Computations of Numerical Solutions in Polymer Flows Using Giesekus Constitutive Model in the hpk Framework with Variationally Consistent Integral Forms

Abstract

This paper presents numerical solutions of boundary value problems (BVPs) for 1-D and 2-D polymer flows using the Giesekus constitutive model in the hpk mathematical and finite element (FE) computational framework utilizing variationally consistent (VC) integral forms. In the mathematical framework used here, h , the characteristic length, p , the degree of local approximation, and k , the order of the approximation space, are independent parameters as opposed to h and p used currently. The order k of the approximation space allows local approximations of higher order global differentiability. The VC integral forms ensure unconditionally stable computational processes, hence eliminating the need for the currently used upwinding methods for stabilizing computations. The work presented in this paper focuses on the following major-areas of investigation using the hpk framework and VC integral forms: (1) investigations of the different choices of stresses as dependent variables in the mathematical models on the performance of the resulting computational processes; (2) computations of numerical solutions for higher Deborah numbers; (3) solutions that are independent of the hpk computational parameters for fixed physics. 1-D fully developed flow, 2-D developing flow, and lid-driven cavity are used as model problems in the numerical studies.

Keywords:

• Giesekus Model
• hpk Framework
• Variational Consistency
• Finite Element Method
• Boundary Value Problems

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.