Numerical simulation of complex flows of non‐Newtonian fluids using the stream tube method and memory integral constitutive equations

Abstract

In this paper a memory integral viscoelastic equation is considered for simulating complex flows of non-Newtonian fluids by stream tube analysis. A formalism is developed to take into account co-deformational memory equations in a mapped computational domain where the transformed streamlines are parallel and straight. The particle-tracking problem is avoided. Evolution in time and related kinematic quantities involved with a K-BKZ integral constitutive model are easily taken into account in evaluating the stresses. Successive subdomains, the stream tubes, may be considered for computing the main flow in abrupt axisymmetric contractions from the wall to the central flow region. The 'peripheral stream tube' close to the duct wall is determined by developing a non-conventional modified Hermite element. A mixed formulation is adopted and the relevant non-linear equations are solved numerically by the Levenberg-Marquardt algorithm. Although the singularity at the section of contraction is not involved explicitly, the results obtained for the peripheral stream tube clearly show the singularity effects and the extent of the recirculating zone near the salient comer. The algorithm is stable even at high flow rates and provides satisfactory solutions when compared with similar calculations in the literature. Y. BEREAUXAND J.-R. CLERMONT Figure 1 . Streamlines in typical meshes used in viscoelastic flow computations axisymmetric cases. Using general meshes for computing the flow of memory fluids such as Maxwell and Curtiss-Bird fluids, approximations for kinematic histories or drift function methods have beenproposed by several authors'-3 for different constitutive equations. Dupont and Crochet! using a K-BKZ constitutive eq~ation,~ proposed parametric equations involving a scalar parameter for each element in order to identify the pathline of a material point. Luo and Tanner6 presented a method in which the integrals of stresses of the same K-BKZ equation were evaluated using a finite element method for flows involving only open streamlines, which were used for building elements updated at each step of the numerical iterative process (Figure 1). Luo an Mitsoulis7 also adopted the same fluid and the same method for the entry flow in a circular abrupt contraction, setting up conventional finite elements to take into account the closed streamlines of the recirculations encountered in such a flow. The fact that, to our knowledge, few numerical simulations (see e.g. Reference 8) of memory integral viscoelastic 3D flows have been attempted up to the present time is due mainly to significant problems in defining accurate parametric equations for warping curves in order to evaluate kinematic quantities and stresses.A different approach for computing the flow of memory integral fluids was proposed by Papanastasiou et al. 9 For the free surface extrusion problem they proposed a Protean co-ordinate system introduced by Duda and Vrentas'O and developed by A d a~h l . '~~'~ In the Protean system one coordinate is the streamfunction. T...

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