On the (de)stabilization of draw resonance due to cooling

Abstract

We study the drawing of a Newtonian viscous sheet under the influence of cooling with temperature dependence of the viscosity. Classically this problem has an instability called draw resonance, when the draw ratio Dr, which is the ratio of the outlet velocity relative to the inlet velocity, is beyond a critical value Drc. The heat transfer from the surface compared to the bulk energy advection is conveniently measured by the Stanton number St. Usual descriptions of the problem are one-dimensional and rigorously apply for St ≤ 1. The model presented here accounts for variations of the temperature across the sheet and has therefore no restriction on St. Stability analysis of the model shows two different cooling regimes: the ‘advection-dominated’ cooling for St ≪ 1 and the ‘transfer-dominated’ cooling for St ≫ 1. The transition between those two regimes occurs at St = O(1) where the stabilizing effect due to cooling is most efficient, and for which we propose a mechanism for stabilization, based on phase shifts between the tension and axial-averaged flow quantities. Away from this transition, the sheet is always shown to be unstable at smaller draw ratios. Additionally, in the limit of St → ∞, the heat exchange is such that the temperature of the fluid obtains the far-field temperature, which hence corresponds to a ‘prescribed temperature’ regime. This dynamical situation is comparable to the isothermal regime in the sense that the temperature perturbation has no effect on the stability properties. Nevertheless, in this regime, the critical draw ratio for draw resonance can be below the classical value of Drc = 20.218 obtained in isothermal conditions.