Polymer solutions in elongational flow: suspension of extensible rods model

Abstract A simple mechanistic model of a polymer solution in strong elongational flow is studied. A molecule is modelled by uniform slender rod of an incompressible rubber-like elastic material imbedded into a coaxial cylindrical cell filled with a viscous fluid (solvent). It is assumed that the polymer solution can be treated as a regular lattice of identical individual cells aligned along the extension direction. So only dynamics of an individual cell is studied. Under these assumptions the cell deforms affinely with bulk flow and the shear stress vanishes at the cell boundary. The rod-fluid interaction within the cell is studied using lubrication-layer approximation. Then the rod extension dynamics is described by a nonlinear parabolic equation for local stretch ratio with a source term responsible for the externally imposed flow. This equation is essentially the same as that derived by Hinch [E.J. Hinch, Phys. Fluids, 20 (1977) 22] for an individual molecule in extensional flow. It should be solved subjected to boundary conditions corresponding to vanishing elastic force in the rod at the rod ends. First, steady-state solutions for extension with constant strain rate are considered. Two steady-state solutions exist at small (subcritical) extension rate, one corresponding to a slightly extended molecule, and another to a highly extended one. It is argued that the second steady state is unstable. The steady states cease to exist beyond a critical strain rate. A `molecule' extends unboundedly in supercritical flow. At constant strain rate the extension asymptotically (after an adjustment period) tends to exponential elongation of the rod affinely with the bulk flow. Then relaxation of a highly extended rod is considered. For an uniformly extended initial state relaxation proceeds non-uniformly, propagating from the rod ends to the center. As a result, the model predicts slow relaxation in the case of high initial extension. The model predictions expressed in terms of the microscopic variables, namely the elastic strain and force distribution along a `molecule' can be related to bulk variables, such as extension rate and effective tensile stress. In particular, the elastic stress proves to be proportional to the elastic force in the rod integrated over its length. More general extension regimes are studied using numerical modelling of the rod dynamics equation. Those include constant-strain-rate extension with subsequent relaxation, step test with switching from high to a lower extension rate, and extension followed by periodic extension regime with zero mean extension rate. Among the effects observed are those predicted by the asymptotic analysis, such as existence of a critical value of the extension rate beyond which the rod extends unboundedly, exponential affine extension after some adjustment period, slow relaxation after strong extension, and hysteresis phenomena: unbounded extension at subcritical extension rate and asymmetric behavior of stress in oscillatory extension regime after preliminary strong extension. It is argued that the simple mechanistic model can be useful in explaining some experimental observations, in particular, unexpectedly successful application of the Maxwell–Oldroyd model to predominantly elongational strong flows.