Microscale Polymer Processing
David's work with the Polymer Fluids Group was carried out as part of the interdisciplinary µPP2 project and focused on experimentally identifying the flow behaviour of polymer melts. This was done by placing the polymer in the Cambridge MultiPass Rheometer (MPR) and capturing stress patterns using optical birefringence and velocity values using Laser Doppler Anemometry (LDA). These parameters were then used to compare and validate Lagrangian finite element simulations performed using Flowsolve.
The Lagrangian finite element software flowSolve is being developed at Leeds University as part of the µPP2 Polymer toolbox for accurate modelling of industrial polymer flows. As the stress of a polymer is a function of its history, flowSolve models the polymer as a series of triangular elements that flow and deform through the geometry. Recently developed POM-POM or ROLIE POLY equations calculate the velocity and stress through these elements and results are compared with experimental work undertaken in the MPR for validation. An example is given on this page of flowSolve simulations modelling the flow of PS680E (polydisperse polystyrene) from DOW chemicals through a constriction using the molecularly based POM-POM equations. Figure 1 sets out the equations used in the POM-POM mode of flowSolve. Figures 2 to 5 show results obtained for a constriction geometry with varying mesh refinements containing from 2000 to 20000 mesh elements (or points, subsequently defined here as p). The stress pattern develops over time due to the stress history of the polymer, so the simulations were run over time steps of approximately 0.01698. Each step took approx 2 mins with p=2000, 8 mins for p=7500 and 3 hours for p=20000. The first two simulations were run for 84 time steps while the last one was used to investigate mesh dependance and so was run for 2 time steps.
|Figure 1 POM-POM equations used by flowSolve|
|Figure 2: This picture illustrates the complex development of the mesh as the polymer flows through the constriction. The Polymer is moving from top to bottom and as the initial elements move through the geometry fresh elements are introduced at the flow inlet. The element colour represents pressure (p=2000, t=0.01698).||Figure 3: This picture illustrates the velocity profile and magnitude corresponding to the same point in time as figure 2. Recirculation zones are seen either side of the constriction close to the main channel wall and the Polymer accelerates as it enters the constriction (p=2000, t=0.01698).|
|Figure 4: This picture illustrates the stress corresponding to the same point in time as figure 2 (p=20000, t=0.01698).||Figure 5: This picture illustrates the stress birefringence patterns corresponding to the same point in time as figure 2. While this image is adequately resolved for comparison with experimental data, a further set of mesh resolutions near the slit wall would be required to fully resolve the birefringence pattern (p=20000, t=0.01698).|
Simulation convergence accuracy
Figure 6 shows the variation in parameter values for this range of mesh sizes. An accurate analysis of convergence accuracy based on Richardson extrapolation was impractical due to the method used by flowSolve to export parameter values and the time dependant nature of the mesh position.
Figure 6: These graphs show the x velocity (top left), y velocity (top right), pressure change (bottom left) and stress (bottom right) along a line in the y-axis one tenth of the way from the slit sidewall. The values of p correspond to the number of mesh elements in a simulation and show the solution progression through progressive mesh refinements.
Stress evolution with time
Given the viscoelastic properties of the polymer, the stress in the system will evolve over time from an initially static beginning to steady state flow conditions. Using the p=7500 simulation this can be seen along two lines in the system, Y=0 and Y=0.9. These correspond to the centre line of the system and a line positioned such that it is close to the side wall. This is seen in figure 7.
Figure 7: These graphs show the stress evolution in time along (left) Y=0 and (right) Y=0.9 (p=7500).
All the simulations shown here were performed on a 800 MHz Pentium processor machine with Windows operating system and 512Mb of RAM.