Viscoelastic subdiffusive flows : modelling, analysis and computation

Abstract

This thesis explores the rheodynamics of viscoelastic subdiffusive fluids and high-lights the potential of fractional calculus in modelling these types of fluids. A novel fractional model is developed to investigate the regions of spatiotemporal instability. Direct numerical simulations are used to capture the macrostructures within the flow, utilizing a new structure tensor, which is physically realizable, contributing to a deeper understanding of this complex fluid dynamics phenomenon. The seven chapters are described as follows: Chapter 1 lays the foundation by introducing subdiffusive fluids and delving into the historical backdrop of fractional models applied to viscoelastic flows. This chapter surveys past analytical outcomes, numerical strategies, and experimental findings, exhibiting a wide range of applications in various fields. In Chapter 2, the fundamental groundwork is established, including a thorough explanation of several types of instability: convective, absolute, and evanescent modes. This chapter elucidates numerical methods, especially finite difference schemes, designed to handle various equation components such as advection and diffusion terms and also discusses the different fractional derivatives and their corresponding numerical approximations. Chapter 3 embarks on a comprehensive investigation of temporal and spatiotemporal linear stability analyses within the viscoelastic subdiffusive plane Poiseuille flow. The Fractional Upper Convected Maxwell model is derived and explored under low to moderate Reynolds numbers (Re) and Weissenberg numbers (We) to uncover regions of topological transition in advancing flow interfaces. The relation between the exponent in the time scale of the microscale models (tα ) and fractional order (α) in stress constitutive equations is derived. Using Brigg’s method of analytic continuation, insights into instability modes as fractional derivative order changes are discussed. The stability studies are limited to two exponents: monomer diffusion in Rouse chain melts, α = 1/2, and in Zimm chain solutions, α = 2/3. The presence of a non-homogeneous environment with hindered flow is revealed by the discovery of an abnormal region of temporal stability at high fluid inertia, highlighting the potential of the model to accurately capture some experimentally observed flow-instability transitions in subdiffusive flows. Chapter 4 presents a theory to quantify the development of the spatiotemporal macrostructures for viscoelastic sub-diffusive flows by decomposing the polymer conformation tensor into the so-called structure tensor. Our method bypasses the traditional arithmetic decomposition’s fundamental flaw, which is that the fluctuating conformation tensor fields might not be positive definite and, as a result, lose their physical significance. By defining and building a geodesic via the inner product on its tangent space, the space of positive definite matrices is converted into a Riemannian manifold using some well-proven results in matrix analysis. Three scalar invariants of the structure tensor are defined by means of this geodesic. The maximum amount of time that the perturbative solution’s evolution may be accurately predicted by linear theory along the Euclidean manifold is found. Chapter 5 introduces a novel family of time-asymptotically stable, implicit-explicit, adaptive, time integration methods (denoted with the θ -method) for the solution of the fractional advection-diffusion-reaction equations. The computationally explicit L1 method is generalised by this class of temporal integration techniques. For a specific range of Peclet numbers, the dispersion relation analysis of the method which takes into account the group velocity and the phase speed, indicates a favorably large region. The one-dimensional fractional diffusion equation is used to validate the method’s correctness and effectiveness. Chapter 6 includes the direct numerical simulations of viscoelastic, subdiffusive, plane Poiseuille flow in the regime of low to moderate Reynolds number and low Weissenberg number. These simulations successfully capture the flow structures by offering (i) a better resolution of the instantaneous regions of elastic shocks (which are the alternating regions of expanded and compressed polymer volume, in comparison with the volume of the mean conformation tensor), and (ii) a higher resolution to identify areas where the mean conformation tensor tends to diverge significantly from the instantaneous conformation tensor, supporting the experimentally observed transition of subdiffusive flows into flow instability. Chapter 7 summarizes the real-life applications involving viscoelastic subdiffusive fluids, focusing on their rheological properties. Fractional models, although instrumental in comprehending intricate phenomena with non-local and memory-dependent behaviour, do present certain limitations. To wrap up, this chapter points out the main challenges in understanding these flows and emphasizes potential future problems that need serious attention in this new category of complex fluids