Master curves for unidirectional flows of FENE-P fluids in rectilinear and curvilinear geometries

We demonstrate that velocity profiles for steady, unidirectional shear flows of the FENE-P (Finitely-Extensible Nonlinear Elastic, with Peterlin closure) fluid, undergoing canonical rectilinear (pressure-driven flow in a rectangular channel or a circular pipe) or curvilinear (in Taylor–Couette or Dean configurations) flows, obey universal master curves that are a function only of the ratio $Wi/L$ , for a fixed solvent to solution viscosity parameter $\beta$. Here, $Wi$ is the Weissenberg number defined as the product of the dumbbell relaxation time and an appropriate shear rate, while $L$ is the ratio of the maximum extension of the polymer to its equilibrium root-mean-square end-to-end distance. The data collapse and the resulting master curves for the velocity profile is a generalization of the recent demonstration of master curves for polymer viscosity and first normal stress coefficient for a FENE-P fluid under steady simple shear flow (Yamani and McKinley, 2023). For pressure-driven channel and pipe flows, we derive simple analytical expressions for the velocity profiles, in the high shear-rate regime of $Wi/L\gg 1$, that readily elucidate the role of finite extensibility of the polymer on the velocity profiles. In the $Wi/L\gg 1$ regime, for all the flows considered, the limit of zero solvent ($\beta =0$) is shown to be singular, owing to the absence of a high-shear plateau in the total solution viscosity, resulting in very different velocity profiles for $\beta =0$ and $\beta \to 0$.