Application of Laplace Transform Technique of Variable Order to the Generalized Caputo Fractional Model of Second Grade Nanofluid

The incorporation of fractional calculus into nanofluid models has proven effective in capturing the complex dynamics of nanofluid flow and heat transfer, thereby enhancing the precision of predictions in this intricate field.In this study, the dynamics of a viscoelastic second-grade nanofluid model are examined through the application of the Laplace transform technique on a vertical plate.Initially, the model is formulated as coupled partial differential equations to describe the second-grade nanofluid system.The governing equations are then rendered dimensionless using appropriate dimensionless parameters.The non-dimensional model is subsequently generalized by introducing a modified Caputo fractional derivative operator.To model a homogenous nanofluid, nanoparticles of Al 2 O 3 in nanometer-sized form are suspended in mineral transformer oil.The Laplace transform is employed to solve the momentum, energy, and mass diffusion equations, providing analytical solutions.Graphical and tabular analyses are conducted to assess the influence of various physical parameters-including the fractional order, nanoparticle volume fraction, and time parameter-on the velocity, thermal, and concentration profiles.The results indicate that increasing the nanoparticle volume fraction, fractional order, and time parameter significantly enhances the rate of heat transfer.Additionally, it is observed that the velocity, temperature, and concentration profiles are notably affected by increasing the volume fraction of nanoparticles.The accuracy and reliability of the obtained solutions are validated through comparisons with existing literature.This work advances the understanding of nanofluid dynamics and presents valuable insights for industrial applications, particularly in enhancing heat transfer performance.