On the stabilizing effects of neck-in, gravity, and inertia in Newtonian film casting

We investigate the influence of the reduction of width along the stretching direction, the so-called neck-in effect, on the draw resonance instability in Newtonian film casting using a linear stability analysis of a model of reduced dimensionality including gravity and inertia forces. Proper scaling reveals the aspect ratio, i.e., the ratio of the initial film half-width to the film length, together with the fluidity and the inlet velocity as independent, dimensionless control parameters. Moreover, we introduce the local Trouton ratio as a measure for the type of elongational deformation, which can be uniaxial, planar, or a combination of both. In the case of purely uniaxial or planar deformations, a one-dimensional model is sufficient. The influence of the control parameters on the draw resonance instability, including a threshold to unconditional stability, is visualized by several stability maps. Special cases of viscous-gravity and viscous-inertia models are analyzed separately due to their practical importance. Gravity appears to influence the aspect ratio at which the critical draw ratio is maximum and amplifies the stabilizing effect of the neck-in. Inertia increases the stabilization due to neck-in, eventually leading to a window of unconditional stability within the analyzed region of aspect ratios. The mechanism underlying the complete suppression of draw resonance is presented, using exclusively steady state analysis. Additionally, the stabilizing mechanisms of gravity and neck-in are revealed. Known alternative stability criteria are extended to the case of finite width and their validity is tested in the presence of inertia, gravity, and finite aspect ratios.