Mathematical modeling of polymer film casting under biaxial stretching taking into account heat transfer

Abstract

Polymer film casting is studied using a modified Vinogradov-Pokrovskii rheological model with heat transfer taken into account. Dependences of the film temperature, velocity, and width on the distance to the extruder exit was obtained and the effect of model parameters on these dependences was examined.Design and manufacture of products from polymer materials using mathematical modeling has a number of advantages, among which are the possibilities of controlling the quality of polymer products and solving a number of optimization problems. A common method for processing polymer materials is the production of polymer films with the use of an extruder through which a polymer melt is extruded [1][2][3][4]. The resulting film falls on a cooling drum. During motion of the film from the extruder to the drum, it is cooled and its width and thickness change. In this process, the film is stretched unevenly, leading to a neck effect [2, 3] which is manifested in the existence of two regions in the curve of the width of the film versus distance to the extruder exit. In the first region, there is a substantial change in the width of the film due to intense deformation, and in the second region, this change is insignificant and the material moves as a whole unit. Since all of these processes occur simultaneously, in their mathematical modeling, one has to simultaneously solve the equations for velocities, stress, and heat transfer. In this work, steady tensile stresses are obtained using a generalized Vinogradov-Pokrovskii rheological model [5] whose parameters are known functions of temperature:(1)Here σ ik is the stress tensor, p is the hydrostatic pressure, η 0 and τ 0 are the initial shear viscosity and relaxation time, ν ik is the velocity gradient tensor, a ik is the symmetric anisotropy tensor of the second rank, I = a jj is the first invariant of the anisotropy tensor, γ ik = (ν ik + ν ki )/2 is the symmetrized velocity gradient tensor, and κ and β are phenomenological parameters of the model that take into account the size and shape of the molecular coil in the equations of dynamics of the macromolecule. Results of calculations of viscometric flows for this rheological model agree well with experimental data [6], which makes it relevant and useful for calculations of more complex flows. For example, in [7,8], steady motion between parallel planes under a constant pressure gradient is considered, and it is shown that model (1) predicts a nonparabolic shape of the velocity profile and the presence of a nonzero pressure gradient in a direction perpendicular to the flow.