Journal of Scientific Computing

, Volume 79, Issue 1, pp 369–388 | Cite as

Numerical Simulations of Viscoelastic Fluid Flows Past a Transverse Slot Using Least-Squares Finite Element Methods

  • Hsueh-Chen LeeEmail author
  • Hyesuk Lee


This paper presents a least-squares (LS) finite element method for linear Phan-Thien–Tanner (PTT) viscoelastic fluid flows. We consider stabilized weights in the LS method for the viscoelastic model and prove that the LS approximation converges to the linearized solutions of the linear PTT model; the convergence is at the optimal rate for the velocity in the \(H^{1}\)-norm and at suboptimal rates for the stress and pressure in the \(L^{2}\)-norm, respectively. For numerical experiments we first consider the flow through a planar channel to illustrate our theoretical results. The LS method is then applied to a flow through the slot channel with two depth ratios and the effects of physical parameters are discussed. Numerical solutions of the channel problem indicate that flow characteristics of the viscoelastic polymer solution are described by the results obtained using the method. Furthermore, we present the hole pressure for various Weissenberg numbers, and compare with that derived from the Higashitani–Pritchard (HP) theory.


Least-squares The PTT model Hole pressure Normal-stress difference Transverse slot Weissenberg number 



The first author gratefully acknowledges the financial support provided in part by the Ministry of Science and Technology of Taiwan under Grant 107-2115-M-160-001-MY2. The second author is grateful for the financial support provided in part by the US National Science Foundation under Grant DMS-1418960.


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Authors and Affiliations

  1. 1.General Education CenterWenzao Ursuline University of LanguagesKaohsiungTaiwan
  2. 2.Department of Mathematical SciencesClemson UniversityClemsonUSA

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