| dc.description.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 | en_US |
