Impact of nanoparticles shapes and fuzzy volume fraction on Al2O3 – H2O nanofluid flow past an unsteady expandable surface

• Investigates the impact of nanoparticle shapes (cylinder vs. brick) and fuzzy volume fraction on the heat transfer dynamics of Al₂O₃-H₂O nanofluids over an unsteady stretching surface. • Employs triangular fuzzy numbers (TFNs) to represent nanoparticle volume fraction, utilizing membership functions to explore variations between 0 and 1. • Highlights the complex interactions between nanoparticle volume fraction fuzziness and heat transfer dynamics in nanofluids. • Provides insights for optimizing heat transfer processes in various fields, including thermal engineering and materials science. • Advanced understanding of nanofluid dynamics, offering a novel approach to modelling and quantifying the influence of different parameters on system behaviour. This study investigates how fuzzy variable volume fractions of Al 2 ​O 3 ​ nanoparticles affect heat transfer in nanofluid flow over a stretching sheet, highlighting nanofluids as a promising thermal performance enhancer. Rapid advancement of nanotechnology highlights nanofluids as effective thermal enhancers. However, understanding how nanoparticle shapes and fuzzy volume fraction variability influence flow and heat transfer characteristics remains limited. To accomplish this, a comprehensive mathematical model is developed, integrating the Navier-Stokes and energy equations to characterize nanofluid flow over an extending surface and examine nanoparticle shape effects. The goal is to simplify the nonlinear partial differential equations (PDEs) through appropriate transformations, converting them into ordinary differential equations (ODEs). The primary aim is to address the complex relationships between various parameters and their impact on nanofluid dynamics and heat transfer processes on an unsteady stretching sheet. This study compares cylinder and brick-shaped nanofluids using membership functions (MFs), where the nanoparticle volume fraction is represented as a triangular fuzzy number (TFN), controlled by the α-cut, varying between 0 and 1. Ultimately, this study highlights the complex interaction between nanoparticle volume fraction fuzziness and heat transfer dynamics in nanofluids flowing over a stretching sheet. The results indicate that temperature distribution is more pronounced in nanofluids with cylindrical nanoparticles than in those with brick-shaped nanoparticles. Additionally, a noticeable increase in fuzzy temperature is observed with higher unsteady parameters and Eckert numbers. This study enhances our understanding of nanofluid dynamics, providing insights to optimize heat transfer, advance thermal engineering, materials science, and design parameters, with practical applications.