Numerical simulation of flow and heat transfer of variable viscosity fractional order Maxwell fluid on stretched plate

This paper investigates the unsteady flow and heat transfer behaviors of viscoelastic fluids over a stretched plate. A novel fractional-order Maxwell constitutive relationship is proposed, which forms a coupled velocity–temperature–stress system and innovatively incorporates the temperature dependence of relaxation time. The governing partial differential equations, characterized by high nonlinearity and coupling effects, are numerically solved via a hybrid method combining the spectral method (for spatial discretization, ensuring high accuracy) and the finite difference method (for temporal discretization, improving stability). Analytical solutions under simplified conditions are derived to validate the reliability and accuracy of the numerical results. Comparative analyses of velocity, temperature, and shear stress fields are conducted across different constitutives, focusing on the influence of the Improved Variable Viscosity Model on flow and heat transfer characteristics. This work provides a reliable theoretical framework and numerical tool for optimizing fluid manipulation and process design in engineering applications involving stretched plate configurations. It enriches the theoretical system of fractional viscoelastic fluid mechanics and offers critical guidance for enhancing the performance of related engineering systems such as polymer processing and heat exchanger design. Moreover, the proposed hybrid numerical method and constitutive model can serve as a reference for subsequent studies on complex fluid flow and heat transfer problems in other similar configurations.