The results of this paper show that one has to be carefully when working with the second-order model for viscoelastic fluids in computational fluid dynamics. Apart from the known possible occurrence of artificial instabilities if limitations of the model are disregarded, additional instabilities may appear, if nonlinear terms of the constitutive equations are neglected. These findings throw a new light on the convergence problem of numerical finite-difference or finite-element schemes applied to compute flows of viscoelastic fluids. Since nonlinear terms are also present in hydrodynamic equations with external fields, such as electric, magnetic, or temperature fields, etc., similar results are to be expected in such systems.
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