Stabilized finite element schemes for computations of viscoelastic free-surface and two-phase flows

Abstract

Viscoelastic flows can be found in a wide range of industrial and commercial applications such as enhanced oil recovery, pesticide deposition, medicinal/pharmaceutical sprays, drug delivery, injection molding, polymer melts, inkjet printing, cosmetics industry and food processing. With the inherent complexity of viscoelastic fluids due to complex secondary flows and transient flow patterns even in very simple geometries and the resulting analytic intractability of the mathematical models in this area, computational approaches are playing an ever increasing role. The focus of this thesis is to develop and implement a stable, efficient and robust finite element scheme for computations of viscoelastic fluid flows. Numerical simulations of incompressible viscoelastic flows involve simultaneous solution of the Navier–Stokes equations and a viscoelastic constitutive equation. Although a considerable progress has been made in this field, many challenges still remain in computations of viscoelastic fluid flows. The constitutive equation is highly advection dominated which may induce both global and local oscillations in the numerical solution. Further, the choice of the approximation spaces for the velocity, the pressure and the viscoelastic stress is restricted by the compatibility conditions. In addition, all numerical schemes that simulate viscoelastic flows encounter a major challenge : the so-called high Weissenberg number problem (HWNP), i.e. difficulty in obtaining mesh-converged numerical solutions even for simple benchmark problems at high Weissenberg numbers. In this work, a new three-field formulation based on the Local Projection Stabilization (LPS) is developed for computations of the coupled Navier–Stokes and Oldroyd-B viscoelastic constitutive equations at high Weissenberg numbers. One-level LPS is based on an enriched approximation space and a discontinuous projection space, where both spaces are defined on a same mesh. It allows us to use equal order interpolation spaces for the velocity and the viscoelastic stress, whereas inf-sup stable finite elements are used for the velocity and the pressure. Since the stabilization terms in LPS are assembled only once, the proposed scheme is computationally efficient in comparison with residual based stabilized numerical schemes. Numerical studies using method of manufactured solutions show an optimal order of convergence in the respective norms. Further, the proposed scheme is validated using two benchmark problems : flow past a cylinder in a rectangular channel and lid-driven cavity flow. Moreover, the numerical results are compared with the results in the literature and the effects of elasticity and inertia are analyzed. In the second part of the work, an accurate and efficient sharp interface arbitrary Lagrangian– Eulerian (ALE) finite element approach is developed for the computations of viscoelastic free-surface flows. As an example, we considered an isothermal 3D-axisymmetric viscoelastic droplet impinging on a horizontal solid surface. The coupled Navier–Stokes and the Giesekus constitutive equations are solved using this numerical scheme. The highlights of the numerical scheme are the tangential gradient operator technique for the curvature approximation and the contact angle inclusion in the variational formulation, the ALE approach with moving meshes to track the free surface, derivation of 3D-axisymmetric variational form using cylindrical coordinates and three-field local projection stabilized formulation. In addition to the mesh convergence study, parametric studies of the Weissenberg number, Newtonian solvent ratio, polymeric viscosity, Reynolds number and equilibrium contact angle are performed to demonstrate the effects of viscoelasticity on the flow dynamics of the droplet on wetting surfaces. In the third part of this work, a finite element scheme using the one-level LPS and ALE approach is developed for computations of viscoelastic two-phase flows. As an example, we considered a 3D-axisymmetric buoyancy driven bubble rising in a liquid column in which either the bubble or the liquid column can be viscoelastic. A comprehensive numerical investigation is performed for a Newtonian bubble rising in a viscoelastic fluid and a viscoelastic bubble rising in a Newtonian fluid. The influence of the viscosity ratio, Newtonian solvent ratio, Giesekus mobility factor and the Eötvös number on the bubble dynamics are analyzed. Interesting flow features such as indentation around the rear stagnation point with a dimpled bubble shape or an extended trailing edge with a cusp-like shape of the bubble are captured by the proposed numerical scheme. In the final part of this work, a coupled ALE-Lagrangian approach is developed for computations of buoyancy driven viscoelastic two-phase flows with insoluble surfactants on the interface. A number of computations are performed for a Newtonian bubble rising in a viscoelastic fluid and a viscoelastic bubble rising in a Newtonian fluid with insoluble surfactants on the interface. The influence of the surfactant elasticity, initial surfactant concentration and Peclet number on the rising bubble dynamics are analyzed