Common to liquids, solids and substances in between the former two is that if a stress is applied to them, they will strain. Stress may be visualized by placing a small amount of fluid between two parallel plates. When one plate slides over the other, forces act on the fluid dependent upon the rate of the plate movement. This causes a shear stress on the liquid. Recall laminar flow of fluids through a tubular vessel. Strain is the response to the stress. If solids are elastic, they deform and return to their original shape. Since fluids are not elastic and, hence, viscous, their deformation is irreversible.

Introduction to rheology

The progress made during the past decade in the application of mixed finite element methods to solve viscoelastic flow problems using differential constitutive equations is reviewed. The algorithmic developments are discussed in detail. Starting with the classical mixed formulation, the elastic viscous stress splitting (EVSS) method as well as the related discrete EVSS and the so-called EVSS-G method are discussed among others. Furthermore, stabilization techniques such as the streamline upwind Petrov‐Galerkin (SUPG) and the discontinuous Galerkin (DG) are reviewed. The performance of the numerical schemes for both smooth and non-smooth benchmark problems is discussed. Finally, the capabilities of viscoelastic flow solvers to predict experimental observations are reviewed. # 1998 Elsevier Science B.V. All rights reserved.

http://www.mate.tue.nl/~hulsen/cr/Baaijens1998.pdf

Mixed finite element methods for viscoelastic flow analysis : a review

Center manifold theory forms one of the cornerstones of the theory of dynamical systems. This is already true for finite-dimensional systems, but it holds a fortiori in the infinite-dimensional case. In its simplest form center manifold theory reduces the study of a system near a (non-hyperbolic) equilibrium point to that of an ordinary differential equation on a low-dimensional invariant center manifold. For finite-dimensional systems this means a (sometimes considerable) reduction of the dimension, leading to simpler calculations and a better geometric insight. When the starting point is an infinite-dimensional problem, such as a partial, a functional or an integro differential equation, then the reduction forms also a qualitative simplification. Indeed, most infinite-dimensional systems lack some of the nice properties which we use almost automatically in the case of finite-dimensional flows. For example, the initial value problem may not be well posed, or backward solutions may not exist; and one has to worry about the domains of operators or the regularity of solutions. Therefore the reduction to a finite-dimensional center manifold, when it is possible, forms a most welcome tool, since it allows us to recover the familiar and easy setting of an ordinary differential equation.

Center Manifold Theory in Infinite Dimensions

Wall slip of molten polymers

Taylor-Couette flow -- the flow between two coaxial co- or counter-rotating cylinders -- is one of the paradigmatic systems in physics of fluids. The (dimensionless) control parameters are the Reynolds numbers of the inner and outer cylinder, the ratio of the cylinder radii, and the aspect ratio. The response of the system is the torque required to retain constant angular velocities, which can be connected to the angular velocity transport through the gap. While the low Reynolds number regime has been very well explored in the '80s and '90s of the last century, in the fully turbulent regime major research activity only developed in the last decade. In this paper we review this recent progress in our understanding of fully developed Taylor-Couette turbulence, from the experimental, numerical, and theoretical point of view. We will focus on the parameter dependence of the global torque and on the local flow organisation, including velocity profiles and boundary layers. Next, we will discuss transitions between different (turbulent) flow states. We will also elaborate on the relevance of this system for astrophysical disks (Keplerian flows). The review ends with a list of challenges for future research on turbulent Taylor-Couette flow. 
The published version in ARFM: Annu. Rev. Fluid Mech. 48:53-80 (2016).

High Reynolds number Taylor-Couette turbulence

The cast film process is studied and a numerical simulation is performed using a finite element method. Two dimensional equations involving the thickness and a mean velocity are used by considering the film as a thin layer. The process is assumed to be isothermal between the die exit and the quick cooling on the chill roll. The width of the film is computed iteratively as a free surface problem by means of a finite element method. The thickness distribution of the film is obtained by a finite volume method. This approach allows modeling of the well known “dog-bone” or “edge bead” defect with a reasonable computation time.

Numerical simulation of the cast film process

A theoretical study of the film casting process has been carried out. In this industrial process, a molten polymer is extruded through a flat die, then stretched in air and cooled on a chill roll. This involves mainly an extensional flow. Between the die and the chill roll, thick edges are formed and a neck-in phenomenon is observed. Above a critical take-up speed, a drawing instability, known as draw resonance, may occur. In this paper, a one-dimensional model adapted from the classical model of the fiber spinning is developed (for a Newtonian or a Maxwell fluid). The influence of the processing parameters (draw ratio, Deborah number, and aspect ratio) on geometry of the lateral free surface (the so-called neck-in phenomenon) is studied. An unattainable zone very similar to the one encountered in fiber spinning is predicted, which only slightly depends on the stretching geometry (initial film width and stretching distance). The onset of draw resonance is studied through the linear stability method. A stability zone, depending on the geometry of the process, the elasticity of the polymer, and the draw ratio, has been obtained. This instability is observed with simultaneous width and thickness film variations. It is proved that the aspect ratio (stretching distance divided by die width) has a strong influence on the onset of the draw resonance instability.

Study of the stability of the film casting process

Film casting process is widely used to produce polymer film: a molten polymer is extruded through a flat die, then stretched in air and cooled on a chill roll. This study is devoted to the extensional flow between the die and the chill roll. The film shows a lateral neck-in as well as a inhomogeneous decrease of the thickness. Thickness as well as width instabilities may be observed above a critical draw ratio. An isothermal and time dependent two-dimensional (2D) membrane model is proposed and compared to a non-constant width 1D model. Newtonian and viscoelastic constitutive equations have been tested. The influence of the processing parameters (draw ratio and aspect ratio) and of the rheology of the polymer (Deborah number) on the film geometry is first determined. The onset of the draw resonance instability is finally studied by linear stability analysis and through the dynamic response to small perturbations. A critical curve splitting the processing conditions into a stable and an unstable zone is derived. It is shown that an increase of the air-gap between the die and the roll improves the stability of the process. Numerical results concerning periodic fluctuations of the flow in unstable conditions are compared with previous experimental results.

Stationary and stability analysis of the film casting process

Theory of steady Ginzburg-Landau equation, in hydrodynamic stability problems

The numerical computation of viscoelastic fluid flows with differential constitutive equations presents various difficulties. The first one lies in the numerical convergence of the complex numerical scheme solving the non-linear set of equations. Due to the hybrid type of these equations (elliptic and hyperbolic), geometrical singularities such as reentrant corner or die induce stress singularities and hence numerical problems. Another difficulty is the choice of an appropriate constitutive equation and the determination of rheological constants. In this paper, a quasi-Newton method is developed for a fluid obeying a multi-mode Phan-Thien and Tanner constitutive equation. A confined convergent geometry followed by the extrudate swell has been considered. Numerical results obtained for two-dimensional or axisymmetric flows are compared to experimental results (birefringence patterns or extrudate swell) for a linear low density polyethylene (LLDPE) and a low density polyethylene (LDPE).

Yves Demay

Papers

Numerical simulation of the cast film process

Study of the stability of the film casting process

Stationary and stability analysis of the film casting process

Theory of steady Ginzburg-Landau equation, in hydrodynamic stability problems

A finite element method for computing the flow of multi-mode viscoelastic fluids : comparison with experiments