New form of the constitutive equations for the Oldroyd-B model, which have physical meaning, is developed to facilitate theoretical analysis. The new equations are used to simulate planar 4∶1 contraction flow of a Maxwell fluid using a third-order upwind finite volume method. The numerical results compare well with the theoretical solutions and the results of other references to show the effectiveness of the numerical method. Numerical experiments suggest that the present method not only converges fairly rapidly, but can also generate a highly resolved approximation to an Oldroyd-B fluid flow at a high Weissenberg number.
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