Theoretical and Computational Methods for Fractional Viscoelastic Models

Abstract

This thesis investigates the behaviour of viscoelastic fluids, such as stress relaxation under constant deformation and time-dependent deformation recovery following load re moval, and explores how to characterize this behaviour using mechanical models. These models demonstrate how fractional viscoelastic models can be derived using spring-pot elements arranged in series and/or parallel. To accurately capture the complex behaviour of viscoelastic fluids, numerical techniques for solving fractional viscoelastic models are developed, as these models frequently employ fractional differential equations to account for the material’s characteristics and memory effects. New fractional viscoelastic models have been derived by extending the single-mode fractional Maxwell model to a multi-mode framework by considering springpots arranged in series and/or parallel to study more complex behaviour. Our theoretical analysis has derived new expressions for the exact solution of these models equations—using the Laplace transform of the Green’s function and expanding in terms of the Mittag-Leffler function (MLF)—as well as for the relaxation time and the dynamic moduli in both single mode and multi-mode settings. This method highlights its effectiveness as a powerful tool for solving various fractional differential equations and boundary value problems in real world applications, while also providing a strong foundation for future studies. Furthermore, an accurate numerical method has been developed to solve two coupled fractional differential equations for Taylor-Couette flow by employing a spectral approxi mation for spatial discretization and a finite difference scheme for temporal discretization. High-order schemes ensure accurate modelling of complex fluid behaviour, and the conver gence properties of the numerical scheme are investigated. Numerical results are presented which highlight the influence of the parameters in the fractional viscoelastic models on the numerical predictions.