Flow front instabilities in an injection moulding process

During an injection moulding process, a process that is often used for the production of plastic objects, molten polymer is injected into a mould. In many cases, the mechanical properties as well as the optical properties of these objects are relevant. Due to internal stresses and distortions of the surface, the optical quality of the product may diminish, e.g. the information on a compact disc can not be read properly. One of the processes that is considered to be responsible for this kind of distortions is an instability of the motion of the flow front during the filling phase of the injection moulding process. There are several mechanisms that may cause this kind of instability, e.g. thermal and/or viscoelastic properties of the polymer. In this PhD-research, we have investigated in what extent thermal properties of the fluid may cause the existence of (anti-symmetric) instabilities of the flow front during the injection moulding process of thin flat plates. In this case, the walls of the mould are flat and parallel to each other. The basic idea in this research is that the viscosity of the fluid is highly dependent on the temperature of the fluid. Since the walls of the mould are cooler than the temperature of the fluid itself, the viscosity of the fluid will be higher near the walls than in the main part of the fluid. The thought behind the development of this kind of instabilities is that when the point of contact of the flow front with one wall (wall 1) lies further downstream than the point of contact with the other wall (wall 2), the temperature of the fluid in a certain cross section will be lower near wall 1 than near wall 2. Therefore, the viscosity near wall 1 will be higher than the viscosity near wall 2. Hence, the velocity near wall 1 will be lower, meaning that the point of contact of the flow front with wall 2 will move downstream faster than the point of contact with wall 1, which causes the situation to reverse. This results in a "wobbling" flow front. The main questions during this research are: Can this mechanism cause the flow front to wobble? and Under what circumstances will the amplitude of the wobbling of the flow front increase or decrease? The research has been done completely by analytical means. Since we only consider thermal effects on the behaviour of the flow front, for the stresses in the fluid we take a generalized Newtonian model, where the viscosity depends on the temperature only. Since the thickness of the plates is much smaller than the width, it may be assumed that the velocity and temperature fields in the unperturbed situation are two-dimensional. We assume that the same holds for the instabilities. Also, we consider a time-interval for which the influence of the inlet is not noticeable in the flow front region, and vice versa, from which it follows that the situation near the inlet, and the situation near the flow front can be considered independently of each other. In order to determine the existence of this kind of flow front instabilities, we need an (analytical) expression for the temperature and velocity fields in the unperturbed situation, i.e. the situation where the flow front is symmetric. To calculate the unperturbed velocity field, we used complex function theory. The flow region is considered to be part of the complex plane. Written in complex variables, the velocity problem can be reduced to the determination of two analytical functions. By mapping the flow region onto a unit circle, we show that this problem is equivalent to solving a Hilbert problem. Since the flow front is a free boundary, its shape has to be determined as well. For this, we use that the flow front is stress-free. To determine the temperature field in the unperturbed situation, we use the fact that the P´eclet number of the fluid is much larger than one, from which it follows that in the main part of the flow region, the temperature is equal to the inlet temperature. Only in thin thermal boundary layers near the walls the temperature will be different due to the fact that the walls are cooled. The temperature problem can be solved using asymptotic approximations and Wiener-Hopf techniques. To determine the existence of anti-symmetric instabilities of the flow front, we use linear perturbation techniques. The symmetric flow front is perturbed with an anti-symmetric time-harmonic function. The ultimate goal is to determine whether, and if so, under what conditions solutions of the perturbed problem exist for which the amplitude of the solution increases in time. In this case, the flow front will be unstable. Similar to the unperturbed problem, the shape of the flow front in the perturbed problem follows from the condition that the flow front is stress-free. This condition leads to an evolution equation from which the shape of the flow front can be determined as a function of the frequency of the perturbation. The frequency of the perturbation follows from the initial condition of the amplitude of the perturbation, and depends on the velocity of the flow front. There exists a critical value for the velocity of the flow front. If the velocity of flow front is greater than this value, then the amplitude of the perturbation decreases, and the solution is stable; if the velocity of the flow front is less than this value, then the solution is unstable. However, if the velocity of the flow front becomes too small, the assumption that the P´eclet number is large no longer holds.