On instabilities of single-integral constitutive equations for viscoelastic liquids

Abstract

Various types of instabilities are exposed in this paper for time-strain separable single-integral viscoelastic constitutive equations (CE's). They were distinguished into two groups and defined as Hadamard and dissipative type of instabilities. As for the Hadamard-type, previously obtained criteria are found to be necessary only. They are necessary and sufficient only for thermodynamic stability. Improved, stricter Hadamard stability criteria are described briefly in this paper, and then applied to study of stability of several CE's. It is shown that the Currie potential with the K-BKZ equation and the model proposed by Papanastasiou et al. are Hadamard unstable. In the case of dissipative stability, the necessary and sufficient condition for stress boundedness in any regular flow with a given history, is proved. Then, this criterion was applied to the neoHookean, Mooney, and Yen and McIntire specifications of the general K-BKZ model, to exhibit unbounded solutions. In addition, Larson-Monroe potential which is later proved to be Hadamard unstable but satisfies the above criterion of boundedness, is shown to have unstable decreasing branch in steady simple shear flow. At present, to the authors' knowledge, there is no viscoelastic single-integral CE of factorable type proposed in the literature which can satisfy all the Hadamard and dissipative stability criteria.