Steady elongational flow from rotarance theory

Abstract

The physics of macromolecular orientation has been used to explain the elasticity of polymeric liquids. Specifically, by first sculpting a rigid bead–rod likeness of the macromolecule, we can then derive its hydrodynamic resistance to orientation. The solution for the orientation distribution function has then been used, by integration in phase space, to get rheological material functions in both (i) small- and (ii) large-amplitude oscillatory shear flow, including its limiting case, and (iii) steady shear flow. However, rheological material functions in steady homogeneous extension from rigid bead–rod theory remain elusive. In this paper, we derive the orientation distribution function, and the rheological material functions, for suspensions of general rigid bead–rod structures. We focus on the time-steady viscosities in extension, and we first do so for general extensional kinematics. We then obtain the viscosities in steady extension for (iv) uniaxial extension, (v) planar extension, and (vi) biaxial extension. We close with a worked example, in which we use our new result for the steady uniaxial extensional viscosity to build a bridge between the macromolecular theory and the Oldroyd framework for rheological constitutive models. We, thus, arrive at a constitutive equation whose parameters are deducible from the moments of inertia of the macromolecule, and thus, deducible from macromolecular architecture alone. Our model is accurate up to third order for time-independent flows and is accurate to second order for time-dependent ones.