Supercritical bifurcation to periodic melt fracture as the 1 st transition to 2D elastic flow instability
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Supercritical bifurcation to periodic melt fracture as the 1st transition to 2D elastic flow instability Youngdon Kwon* School of Chemical Engineering, Sungkyunkwan University, Seobu-ro 2066, Suwon, Gyeonggi-do 16419, Korea (Received August 19, 2020; accepted September 21, 2020) This study, employing a numerical approximation, computationally describes 2D melt fracture as elastic instability in the flow along and outside a straight channel. In the preceding research (Kwon, 2018, Numerical modeling of two-dimensional melt fracture instability in viscoelastic flow, J. Fluid Mech. 855, 595-615) several types of unique instability and corresponding bifurcations such as subcritical and chaotic transitions have been illustrated with possible mechanism presumed. However, the 1st bifurcation from stable steady to unstable periodic state could not be accurately characterized even though its existence was proven evident. The analysis herein aims at verification of this 1st transition to temporally (and also spatially) periodic instability, utilizing the same numerical technique with attentive control of flow condition. As a result of scrutinizing the solutions, the steady elastic flow described by the Leonov rheological model passes through supercritical Hopf bifurcation at the Deborah number of 10.42 and then transforms to the state of the 1st weak periodic instability. It has also been confirmed that near this bifurcation point it takes extremely long to completely develop into either steady state (in the stable case) or periodic instability, which obstructed immediate characterization of the transition in the previous work. Keywords: high Deborah number, bifurcation, viscoelasticity, melt fracture
1. Introduction Viscoelastic liquid, when it is pumped out from an orifice, exhibits a peculiar phenomenon called die (or extrudate) swell or Barus effect, which indeed expresses relaxation of elasticity accumulated in the Poiseuille flow and cannot be observed for viscous liquid. Its intensity, i.e. transverse size of extrudate increases normally with flow rate, and in addition orifice geometry, temperature, even cooling air friction, etc. as well as viscoelastic property determine its detailed characteristics. As the flow rate increases in the general viscoelastic flow, nonlinear dynamic effects also strengthen and unique phenomena like various elastic instabilities appear (Boger and Walters, 1993; Larson, 1992; Shaqfeh, 1996). The most representative for the extrudate instability is melt fracture, that is, the distortion of liquid output with undulating free surface (Denn, 2001; Leonov and Prokunin, 1994; Tordella, 1956). In some occasions as an interfacial instability extrudate surface accompanies longitudinal scratch usually at the onset of melt fracture (Piau et al., 1990, 2000), however in most cases distortion of the melt is displayed as sharkskin and gross melt fracture frequently with a periodic pattern (Boger and Walters, 1993). Early studies on melt fracture before 2000 are quite in detail su
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