Shape factor and temperature-dependent viscosity analysis for the unsteady flow of magnetic Al $$_2$$ O $$_3$$ –TiO $$_2/$$ C $$_2$$ H $$_6$$ O $$_2$$ –H $$_2$$ O using Legendre wavelet technique

The unsteady flow of fluids is crucial for real-world applications, efficiency and performance optimisation in various sectors, such as engineering, environmental impact research and developing technologies. Shape of nanoparticles in hybrid nanofluids is important for optimising energy applications and customising the performance of the nanofluid as it has effect on heat transport, material characteristics and stability. Considering the importance of unsteady flow and the shape factor of the nanoparticles, the present study aims to explore the solution of the unsteady flow problem of hybrid nanofluid over a stretching surface embedded within the porous medium. This study deals with the shape factor analysis by considering four shapes: brick, lamina, platelet and blade. The Legendre wavelet collocation technique is implemented to obtain the solution of the problem. It is revealed by creating a pie chart that the thermal conductivity is found to be maximum for lamina-shaped nanoparticles (i.e., 32%) while it is minimum for brick-shaped nanoparticles (i.e., 20%). The rate of heat transfer enhancement is also presented by the waterfall graph. The graph disclosed that on increasing the volume fraction of TiO\(_2\) from 1 to 10%, the rate of heat transfer is enhanced by 39.63%. The velocity profiles are inversely related to the temperature-dependent viscosity and velocity slip parameter.