Effect of inertia and gravity on the draw resonance in high-speed film casting of Newtonian fluids

Abstract

The interplay between inertia and gravity is examined for Newtonian film casting in this study. Both linear and nonlinear stability analyses are carried out. Linear stability analysis indicates that while both inertia and gravity enhance the stability in film casting, inertia plays a more dominant role regarding the critical draw ratio. In contrast, the disturbance frequency is more sensitive to the effect of gravity. The nonlinear results show that at the critical draw ratio, the system oscillates harmonically, indicating the onset of a Hopf bifurcation. For a draw ratio above criticality, finiteamplitude disturbances are amplified, and sustained oscillation is achieved. It is found that the growth rate increases with draw ratio, but decreases with inertia and gravity, which suggests that initial transients tend to take longer to die out for a fluid with inertia and gravity. Transient post-critical calculations show that the nonlinearity can be effectively halted by inertia and gravity. The oscillation frequency (film-thickness amplitude) decreases (increases) with draw ratio. However, the film oscillates more frequently but less fiercely with stronger inertia and gravity effects. The rupture of the film is also examined, and is found to be delayed by inertia and gravity. Interestingly, although the oscillation amplitude is found to be weakest at the chill roll, it is at this location that the film tends to rupture first.

Newtonian pizza: spinning a viscous sheet

“…Though the stretching of liquid sheets in many industrial applications is essentially unidirectional, two-dimensional stretching is also used in practice (Pilkington 1969), as recently modelled by Scheid et al (2009). In either case, the main driving force is set by the boundary conditions, though gravity if aligned with the stretching direction can also contribute to the thinning process (Cao, Khayat & Puskas 2005). In this paper, we investigate the use of centrifugal force to stretch a thin circular viscous sheet, inspired by the example of a chef spinning pizza dough around his hand.…”

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

“…It can be seen that increasing the Reynolds number lowers the velocity values over the entire domain, leading to a sharp increase near the outlet such that the boundary condition is fulfilled, which has already been reported before. 11 This behavior is caused by the damping effect of inertia, which hinders the equal distribution of the force accelerating the fluid along the x-axis, as shown in Fig. 10(b).…”

“…This model had been extended afterwards to viscoelastic fluids, 3,6 non-isothermal effects, 2,12 and additional external forces like gravity and inertia. 11,14 Bechert, Schubert, and Scheid 14 introduced the dimensionless fluidity and inlet velocity as control parameters and proposed a stability map for these two parameters. Explicit empirical expressions for the critical draw ratio were presented and for the first time, a region of unconditional stability, within a) Author to whom correspondence should be addressed.…”

“…The results of Silagy, Demay, and Agassant 18 show that the effect of finite film width is not negligible for realistic film geometries. Additionally, it had been demonstrated by various authors 11,14,22 that both inertia and gravity may have a big influence on the stability behavior. However, no stability analysis including both finite width and additional forces except viscous ones exists up to now.…”

Non-Newtonian effects on draw resonance in film casting

“…Many theoretical studies on this phenomenon were undertaken in the past decades, starting with a linear stability analysis of a purely viscous, one-dimensional model for infinite width film casting of a Newtonian fluid [2]. This model had been extended afterwards in several works to cover additional effects like inertia and gravity [3,4], neck-in [5,6], and cooling [7]. In many cases, viscoelastic materials like polymer melts are processed and the non-Newtonian flow properties need to be taken into account in the modeling.…”