Nature of continuous spectra in wall-bounded shearing flows of FENE-P fluids

Abstract

Owing to the spatially local nature of the constitutive equations typically used to model polymeric stresses, the differential operators governing the linearized dynamics of bounded viscoelastic shearing flows have singular points. As a result, the eigenspectra of such shearing flows contain, in addition to discrete eigenvalues, continuous spectra (CS) comprising singular eigenfunctions. A clear understanding of the theoretical CS loci is crucial in discriminating physically genuine (discrete) eigenvalues from the poorly approximated numerical CS. For rectilinear shear flows of Oldroyd-B fluids, the CS are a pair of line segments, with lengths equal to the base-state range of velocities. In this study, we provide the first comprehensive account of the nature of the CS for both rectilinear and curvilinear shearing flows of the FENE-P fluid. In stark contrast to the CS for the Oldroyd-B fluid mentioned above, we show analytically that there are up to six distinct continuous spectra for shearing flows of FENE-P fluids. When the finite extensibility parameter $L > 50$, as appropriate for large molecular weight polymers used in experiments, three of the CS are nearly identical, and independent of the solvent-to-solution viscosity ratio ($\beta$). The other three CS are $\beta$-dependent, with one of them being the analogue of the solvent (viscous) continuous spectrum in the Oldroyd-B fluid. The remaining two $\beta$-dependent CS are novel features of the FENE-P spectrum, and can have phase speeds outside the base range of velocities, including negative ones. The complexity of the CS predicted here for shearing flows of FENE-P fluids is expected to carry over to other nonlinear viscoelastic models that exhibit a shear-thinning rheology.