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

TL;DR

The paper addresses how continuous spectra (CS) arise in linearized wall-bounded viscoelastic flows modeled by the FENE-P constitutive equation, revealing a richer CS structure than in Oldroyd-B. It develops analytical predictions for up to six CS across rectilinear (plane Couette, Poiseuille) and curvilinear (Dean, Taylor–Couette) flows and validates them against full numerical spectra, showing that for $L>50$ three CS are nearly identical and β-independent while three others depend on $β$ (including a solvent-like CS and two novel wing CS that can exceed the base velocity range). The study also uncovers qualitative differences between axisymmetric and non-axisymmetric disturbances in curvilinear flows, such as wing-like CS in Dean and truncated wings in Taylor–Couette, and demonstrates CS collapse along fixed $Wi/L$ lines. These insights clarify when linear dynamics are CS-dominated and lay groundwork for nonmodal analyses and extensions to other shear-thinning models, including the potential impact of diffusion on the CS-discrete mode interplay.

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 ($β$). The other three CS are $β$-dependent, with one of them being the analogue of the solvent (viscous) continuous spectrum in the Oldroyd-B fluid. The remaining two $β$-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.

Figures (25)

• Figure 1: Eigenspectra for viscoelastic Dean flow for $n = 1$, $\alpha = 7$, $\beta = 0.5$, $W\!i = 50, \epsilon = 0.1$. Here, $\omega$ is the complex frequency made dimensionless using the relaxation time, and $W\!i$ is a suitably defined Weissenberg number discussed below in Sec. \ref{['sec:probformulation']}. The spectra in panels (a) and (b) are for the FENE-P fluid with $L = 100$. The spectra in panels (c) and (d) are for the Oldroyd-B fluid for the same set of parameters. Note that the constitution of the densely filled curves, shown later to be the CS, appears to be independent of $Re$.
• Figure 2: Numerical eigenspectra for plane Couette flow showing the independence of CS with respect to the spanwise wavenumber $l$; data for $k = 0.1$, $W = 200, L = 100, \beta = 0.5$.
• Figure 3: Schematic location of the various CS for plane Couette flow for $\beta \rightarrow 1$. For $L \gtrsim 50$, CS1a, 1b, and 1c are nearly identical. Their vertical loci are shown to distinct in this schematic for the purposes of illustration.
• Figure 4: Numerical eigenspectra for plane Couette flow at different $N$'s illustrating the slow convergence of the various CS balloons. Data for $W\!i = 200, \beta = 0.5, k = 0.1, Re = 0$.
• Figure 5: Effect of wavenumber $k$ on the analytical CS and numerical spectra for plane Couette flow. Data for $W\!i = 200, L= 100, \beta = 0.98, Re = 0, l = 0$.