2D Maxwell model

Abstract

A polymer is considered to be an ensemble of elastic elements with friction between them. Newton's law relating shear stress and strain is assumed to be valid even at high shear rate flow of the polymer, but the viscosity is assumed to be dependent on variables other than the shear rate. In particular, it is assumed to be dependent on the deformation of the elastic elements. The case of isothermal, incompressible flow of a polymer through a duct, whose cross-sectional height (x 3 direction) is much larger than its cross-sectional width (x 2 direction), is analyzed. The assumptions of the theory indicate a dependence of viscosity on the deformation of the elastic elements. For a linear material, Hooke's Law gives the relation between the shear deformation of the elements and the shear stress; so the dependence of the viscosity on the pressure gradient can be found. Application of Newton's law then enables the velocity profile and other flow properties to be found in terms of this variable. For a nonlinear material, a power-series relation between the shear deformation of the elements and the shear stress is assumed. We show how to find the parameters in this series by a simple curve fitting procedure beginning from linear behavior. Once this dependence of the viscosity on the pressure gradient is found, the application of Newton's law enables the velocity profile and other flow properties to be found.