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Abstract
This thesis presents a new numerical algorithm for the simulation of two‐dimensional multiphase viscoelastic
flows. The simulation of viscoelastic flows has both a scientific importance and practical implications in polymer
processing. This work has put the emphasis on the extrusion of polymeric materials, where viscoelastic effects
cause dynamical instabilities, despite the very simple geometry. This thesis reviews the popular differential
constitutive models derived from molecular theories of dilute polymer solutions, polymer networks, and
entangled polymer melts, as well as the inelastic phenomenological models describing shear–thinning and
viscoplastic (yield stress) fluids, based on the generalized Newtonian fluid model. In addition, the numerical
issues related to the high Weissenberg number problem, and its remedy with the log–conformation
representation, are discussed.
The proposed algorithm utilizes a new streamfunction/log–conformation scheme. The drawbacks of the
classical velocity–pressure decoupled method, which is by far the most popular approach, are remedied with the
pure streamfunction formulation, which is derived from the pressureless vorticity‐based methods. The implicit
pure streamfunction formulation is formally more accurate than the velocity–pressure decoupled method,
because it is immune of decoupling errors. Moreover, the absence of decoupling enhances the stability of the
calculation. The governing equations (conservation laws and constitutive models) are discretized with the finite–
volume method, on a Cartesian grid. Discrete curl operators are applied to the discretized momentum equations
in order to obtain the matrix system of the discrete streamfunction variables. The coupling of the
streamfunction/log–conformation scheme with adaptive under‐relaxation and adaptive time‐stepping yield a
robust and efficient viscoelastic flow solver algorithm. The potential extension of the method to threedimensional
simulations is also discussed in this thesis.
Bi‐phasic/free‐surface flows are modelled with the Volume–of–Fluid (VOF) method, and the standard
piecewise–linear–interface–construction technique. In addition, a new Cellwise Conservative Unsplit (CCU)
advection scheme is presented. The CCU scheme updates the liquid volume fractions based on cellwise
backward‐tracking of the liquid volumes. The algorithm calculates non‐overlapping and conforming adjacent
donating regions, which ensures the boundedness and conservativeness of the liquid volume. As a result, the CCU
advection scheme is overall more accurate in classical benchmark tests, than the other state‐of‐the‐art multidimensional
VOF–advection schemes. In complex flows, the convergence rate of the CCU scheme with mesh
refinements is between 2 and 3. Moreover, the remaining geometrical errors are mostly due to the inability of
the standard piecewise linear interface to represent subgrid material topologies (i.e. high curvatures and thin
material filaments), rather than the proposed CCU advection scheme.
This thesis reports examples of numerical simulations of the Oldroyd–B liquid, calculated with the proposed
streamfunction/log–conformation/VOF–CCU methodology, implemented in Matlab. A thorough investigation of
the viscoelastic flow in the lid‐driven cavity is conducted. The streamfunction/log–conformation shows secondorder
accuracy and numerical stability at very large time‐step increments, which demonstrates the robustness of
vi
the scheme. The numerical results at moderate Weissenberg numbers are in good agreement with the literature.
Moreover, the enhancement of numerical stability, with the streamfunction/log–conformation scheme, makes it
possible to simulate elastic instabilities at high Weissenberg numbers. Quasi‐periodic elastic instabilities at the
upstream corner appear to be a mechanism that dissipates the stored elastic energy. The simulations of
viscoelastic flows in the planar 4:1 contraction are also in good agreement with data in the literature. Finally,
preliminary simulations of extrudate swelling show that the fracture melt extrusion defect could be caused by
instabilities in the stress layer at the surface of the die, triggered at moderate Weissenberg numbers.
Original language  English 

Place of Publication  Kgs. Lyngby 

Publisher  Department of Energy Conversion and Storage, Technical University of Denmark 
Number of pages  234 
Publication status  Published  2015 
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 1 Finished

Numerical modelling of extrusion of functionally graded ceramic materials
Comminal, R. B., Hattel, J. H., Pryds, N., Spangenberg, J., Walther, J. H., Alves, M. A. M. & Kupferman, R.
01/01/2011 → 24/08/2015
Project: PhD