural element of the network corresponds to a physically motivated quantity in the governing formulation, why the diagnostic outputs mean squared error (MSE), regression coefficient, error histogram, and training-state curves constitute independent corroboration of the numerical solver, and how the network functions as a low-cost surrogate for parametric exploration across a high-dimensional input space. Statistical evidence indicates a global regression coefficient of R = 0.99992 and a best-epoch MSE of 3.3679 × 10 −4 , confirming that the trained network faithfully encodes the coupled nonlinear dynamics of the present nanofluid system.