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.
REFERENCES
Some authors use a reciprocal definition of the aspect ratio.18,20,21
Note that the fluidity F is defined here in a slightly different manner than in our previous work,14 as the Trouton ratio, which is always 4 in the latter case, is not constant here and therefore cannot be included in the definition.
The stability map presented in our previous work14 differs slightly from the one shown here, as the definition of the fluidity differs by a factor of 4.
Note that a complete suppression of draw resonance does not always imply that steady processing is technically achievable, as the strong acceleration near the outlet in the case of high inertia force leads to strong tension within the fluid, which may lead to film break-up.
Strictly speaking, the term “unity-throughput” is only correct for Q = 1 and “Q-throughput” would be more adequate in general, as according to boundary conditions (7), the steady throughput is equal to Q. Nevertheless, we keep the term consistent with Kim et al.,26 as we will use exclusively the alternative scaling with boundary conditions (8) throughout this section.
