A Simple Model of the Melt Fracture Instability.pdf

A Simple Model of the Melt Fracture InstabilityJ. M. GreenbergDepartment of Mathematics and StatisticsUniversity of Maryland, Baltimore CountyBaltimore, MD 21228Y. DemayInstitut Non-Lineaire de NiceU.M.R.-C.N.R.S. no19Universite de NiceParc Valrose, 06034 Nice (France)AbstractIn this note we develop and analyze a system of equations describing a moltenlinear polymer being extruded in a capillary rheometer which is operating underthe controlled condition that the mean velocity at the capillary inlet is maintainedat a constant value. Slipping of the melt at the pipe wall is permitted and anevolution equation for the boundary slip parameter is postulated. The combinedsystem governing the mean cross-sectional velocity and slip parameter is shownto exhibit relaxation oscillations similar to those observed in actual rheometers.1. Introduction.Some molten polymers are highly viscous uids and measurement of their viscosity isan important tool for industrial applications. Viscosity is estimated as relationshipbetween shear stress and shear rate in various kinds of ows. For example, a classicalapparatus is the cone-plate (or plate-plate) rheometer. In this case the molten polymeris sheared at a controlled temperature between the plate and cone (or plate and plate).This research was partially supported by the U.S. National Science Foundation, the U.S. Depart-ment of Energy and U.S. Air Force Oce of Scientic Research.1 2The relationship between torque and angular velocity is nonlinear for molten polymersin general. The viscosity is constant at low shear rates (Newtonian plateau) and istypically a power law function of the shear rate above a certain threshold. Theserheometers allow simultaneous measurements of viscoelastic eects such as the rstnormal stress dierence. These viscoelastic eects are an important property of moltenpolymers (see [1]).Another classical apparatus is the capillary rheometer. In such a device, the moltenpolymer is pushed from a reservoir and t