Numerical Simulations of BVPs and IVPs in Fiber Spinning Using Giesekus Constitutive Model in hpk Framework

Abstract

This paper presents numerical simulations of 1-D boundary value problems (BVPs) and initial value problems (IVPs) in fiber spinning using Giesekus constitutive model in hpk mathematical and computational framework with variationally consistent (VC) and space-time variationally consistent (STVC) integral forms. In hpk framework, h, the characteristic length, p, the degree of local approximations, k, the order of the approximation space in space as well as space and time, are independent variables in all finite element (FE) computational processes as opposed to h and p in currently used FE processes. k, the order of the approximation space, allows higher order global differentiability local approximations in space as well as in time that are necessitated by the mathematical models and the higher order global differentiability features of the theoretical solutions. VC integral forms for the BVPs and STVC integral forms for IVPs yield unconditionally stable computational processes. It is shown that for 1-D BVPs and IVPs the mathematical models used in the published work for fiber spinning are deficient in incorporating the desired physics, utilize redundant dependent variables, and hence can lead to spurious numerical solutions. Within the assumptions used in the literature, a new mathematical model is presented in this paper that is free of the problems described here. Numerical studies are presented for BVPs and IVPs for four different fluids described by the Giesekus constitutive model for high draw ratios. In all numerical studies presented in the paper, the stationary states of the evolutions are shown to be in extremely good agreement with the solutions of the corresponding BVPs.

Keywords:

• 1-D Fiber Spinning
• Boundary Value Problems
• Finite Element Method
• Giesekus Constitutive Model
• hpk Framework
• Initial Value Problems
• Space-Time Variational Consistency
• Variational Consistency

This work has been supported by the DEPSCoR/AFOSR and AFOSR under grant numbers F49620-03-1-0298 and F49620-03-1-0201 to the University of Kansas Department of Mechanical Engineering and Texas A&M University Department of Mechanical Engineering. The seed grant from ARO under the grant number FED46680 is gratefully acknowledged. The financial support provided by the first and fourth authors' endowed professorships is gratefully acknowledged. The fellowships provided by the School of Engineering and the Department of Mechanical Engineering of the University of Kansas are also acknowledged. The computational facilities for this work have been provided by the Computational Mechanics Program (CMP) of the Department of Mechanical Engineering of the University of Kansas.