Gass-Assisted Displacement of Non-Newtonian Fluids

During the resent years several publications (for instance Hyzyak and Koelling, J. Non-Newt. Fluid Mech. 71,73-88 (1997) and Gauri and Koelling, Rheol. Acta, 38, 458-470 (1999)) have concerned gas assisted displacement of viscoelastic fluids (polymer melts and polymeric solutions) contained in a circular cylinder. This is a simple model system used to investigate the gas-fluid displacement, as the problem is reduced to an axis-symmetric flow problem. The understanding of this process is relevant for the geometrically much more complex polymer processing operation Gas-assisted injection moulding (GAIM). This is a process, where a mould is filled partly with a polymer melt followed by the injection of inert gas into the core of the polymer melt. The numerical analysis of the fluid flow concerning the experimental observations data in these publications is all based on Newtonian or general Newtonian fluid models. As polymer melts and polymeric solutions are viscoelastic fluids an increased understanding of the displacement process can be achieved performing numerical simulation based on a viscoelastic model. This is especially important in processes that are dominated by stretch (e.g. elongation) of the fluid, as the GAIM. The stretch occurs in the fluid being displaced in front of the gas. Here we will focus on the work by Hyzyak and Koelling, J. Non-Newt. Fluid Mech. 71,73-88 (1997) and Gauri and Koelling, Rheol. Acta, 38, 458-470 (1999). They performed displacement experiments on diluted solutions of linear polymers, normally referred to as Booger fluids. These fluids have almost constant shear viscosities and elongational viscosities several order of magnitudes larger than the shear viscosities, at high Deborah numbers. The simplest possible model to describe the constitutive equation of Boger fluids is the Oldroyd-B model. This model has, with success, been able to describe the complex flow behaviours of Boger fluid. Though, refinements in the flow analysis can be obtained using more complex constitutive models. To keep the flow analysis as simple as possible the Oldroyd-B constitutive model will be used throughout this paper. A numerical method is needed in order to calculate the flow of the viscoelastic fluid during the displacement. To model the displacement numerically, the time-dependent finite element method from Rasmussen [1] is used. This method has second order convergence both in the time and the spatial discretization. The non-dimensional geometrical groups in this displacement are the Deborah and the surface elasticity number. The Deborah number is in a general definition (e.g. independent of constitutive equation) given as De=(2·U/R)·Ø1(2·U/R)/(2·çp(2·U/R)). Here U, R, Ø1 and çp are the (average) velocity of the gas, the radius of the cylinder, the first normal stress coefficient and the polymer contribution to the shear viscosity, respectively. Furthermore, the surface elasticity number is given as the ratio of the surface tension stresses relative to the elastic modulus. Using the above definitions good agreements between the Oldroyd-B displacement simulations and the experiments from by Hyzyak and Koelling, J. Non-Newt. Fluid Mech. 71,73-88 (1997) and Gauri and Koelling, Rheol. Acta, 38, 458-470 (1999) are obtained, comparing the fractional coverage defined as m=(R2-R02)/R2 where R0 is the radius of the penetrating gas front.