Elasto-inertial wall mode instabilities in viscoelastic plane Poiseuille flow

Abstract

A linear stability analysis of plane Poiseuille flow of an upper-convected Maxwell (UCM) fluid, bounded between rigid plates separated by a distance $2L$, has been carried out to investigate the interplay of elasticity and inertia on flow stability. The stability is governed by the following dimensionless groups: the Reynolds number $Re=\unicode[STIX]{x1D70C}U_{max}L/\unicode[STIX]{x1D702}$ and the elasticity number $E\equiv W/Re=\unicode[STIX]{x1D706}\unicode[STIX]{x1D702}/(\unicode[STIX]{x1D70C}L^{2})$, where $W=\unicode[STIX]{x1D706}U_{max}/L$ is the Weissenberg number. Here, $\unicode[STIX]{x1D70C}$ is the fluid density, $\unicode[STIX]{x1D702}$ is the fluid viscosity, $\unicode[STIX]{x1D706}$ is the micro-structural relaxation time and $U_{max}$ is the maximum base-flow velocity. The stability is analysed for two-dimensional perturbations using both pseudo-spectral and shooting methods. We also analyse the linear stability of plane Couette flow which, along with the results for plane Poiseuille flow, yields insight into the structure of the complete elasto-inertial eigenspectrum. While the general features of the spectrum for both flows remain similar, plane Couette flow is found to be stable over the range of parameters examined ($Re\leqslant 10^{4},E\leqslant 0.01$). On the other hand, plane Poiseuille flow appears to be susceptible to an infinite hierarchy of elasto-inertial instabilities. Over the range of parameters examined, there are up to seven distinct neutral stability curves in the $Re$–$k$ plane (here $k$ is the perturbation wavenumber in the flow direction). Based on the symmetry of the eigenfunctions for the streamwise velocity about the centreline, four of these instabilities are antisymmetric, while the other three are symmetric. The neutral stability curve corresponding to the first antisymmetric mode is shown to be a continuation (to finite $E$) of the Tollmien–Schlichting (TS) instability already present for Newtonian channel flow. As $E$ is increased beyond $0.0016$, a new elastic mode appears at $Re\sim 10^{4}$, which coalesces with the continuation of the TS mode for a range of $Re$, thereby yielding a single unstable mode in this range. This trend persists until $E\sim 0.0021$, beyond which this neutral curve splits into two separate ones in the $Re$–$k$ plane. The new elastic mode which arises out of this splitting has been found to be the most unstable, with the lowest critical Reynolds number $Re_{c}\approx 1210.9$ for $E=0.0066$. The neutral curves for both the continuation of the original TS mode, and the new elastic antisymmetric mode, form closed loops upon further increase in $E$, which eventually vanish at sufficiently high $E$. For $E\ll 1$, the critical Reynolds number and wavenumber scale as $Re_{c}\sim E^{-1}$ and $k_{c}\sim E^{-1/2}$ for the first two of the symmetric modal families, and as $Re_{c}\sim E^{-5/4}$ for first two of the antisymmetric modal families; $k_{c}\sim E^{-1/4}$ for the third antisymmetric family. The critical wave speed for all of these unstable eigenmodes scales as $c_{r,c}\sim E^{1/2}$ for $E\ll 1$, implying that the modes belong to a class of ‘wall modes’ in viscoelastic flows with disturbances being confined in a thin region near the wall. The present study shows that, surprisingly, even in plane shear flows, elasticity acting along with inertia can drive novel instabilities absent in the Newtonian limit.

Linear instability of viscoelastic pipe flow

“…In figure 18(b), the upper branch has a non-monotonic behaviour as E is increased, with a secondary minimum emerging at a higher Re. This feature of multiple minima is reminiscent of a similar phenomenon observed, albeit for wall modes, in the UCM limit for plane channel flow (see figure 15 of Chaudhary et al 2019). The two minima move apart with increasing E, and for E ≥ 0.8, the junction of the two distinct lobes in a given neutral curve moves out of the range of Re examined.…”

“…For the values of β examined here, both the axial and radial eigenfunctions exhibit a rather smooth variation with r, unlike the rapid, oscillatory variation (not shown) characteristic of wall modes (c r → 0) for β → 0. The latter are analogous to wall modes in viscoelastic channel flow whose structures was examined in detail by Chaudhary et al (2019) (see figure 20 therein); the overall similarity of the pipe and channel flow UCM spectra was discussed previously.…”

“…Gorodtsov & Leonov 1967;Lee & Finlayson 1986;Renardy & Renardy 1986;Ho & Denn 1977;Sureshkumar & Beris 1995) have examined the stability of viscoelastic plane Couette and Poiseuille flows. A detailed survey of the literature on viscoelastic plane shearing flows has been presented in Chaudhary et al (2019), and herein we restrict ourselves to summarizing the principal conclusions of Garg et al (2018). Garg et al (2018) showed that viscoelastic pipe flow is indeed linearly unstable in parameter regimes where experiments (Samanta et al 2013;Chandra et al 2018) observe an instability.…”

“…At still higher E ∼ 10 −3 (see figures 3e and 3 f ), the modes originally on the ring collapse, wrapping around the CS in an irregular manner. To understand the origin of the ring structure, it is relevant to recall a prominent feature of the viscoelastic spectra (at non-zero Re) in the UCM limit (β = 0): an infinite sequence of discrete modes corresponding to damped shear waves in a viscoelastic fluid (discussed in § 3.2), and are referred to as the high-frequency Gorodtsov-Leonov ('HFGL') modes (Gorodtsov & Leonov 1967;Kumar & Shankar 2005;Chaudhary et al 2019). This sequence corresponds to c i = −1/(2kW), and extends to infinity in either direction parallel to the c r axis.…”

“…The spectrum for β = 0 (figure 11a) now has a more intricate structure, necessitating an enlarged view of the phase speed interval (0, 1). The features of the high-Re UCM channel-flow spectrum were first explained in Chaudhary et al (2019), and include CS1 (which appears as a balloon owing to the finite resolution), the HFGL and additional discrete modes with c r ∈ [0, 1] that lie on either side of the HFGL line, roughly along the contours of an 'hourglass'. These features of the UCM pipe-flow spectrum, in figure 11(a), are analogous to the channel flow case described previously.…”