Mathematical characteristics of the pom-pom model

Abstract

Recently, in order to describe the complex rheological behavior of polymer melts with long side branches like low density polyethylene, new constitutive equations called the pom-pom equations have been derived in the integral/differential form and also in the simplified differential type by McLeish and Larson on the basis of the reptation dynamics with simplified branch structure taken into account. In this study, mathematical stability analysis under short and high frequency wave disturbances has been performed for these constitutive equations. It is proved that the differential model is globally Hadamard stable, as long as the orientation tensor remains positive definite or the smooth strain history in the flow is previously given. However both versions of the model are Hadamard unstable if we neglect the arm withdrawal in the case of maximum backbone stretch. It is also dissipatively unstable, since the steady shear flow curves exhibit non-monotonic dependence on shear rate. Additionally, in the flow regime of creep shear flow where the applied constant shear stress exceeds the maximum achievable value in the steady flow curves, the constitutive equations exhibit severe instability that the solution possesses strong discontinuity at the moment of change of chain dynamics mechanisms.