Approximation error in finite element calculation of viscoelastic fluid flows

Abstract Numerical calculations of complex, two-dimensional flows of viscoelastic fluids fail when the elastic contribution to the stresses, measured by a Deborah number, becomes comparable to the viscous contribution. Reasons for the limit on Deborah number are explored by a sequence of finite element calculations with contravariant convected Maxwell and second-order fluid models. The possibilities of bifurcation or loss of a steady, two-dimensional flow field are ruled out by employing continuation methods for flow through a planar contraction and by existence and uniqueness proofs for a second-order fluid in a driven cavity. Error in the finite element approximation to steep gradients of stress causes the breakdown of all calculations. This is most clearly seen in the calculations for a second-order fluid, because the exact flow field for any Deborah number is known.

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