The important industrial process of casting polymeric films suffers from the "draw resonance" instability that appears as sudden oscillations in the product dimensions. This instability influences the quality of the end-product and negatively limits productivity and efficiency of the process. The draw resonance originates when a material is being processed beyond the limits of its intrinsic properties. Research is conducted with the intention to find those process and material properties that allow to optimize the production process while keeping it stable. This paper concentrates on a non-isothermal analysis of the stability of the film casting. The mathematical model of the process is given by a quasi-linear system of first order PDEs with two point boundary conditions. The constitutive polymer behavior is approximated by the modified Giesekus model. Linear stability analysis combined with the Laplace transformation of the resulting linear system is applied to find parameters that determine mathematical and thus process instability. It all comes down to determining the spectrum of a compact operator; corresponding eigenfunctions can be regarded as the characteristic modes of the system. For implementation, the modification of Galerkin approach is used. The major advantage of the mathematical and numerical method is that the full spectrum is calculated in a matter of seconds. Our results agree perfectly with the ones from literature for isothermal case, and with the experimental data for the non-isothermal case. The results also indicate that non-isothermality is highly important and cannot be excluded from modeling.
Zavinska, O., Claracq, J., & Eijndhoven, van, S. J. L. (2008). Non-isothermal film casting : determination of draw resonance. Journal of Non-Newtonian Fluid Mechanics, 151(1-3), 21-29. https://doi.org/10.1016/j.jnnfm.2008.01.003