Lateral Shaping and Stability of a Stretching Viscous Sheet

We study the stability of an isothermal stretching viscous sheet with lateral shaping. We propose a one-dimensional model for the dynamics and consider two types of boundary conditions: (i) Specifying the tension at the edges, we show that a pure outward normal tension (Sn) is usually unfavorable to the draw resonance instability as compared to the case of stress-free lateral boundaries. Alternatively, a pure streamwise tangential tension (St) is stabilizing. (ii) Specifying the lateral velocity at the edges, we show that the stability properties is problem specific but can be rationalized based on the induced tension (Sn,St).