4.1.1 The continuous spectrum

Equations (3.8)–(3.13) can be combined together to yield a single fourth-order differential equation in $\tilde{v}_{z}$ , as shown below in (4.1) of § 4.1.3. The highest-order derivative of this fourth-order equation is multiplied by the factor $\{1+\text{i}kW(U-c)\}$ (Graham Reference Graham1998; Wilson, Renardy & Renardy Reference Wilson, Renardy and Renardy1999; Grillet et al. Reference Grillet, Bogaerds, Peters and Baaijens2002; Chokshi & Kumaran Reference Chokshi and Kumaran2009). The origin of the continuous spectrum in viscoelastic flows can be traced to the singularity which arises when the factor $\{1+\text{i}kW(U-c)\}$ equals zero. The location of the continuous spectrum is thus given by $c_{r}=U(z)$ and $c_{i}=-1/(kW)$ in the complex $c$ -plane. Thus, $c_{i}=-1/(kW)$ and $c_{r}\in [-1,1]$ and $[0,1]$ for plane Couette and plane Poiseuille flow respectively (Renardy & Renardy Reference Renardy and Renardy1986; Sureshkumar & Beris Reference Sureshkumar and Beris1995b ; Wilson et al. Reference Wilson, Renardy and Renardy1999; Sureshkumar Reference Sureshkumar2000; Chokshi & Kumaran Reference Chokshi and Kumaran2009). Physically, the origin of the continuous spectrum may be attributed to the local nature of the constitutive relation for the polymeric stress, with the corresponding eigenfunctions being singular (Sureshkumar & Beris Reference Sureshkumar and Beris1995b ; Graham Reference Graham1998; Sureshkumar Reference Sureshkumar2000) (Author’s paper in preparation: Elastic instability of a vortex column.). Consequently, the spectral method, which uses a finite number of polynomials to approximate the dynamical variables, does not resolve the singular eigenfunctions as accurately as those corresponding to the discrete eigenvalues. The result is a ballooning up of the (expected) ‘line’ of eigenvalues. The spread of the continuous spectrum balloon along the imaginary axis shrinks by a small amount with increase in the number of collocation points $N$ (figure 4). Upon filtering (based on a specified tolerance), the balloon disappears with the other unconverged discrete eigenvalues in its vicinity. Inclusion of slow diffusive dynamics of the polymer concentration field (Sureshkumar & Beris Reference Sureshkumar and Beris1995a ; Chokshi & Kumaran Reference Chokshi and Kumaran2009), and thence, of the polymeric stress, introduces non-locality in the constitutive relation thereby regularizing it, in turn leading to the disappearance of the continuous spectrum.

Figure 4.

A magnified view of the (unfiltered) continuous spectrum for (a) plane Couette and (b) plane Poiseuille flows for $E=0.005,k=2.0,Re=5200$ with $N=500$ , $600$ , $700$ and $800$ .

4.1.2 High-frequency Gorodtsov–Leonov modes

The set of discrete eigenvalues in figures 2 and 3 corresponding to the horizontal line $c_{i}=-1/(2kW)$ , that appears to extend indefinitely (i.e. $c_{r}\rightarrow \pm \infty$ ) on either side of the hourglass structure, was first analysed by Gorodtsov & Leonov (Reference Gorodtsov and Leonov1967), and are accordingly termed the HFGL modes. The decay rates of the HFGL eigenmodes are independent of the base velocity profile, with $c_{i}=-1/(2kW)$ (Gorodtsov & Leonov Reference Gorodtsov and Leonov1967; Renardy & Renardy Reference Renardy and Renardy1986), although the precise values of the wave speeds ( $c_{r}$ ) are dependent on the velocity profile. The original analysis, for plane Couette flow, was restricted to the limit $Re\ll W\sim O(1)$ and $\sqrt{ReW}\ll 1$ , in which case $c_{r}\sim O(1/\sqrt{ReW})\gg 1$ , with $\sqrt{ReW}$ being the viscoelastic Mach number. In dimensional terms, $c_{r}\sim O(\sqrt{\unicode[STIX]{x1D702}/(\unicode[STIX]{x1D70C}\unicode[STIX]{x1D706})})\sim O(\sqrt{G/\unicode[STIX]{x1D70C}})\gg U$ , where $G$ is the shear modulus of the polymeric liquid. Thus, in the limit analysed, these modes represent a class of weakly damped supersonically propagating shear waves in a viscoelastic fluid. Gorodtsov & Leonov (Reference Gorodtsov and Leonov1967) showed that the $n\text{th}$ HFGL eigenvalue is given by $c_{i,n}=-1/(2kW)$ , with $c_{r,n}=\pm A_{n}/(k\sqrt{ReW})$ , where $A_{n}=n\unicode[STIX]{x03C0}/2$ for $n\gg 1$ (in the present notation, where we consider the domain $z\in [-1,1]$ ). Rather interestingly, we find that the expression $(n+2)\unicode[STIX]{x03C0}/2$ is a more accurate expression for $A_{n}$ , if the HFGL eigenvalue with the lowest $|c_{r}|$ is designated to be $n=0$ . As shown in figure 5(a), the results from the spectral code (for $Re=10^{-4},W=0.9$ ) are in excellent agreement with the asymptotic predictions above. Further, figure 5(b) shows that the difference between successive (scaled) eigenvalues approach $\unicode[STIX]{x03C0}$ and that $c_{i,n}\rightarrow -1/(2kW)$ for higher values of $n$ . The numerical results shown in figure 5 splay off the theoretical prediction owing to the invariable loss of convergence (discussed below in more detail) for the higher modes, but this divergence is postponed to a larger value of $n$ as $N$ increases. The agreement of the difference between successive (scaled) eigenvalues with the theoretical prediction of Gorodtsov & Leonov (Reference Gorodtsov and Leonov1967), on the scale of the plot shown, seems uniformly good for all $n$ , although the theoretical prediction is only valid for $n\gg 1$ . A zoomed view (not shown) does reveal a discrepancy for the first few values of $n$ .

Figure 5.

HFGL modes in plane Couette flow for $Re=10^{-4},W=0.9$ and $k=0.7$ : (a) variation of $c_{r}$ with mode number $n$ and (b) the absolute value of the difference between successive scaled eigenvalues approaches $\unicode[STIX]{x03C0}$ and $2kWc_{i,n}\rightarrow -1$ as $n\gg 1$ . Inset (A) shows the zoomed view of the region near small values of $n$ , and for this case, the estimate, $(n+2)\unicode[STIX]{x03C0}/2$ , for the scaled eigenvalue is accurate even for $n\sim 1$ . Inset (B) shows the zoomed view of the region where modes splay out for $N=500$ . The inset inside panel (b) shows the scaled growth rate versus $n$ .

Figure 6.

A zoomed view of the HFGL line for plane Poiseuille flow for $E=0.005,k=2,Re=5200$ with $N=500$ , $600$ , $700$ and $800$ . (a) The unfiltered eigenspectra. (b) The filtered eigenspectra.

Results from the asymptotic analysis of Gorodtsov & Leonov (Reference Gorodtsov and Leonov1967) continue to provide a good approximation to the HFGL part of the spectrum even for $Re\gg W\sim O(1)$ , provided $A_{n}$ is sufficiently large; such modes still conform to the basic assumption, in the original analysis, of being highly supersonic (i.e. $|c|\gg 1$ ), and their decay rates are still given by $c_{i}=-1/(2kW)$ . This is illustrated most clearly in figure 6 which shows the HFGL part of the eigenspectrum corresponding to figure 3(b), where the Reynolds number ( $Re=5200$ ) is several orders of magnitude larger than that in figure 5, and $W=26$ , which explicitly violates the HFGL assumption of $Re\ll W$ . This feature is further explored in the following subsection, wherein it is shown that HFGL modes with $|c_{r}|\gg 1$ for $Re\gg W\sim O(1)$ are well predicted by a WKBJ analysis. Note that although the highly supersonic modes again have decay rates close to $c_{i}=-1/(2kW)$ , for $c_{r}$ near and in the interval of base-state velocities, $c_{i}$ deviates from the aforementioned analytical estimate, and multiple kinks are observed in the HFGL line (figures 2 and 3).

Representative eigenfunctions, corresponding to the HFGL modes from the eigenspectra (figure 2), for plane Couette and Poiseuille flows for $Re=10^{4},k=1$ and $E=0.001$ are shown in figures 7(a) and 7(b), respectively. In these and the following figures, the eigenfunctions (with real and imaginary parts denoted by the subscripts ‘ $r$ ’ and ‘ $i$ ’, respectively) are normalized so that their maximum absolute value is unity. The eigenfunctions are similar for both the flows and exhibit highly oscillatory behaviour contained within an amplitude envelope that exhibits a secondary longer-wavelength modulation. For both plane Couette and Poiseuille flows, the features in the HFGL eigenfunctions become increasingly fine grained with increase in $|c_{r}|$ (figure not shown). The fine-scaled oscillations cause the eigenvalues with higher $|c_{r}|$ to splay out from their analytical estimates of $c_{i}=-1/(2kW)$ (figure 6 a) for a given number of collocation points $N$ . The smallest length scale resolved is limited by the chosen value of $N$ , and HFGL modes with $c_{r}$ higher than an $N$ -dependent threshold cannot be predicted accurately. Equivalently, in figure 6(b), the greater horizontal ‘extent’ of the HFGL line for higher $N$ illustrates the improved resolution of the fine-scaled higher $|c_{r}|$ modes.

Figure 7.

Real and imaginary parts of normalized $x$ -velocity eigenfunctions corresponding to the HFGL modes in (a) plane Couette flow, and (b) plane Poiseuille flow for $E=0.001,k=1,Re=10^{4}$ and $N=500$ . These eigenfunctions correspond to eigenvalues that belong to the eigenspectra shown in figures 2(a) and 2(b), respectively.

4.1.3 Asymptotic analysis of HFGL modes at high $Re$

The governing linearized momentum and constitutive relations (3.8)–(3.13) are combined to yield a single fourth-order differential equation for the wall-normal velocity, $\tilde{v}_{z}$ , by eliminating the other independent fields,

(4.1) $$\begin{eqnarray}\displaystyle & & \displaystyle \mathscr{D}^{4}\tilde{v}_{z}+\mathscr{D}^{3}\tilde{v}_{z}(2\text{i}kWU^{\prime }-2U^{\prime }\unicode[STIX]{x1D6F4}^{-1})\nonumber\\ \displaystyle & & \displaystyle \quad +\,\mathscr{D}^{2}\tilde{v}_{z}(-2k^{2}(1+W^{2}(U^{\prime })^{2})+3\text{i}kWU^{\prime \prime }-\unicode[STIX]{x1D6F4}^{-1}(4\text{i}kW(U^{\prime })^{2}+3U^{\prime \prime })+2\unicode[STIX]{x1D6F4}^{-2}(U^{\prime })^{2})\nonumber\\ \displaystyle & & \displaystyle \quad +\,\mathscr{D}\tilde{v}_{z}(-2\text{i}k^{3}WU^{\prime }+8k^{2}W^{2}U^{\prime }U^{\prime \prime }-\unicode[STIX]{x1D6F4}^{-1}(2k^{2}(1-2U^{\prime }-2W^{2}(U^{\prime })^{3})+8\text{i}kWU^{\prime \prime }U^{\prime })\nonumber\\ \displaystyle & & \displaystyle \quad +\,\text{i}k^{-1}\unicode[STIX]{x1D6F4}^{-2}(4k^{2}W(U^{\prime })^{3}-4\text{i}kU^{\prime \prime }U^{\prime }))\nonumber\\ \displaystyle & & \displaystyle \quad +\,\tilde{v}_{z}\left(-\text{i}k(\text{i}k^{3}+2\text{i}k^{3}W^{2}(U^{\prime })^{2}+4\text{i}kW^{2}(U^{\prime \prime })^{2}+4\text{i}kW^{2}U^{\prime \prime \prime }U^{\prime }-8\text{i}kU^{\prime \prime }U^{\prime }+k^{2}WU^{\prime \prime })\right.\nonumber\\ \displaystyle & & \displaystyle \quad -\,\unicode[STIX]{x1D6F4}^{-1}(2\text{i}k^{3}WU^{\prime }(1-U^{\prime })+\text{i}kU^{\prime \prime }(\text{i}k+2\text{i}kW^{2}(U^{\prime })^{2}+3WU^{\prime \prime }))\nonumber\\ \displaystyle & & \displaystyle \quad \left.+\,\text{i}k^{-1}\unicode[STIX]{x1D6F4}^{-2}(2k^{2}(U^{\prime })^{2}(\text{i}k+2\text{i}kW^{2}(U^{\prime })^{2}+WU^{\prime \prime })+4k^{2}WU^{\prime \prime }(U^{\prime })^{2})\right)\nonumber\\ \displaystyle & & \displaystyle =-k^{2}ReW(U-c)\unicode[STIX]{x1D6F4}((\mathscr{D}^{2}-k^{2})\tilde{v}_{z}+U^{\prime \prime }(U-c)^{-1}\tilde{v}_{z}),\end{eqnarray}$$

where $\unicode[STIX]{x1D6F4}=U-c-\text{i}k^{-1}W^{-1}.$

We consider here the limit $Re\gg 1$ , and define the small parameter $\unicode[STIX]{x1D716}=1\!/\!(\text{i}k(ReW)^{1/2})\ll 1$ in the ensuing analysis. In this limit, to achieve a dominant balance between the inertial, viscous and elastic forces, it is necessary to rescale the lengths using $O(ReW)^{-1/2}\ll 1$ , implying that we look for rapidly varying solutions, on the length scale of $(ReW)^{-1/2}$ , using the WKBJ ansatz (Lin Reference Lin1945; Drazin & Reid Reference Drazin and Reid2004; Bender & Orszag Reference Bender and Orszag2013) ansatz, as follows:

(4.2) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{v}_{z}=\exp \left(\int g\,\text{d}z\right), & \displaystyle\end{eqnarray}$$

with

(4.3) $$\begin{eqnarray}\displaystyle & \displaystyle g=g_{0}/\unicode[STIX]{x1D716}+g_{1}+g_{2}\unicode[STIX]{x1D716}+\cdots \,. & \displaystyle\end{eqnarray}$$

The equation governing $\tilde{v}_{z}$ to $O(1)$ in (4.3) can be derived from (4.1) and is given by,

(4.4) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D716}^{2}(\mathscr{D}^{4}\tilde{v}_{z}+\mathscr{D}^{3}\tilde{v}_{z}(2\unicode[STIX]{x1D716}^{-1}U^{\prime }E^{1/2}-2U^{\prime }\unicode[STIX]{x1D6F4}^{-1}))=\mathscr{D}^{2}\tilde{v}_{z}((U-c)\unicode[STIX]{x1D6F4}-2(U^{\prime })^{2}E). & & \displaystyle\end{eqnarray}$$

Substituting equation (4.3) in (4.4), then leads to the following expression for $\tilde{v}_{z}$ :

(4.5) $$\begin{eqnarray}\displaystyle \tilde{v}_{z} & = & \displaystyle {\displaystyle \frac{1}{((U-c)\unicode[STIX]{x1D6F4}-(U^{\prime })^{2}E)^{1/2}(-U^{\prime }E^{1/2}\pm ((U-c)\unicode[STIX]{x1D6F4}-(U^{\prime })^{2}E)^{1/2})^{2}}}\nonumber\\ \displaystyle & & \displaystyle \quad \exp \left(\unicode[STIX]{x1D716}^{-1}E^{1/2}U\pm \text{i}k\unicode[STIX]{x1D716}^{-1}\int ((U-c)\unicode[STIX]{x1D6F4}-E(U^{\prime })^{2})^{1/2}\,\text{d}z\right.\nonumber\\ \displaystyle & & \displaystyle \quad \left.+\int U^{\prime }g_{0}\unicode[STIX]{x1D6F4}^{-1}(g_{0}+U^{\prime }E^{1/2})^{-1}\,\text{d}z\right),\end{eqnarray}$$

where

(4.6) $$\begin{eqnarray}\displaystyle g_{0}=-U^{\prime }E^{1/2}\pm ((U-c)\unicode[STIX]{x1D6F4}-(U^{\prime })^{2}E)^{1/2}. & & \displaystyle\end{eqnarray}$$

Using continuity, we obtain the following expression for the axial velocity, simplified to the leading order in $\unicode[STIX]{x1D716}$ and $E^{1/2}$ :

(4.7) $$\begin{eqnarray}\displaystyle \tilde{v}_{x}={\displaystyle \frac{\unicode[STIX]{x1D716}^{-1}}{((U-c)\unicode[STIX]{x1D6F4})^{3/2}}}\exp \left(\unicode[STIX]{x1D716}^{-1}E^{1/2}U\pm \unicode[STIX]{x1D716}^{-1}\int (U-c)^{1/2}\unicode[STIX]{x1D6F4}^{1/2}\,\text{d}z\right)+O(\unicode[STIX]{x1D716})+O(E^{1/2}). & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Owing to the small transverse length scale, the flow associated with these elasto-inertial eigenfunctions is a nearly unidirectional flow, with a dominant tangential component. One therefore needs to satisfy only the no-slip condition for $v_{x}$ at leading order. Using the boundary conditions for $\tilde{v}_{x}$ , then leads to the following dispersion relation for $(ReW)^{1/2}\gg 1$ and $E^{1/2}\ll 1$ :

(4.8) $$\begin{eqnarray}\displaystyle k(ReW)^{1/2}\int _{-1}^{1}(U-c)^{1/2}\unicode[STIX]{x1D6F4}^{1/2}\,\text{d}z=n\unicode[STIX]{x03C0}. & & \displaystyle\end{eqnarray}$$

The above (4.8) is solved by $c=c_{r}+\text{i}c_{i}$ , where $c_{i}=-1/(2kW)$ and

(4.9) $$\begin{eqnarray}\displaystyle \int _{-1}^{1}\left((U-c_{r})^{2}+{\displaystyle \frac{1}{(2kW)^{2}}}\right)^{1/2}\,\text{d}z={\displaystyle \frac{n\unicode[STIX]{x03C0}}{k(ReW)^{1/2}}}. & & \displaystyle\end{eqnarray}$$

Figures 8 and 9 reveal a good comparison between the numerical and asymptotic results. In our analysis, we have assumed that (4.5) is valid throughout the domain which requires that the singularities of the expression are well away from the real axis. This is true for $|c_{r}|>1$ . When $|c_{r}|<1$ , there is a critical layer in the channel where (4.5) breaks down and a matched-asymptotic analysis with an inner turning point region is required (Lin Reference Lin1945; Drazin & Reid Reference Drazin and Reid2004; Bender & Orszag Reference Bender and Orszag2013) to accurately capture the eigenvalues. This is reflected in figures 8(a) and 8(c), where the differences between asymptotic and numerical results become substantial as $c_{r}\rightarrow 1$ .

Figure 8.

Comparison of the eigenvalues for the HFGL part of the spectrum obtained from the asymptotic expression with the numerically obtained eigenvalue for $Re=5000,W=10,k=1$ for Couette flow; $1500$ collocation points were taken for the numerics. (a)  $1<c_{r}<2.6$ . (b)  $c_{r}\sim 2.5$ . (c)  $c_{r}\sim 1.1$ .

Figure 9.

Comparison of the axial velocity eigenfunctions from the asymptotic analysis and the numerical solution for the HFGL eigenvalue $c_{r}=2.458$ and $Re=5000,W=10,k=1$ for Couette flow; $1500$ collocation points were taken for the numerics. While panel (a) shows the entire domain, the other panels (b–d) zoom into smaller intervals to show the rapid oscillations more clearly.

Figure 10.

Filtered eigenspectra illustrating the hourglass modes for (a) plane Couette flow, and (b) plane Poiseuille flow for $E=0.005,k=2,Re=5200$ with $N=500$ , $600$ , $700$ and $800$ , and normalized $x$ -velocity eigenfunctions corresponding to the least stable modes for (c) plane Couette flow in panel (a) and (d) plane Poiseuille flow in panel (b).

4.1.4 Hourglass modes

As seen in figures 2 and 3 (depicting unfiltered spectra), and in figures 10(a) and 10(b) (depicting filtered spectra), the HFGL line lies between the two ‘bulbs’ of an (approximately) hourglass-shaped structure in the interval $c_{r}\in [0,1]$ , for both plane Couette and Poiseuille flows. It is necessary to plot (as in figure 10) the filtered spectra for parametric regimes in which the ballooned up continuous spectrum tends to mask the physically genuine eigenvalues belonging to the lower bulb of the hourglass structure. The hourglass structure is expectedly symmetric about $c_{r}=0$ for plane Couette flow, but is not for plane Poiseuille flow where fewer eigenvalues are present for $c_{r}$ close to $1$ compared to $c_{r}$ close to zero. The lower bulb of the hourglass lies between the HFGL line ( $c_{i}=-1/(2kW)$ ) and the continuous spectrum ( $c_{i}=-1/(kW)$ ). The upper bulb lies between the HFGL line and the $c_{i}=0$ axis for plane Couette flow, while for plane Poiseuille flow, the low- $c_{r}$ branch of the upper bulb crosses the $c_{i}=0$ line, leading to instabilities. In figures 10(a) and 10(b), a higher value of $N$ ( ${\sim}800$ ) is required to capture the lower bulb of the hourglass compared to that required ( $N\sim 500$ ) for the upper bulb, due to the encroachment of the lower bulb by the ballooned-up continuous spectrum. Indeed, only at $N=800$ , is the lower bulb of the hour glass structure reasonably complete in figure 10(a) for plane Couette flow, and is beginning to get populated in figure 10(b) for plane Poiseuille flow.

The eigenfunctions corresponding to the least stable eigenvalues in the upper bulb of the hourglass structure are shown in figures 10(c) and 10(d). While the oscillations of the HFGL eigenfunctions (in figure 7) span the entire channel domain, those for the least stable hourglass modes exhibit localized large-amplitude oscillations close to the wall. However, as we proceed down the hourglass structure to modes with $c_{r}\sim 0$ (for plane Couette) or $c_{r}\sim 0.5$ (for plane Poiseuille), there is a gradual transition in the structure of the eigenfunctions from wall-localized oscillations (characteristic of the least stable or unstable modes; see § 4.2 for the eigenfunctions corresponding to the unstable mode) to de-localized oscillations similar to the (stable) HFGL modes. The maximum amplitude of the hourglass eigenfunctions occurs in the vicinity of the critical point $z_{c}=(1-c_{r})^{1/2}$ where the phase speed $c_{r}$ equals the local base-flow velocity indicated by a vertical line in figures 10(c) and 10(d). The least stable eigenfunctions of the hourglass structure are wall modes with phase velocities $c_{r}\rightarrow \pm 1$ (for plane Couette flow) and $c_{r}\ll 1$ (for plane Poiseuille flow) in the limit of $Re\gg 1$ (Schmid & Henningson Reference Schmid and Henningson2001). As mentioned above, and discussed in more detail below, it is the wall modes belonging to the upper bulb of the hourglass that turn unstable in plane Poiseuille flow as $Re$ and $E$ are increased.

4.1.5 Evolution of the elasto-inertial eigenspectrum with $E$

Figure 11(a) shows the Newtonian eigenspectrum for plane Couette flow ( $E=0$ ) which has the familiar Y-shaped structure with three distinct branches (Schmid & Henningson Reference Schmid and Henningson2001). Following Mack (Reference Mack1976), and as mentioned (briefly) at the beginning of this section, we label the vertical branch with phase speeds equal to the average velocity of the base flow ( $c_{r}=0$ ) as the ‘ $S$ ’-branch, and the two symmetrical arms of the Y-shaped spectrum consisting of modes, with $c_{r}$ progressively approaching the velocity of the two walls, as the ‘ $A$ ’-branches. The $S$ -branch in Newtonian plane Couette flow extends down to $c_{i}=-\infty$ . In figure 11(a) there is a region at the intersection of the three branches where the eigenvalues are not convergent. This lack of convergence is well known, and has been attributed to non-orthogonality of the eigenvectors, which renders the eigenspectrum in that region very sensitive to numerical (round-off or truncation) errors in the discretized linear operator (Schmid & Henningson Reference Schmid and Henningson2001).

Figure 11.

Evolution of the eigenspectrum for plane Couette flow with increase in $E$ : unfiltered eigenspectrum for $N=450$ and $500$ for (a) Newtonian fluid ( $E=0$ ), (b) $E=10^{-4}$ , (c) $E=5\times 10^{-4}$ , (d) $E=10^{-3}$ and (e) $E=5\times 10^{-3}$ , for $Re=5000$ and  $k=1$ .

The Newtonian limit of $E\rightarrow 0$ is a singular one, so even for $E\ll 1$ , in sharp contrast to the Newtonian spectrum, the vertical extent of the elasto-inertial UCM spectrum is finite (of $O(1/E)$ ), because the fastest relaxation is limited by the intrinsic relaxation time of the polymer. Thus, as shown in figures 11(a)–11(e), as $E$ is increased for a fixed $Re$ , the extent of the eigenspectrum along the $c_{i}$ -axis decreases as $1/E$ . An inverted Y-structure also appears below the truncated $S$ -branch ending in the continuous spectrum balloon. The HFGL modes are evidently absent for $E=0$ . For any non-zero $E$ , the HFGL line appears midway between the Y and inverted Y structures (figure 11 b). Figures 11(b)–11(e) show that for increasing values of $E$ the upper and lower Y structures gradually transform to the approximately hourglass-shaped structure mentioned earlier, with the vertical extent of this structure again shrinking with increasing $E$ (and $k$ ; figures 11 b–11 e). In addition, the ballooned-up continuous spectrum expands along $c_{i}$ -axis with increase in $E$ , presumably because the eigenfunctions for the polymer stress components becoming more singular with increasing $E$ . The hourglass structure is found to have more eigenvalues for higher $E$ , but none of these modes become unstable for plane Couette flow. A similar, but not identical, scenario prevails for the plane Poiseuille spectrum. In this case, however, as $E$ is increased, the modes along the low- $c_{r}$ branch eventually become unstable; the modes along the high- $c_{r}$ branch remain stable. The unstable wall modes above are discussed in more detail in the next section.

Figure 12.

Spectrum for plane Poiseuille flow for Newtonian ( $E=0$ ) and Oldroyd-B ( $E=0.005$ ) fluids: $Re=6000$ and $k=1$ . For the Newtonian fluid, there is only one unstable (antisymmetric) mode. For the UCM fluid, there are two unstable modes, with the most unstable mode ( $A1(b)$ ) being antisymmetric and the other mode being symmetric ( $S1$ ). (a) Newtonian, $E=0$ . (b) UCM, $E=0.005$ .

4.2.1 Effect of fluid elasticity on the Tollmien–Schlichting mode

We examine here the role of fluid elasticity on the neutral curve for the TS mode in the $Re$ – $k$ plane (the elastically modified TS-mode, referred to as mode $A1(a)$ hereafter). Figures 15(a) to 15(d) show the neutral stability curves obtained by tracing the loci of the points with $c_{i}=0$ that separate stable and unstable regions; the flow being unstable inside the neutral curves. The minimum Reynolds number on the neutral curve is the critical Reynolds number $Re_{c}$ and the corresponding wavenumber is the critical wavenumber $k_{c}$ , both being functions of $E$ ; thus, for the Newtonian case, $E=0$ , one has $Re_{c}=5772.2$ with $k_{c}=1.02$ . To begin with, as $E$ increases, the neutral stability curve shifts towards lower Reynolds numbers and slightly higher wavenumbers and hence the flow becomes more unstable (figure 15 a). This is consistent with the trends first reported by Sureshkumar & Beris (Reference Sureshkumar and Beris1995b ), who, however, did not report the neutral curves shown in figure 15(a). For $E>0.0005$ , the neutral curve starts to develop a two-lobed structure (figure 15 b). Given the upper bound, $Re=10^{4}$ , that applies to the current study, we find that for $E\geqslant 0.0016$ , another unstable mode emerges which is not present in the Newtonian limit $E=0$ . This new unstable mode, labelled $A1(b)$ henceforth, appears only in a UCM fluid, and is therefore an elasto-inertial mode. The neutral curves corresponding to $A1(a)$ and $A1(b)$ for $E=0.0016$ (figure 15 b) show that for $Re>9543$ , the unstable modes $A1(a)$ and $A1(b)$ are distinct, in that, the variation of $c_{r}$ and $c_{i}$ for these modes exhibit two different curves. However, the two modes coalesce to form a single mode for $Re<9543$ (below the cusp-like region in figure 15 b) yielding a single unstable region. The $Re$ at which this modal coalescence between $A1(a)$ and $A1(b)$ occurs, as indicated by the cusp in the $Re-k$ plane, decreases as $E$ is further increased. Beyond $E=0.0019$ , the two-lobed structure develops a pair of distinct minima, and a second cusp appears for lower $Re$ (figure 15 c); however, only a single unstable mode is found until $E=0.0021$ . Thus, in the region below the upper cusp in the $Re-k$ plane, the two modes $A1(a)$ and $A1(b)$ become identical, and the two unstable regions fuse into one, yielding a single $c_{r}$ and $c_{i}$ versus $k$ curve. The $Re$ at which the lower cusp appears in the $Re-k$ plane increases with increase in $E$ , resulting in the upper and lower cusps approaching each other (figure 15 c). For $0.0021<E<0.0022$ (data not shown) the two cusps meet each other, leading to the neutral curve splitting into two separate ones for $E\gtrsim 0.0022$ as shown in figure 15(c), and concomitantly, the modes become separate once again.

Figure 16.

Shrinking of the neutral stability curves and the formation of closed neutral stability ‘loops’ for eigenmode $A1(b)$ with increase in $E$ from $0.003$ to  $0.008$ .

The aforementioned coalescence and bifurcation of the neutral curves was interpreted by Sureshkumar & Beris (Reference Sureshkumar and Beris1995b ) as two separate eigenfamilies in their study. The $x$ -velocity eigenfunction for the original TS instability, as already seen, is antisymmetric about the channel centreline (figure 13 a). The $x$ -velocity eigenfunctions that result, after the splitting of the original neutral curve, are therefore both antisymmetric (figure 13 c). The labelling of the two eigenfamilies (1 and 2 in the nomenclature of Sureshkumar & Beris (Reference Sureshkumar and Beris1995b )) as $A1(a)$ and $A1(b)$ is justified because, as demonstrated above, for a range of $Re$ , $E$ and $k$ , the two eigenmodes become identical. In light of the above observation, the nomenclature used for labelling the modes in this section, and the new elasto-inertial modes examined in later sections, is as follows. The modes are numbered in increasing order of their critical Reynolds number, with the symmetry of the mode (about the centreline) being indicated by either an $A$ or an $S$ ; $A$ here refers to antisymmetric and $S$ to symmetric modes. Thus, the alphabet $A$ in $A1$ refers to antisymmetry of the mode, and ‘1’ refers to the mode with the lowest critical Reynolds number ( $Re_{c}$ ).

Figure 15(d) shows that for $E\gtrsim 0.0024$ , the neutral curve for mode $A1(a)$ becomes a closed loop which, with a further increase in $E$ , shrinks in extent, while simultaneously moving towards higher $Re$ . It ceases to exist at $E>0.003$ . The neutral stability curve for the eigenmode $A1(b)$ shifts towards lower $Re$ as $E$ increases further, making it more unstable, with a lowest value of $Re_{c}=1210.9$ at $E=0.0066$ . From figure 16, it is clear that for $E=0.006,0.007$ and $0.008$ , the neutral curve for eigenmode $A1(b)$ also becomes a closed curve, which again shrinks, eventually ceasing to exist for $E>0.008$ .

Porteous & Denn (Reference Porteous and Denn1972) were among the first to analyse the stability of viscoelastic plane Poiseuille flow and reported neutral curves for different $E$ . We have confirmed that their neutral curves correspond to modes $A1(a),A1(b)$ (and the mode $A2$ discussed below) found in this study. In the results of Porteous & Denn (Reference Porteous and Denn1972), the neutral curve for eigenmode $A1(a)$ did not exist at $E=0.005$ , and the reason for this is clear from figures 15(c) and 15(d), which show that the neutral curve disappears for $E>0.003$ . This disappearance also explains the abrupt ending of the curve depicting the variation of $Re_{c}$ with $E$ for eigenfamily 1 in figure 4 of Sureshkumar & Beris (Reference Sureshkumar and Beris1995b ). These authors extended the results of Porteous & Denn (Reference Porteous and Denn1972) for the $A1(a)$ and $A1(b)$ modes and showed the existence of a minimum critical Reynolds number $Re_{c}$ for mode $A1(a)$ at $E\approx 0.0025$ . For mode $A1(b)$ , the data presented by Sureshkumar & Beris (Reference Sureshkumar and Beris1995b ) were limited to $E\leqslant 0.0035$ .

Figure 17.

Neutral stability curves (a) $Re$ versus $k$ , and (b) $c_{r}$ versus $k$ , for $E=0.005$ . Points marked on panel (a) represent the $(k,Re)$ pairs for which the eigenspectra and eigenfunctions for the unstable modes are discussed further. The neutral curve for mode $A1(b)$ corresponds to the mode that splits from the continuation of the Newtonian TS mode to finite $E$ . At $E=0.005$ , the neutral curve of the original TS mode (mode $A1(a)$ in our nomenclature) does not exist. The neutral curves for modes $A2$ – $A4$ and $S1$ – $S3$ belong to the new elasto-inertial instabilities found in this study.

4.2.2 New elasto-inertial unstable eigenmodes

Within the parameter range $0<Re\leqslant 10^{4}$ and $0<E\leqslant 0.01$ , in addition to the modes $A1(a)$ and $A1(b)$ discussed above, we identified three new unstable antisymmetric ( $A2$ – $A4$ ) modes and three symmetric ones ( $S1$ – $S3$ ). A representative sample of $k$ and $Re$ values for $E=0.005$ , and the unstable modes that are found in this study, are mentioned in table 1. The neutral stability curves for all instabilities identified in this work are shown in figure 17(a) for $E=0.005$ . At $E=0.005$ , out of the seven eigenmodes tracked in this study, the eigenmode $A1(b)$ is the most unstable. Antisymmetric eigenmodes typically become unstable at a lower $Re_{c}$ in comparison to the symmetric modes.

Figure 18.

Neutral stability curves (a) $Re$ versus $k$ , and (b) $c_{r}$ versus $k$ , for $E=0.0035$ .

Table 1.

Unstable eigenmodes found for various combinations of $k$ and $Re$ for a fixed value of the elasticity number $E=0.005$ .

In order to demonstrate the effect of decreasing $E$ , another set of neutral curves for $E=0.0035$ is shown in figure 18(a). It is evident that, as $E$ decreases, the neutral curves shift towards higher $Re$ , while also increasing in extent along the $k$ -axis. Thus, for $E=0.0035$ , the neutral curves for modes $S3$ and $A4$ have moved to $Re>10^{4}$ . Figures 17(b) and 18(b) show the corresponding variation of $c_{r}$ with $k$ along the neutral curves. One important feature that emerges from these figures is that the wave speeds of all unstable modes lie within the range of base-flow velocities $0\leqslant c_{r}\leqslant 1$ . For large $Re$ , the $c_{r}$ are relatively small ( $c_{r}\lesssim 0.3$ ) implying that the mechanism underlying the instability involves the confining walls in an essential way. This is discussed further in § 4.2.4.

Figure 19.

Filtered eigenspectrum (a) for $E=0.005,k=1.6,Re=4000$ (location 1 in figure 17 a) for $N=500$ . Panel (b) shows the normalized $\tilde{v}_{x}$ eigenfunction corresponding to the unstable mode $0.3784+0.00552\text{i}$ shown in (a).

We now present a few representative eigenspectra and eigenfunctions, for the unstable modes, for the $(k,Re)$ combinations marked by circles within the different neutral curves in figure 17(a). The eigenspectrum for $E=0.005$ , $k=1.6$ and $Re=4000$ , corresponding to point 1 in figure 17(a), is shown in figure 19(a). Only one unstable mode is present for the said parameters, and this is the eigenmode $A2$ . The corresponding streamwise velocity ( $\tilde{v}_{x}$ ) eigenfunction appears in figure 19(b). The eigenspectrum for $E=0.005$ , $k=1.3$ and a higher $Re=8000$ (point 2 in figure 17 a) shows three unstable modes (figure 20 a). The corresponding eigenfunctions, in decreasing order of their growth rates, are shown in figures 20(b), 20(c) and 20(d). The leading mode is the eigenmode $A2$ , and the two other eigenvalues are the new unstable eigenmodes, namely $A3$ and $S2$ , identified here for the first time. The eigenspectrum for $E=0.005$ , $k=1.67$ and $Re=9800$ (corresponding to point 3 on figure 17 a) again shows three unstable modes (figure 21(a)). A higher value $N=900$ is required for an accurate resolution of these unstable modes owing to the finer-scaled structures in the corresponding eigenfunctions for this set of parameters. The most unstable eigenvalue belongs to the family $A3$ whereas the two other eigenvalues belong to new families $A4$ and $S3$ . The eigenfunctions for the three unstable eigenmodes of this eigenspectrum are shown in figures 21(b), 21(c) and 21(d).

Figure 20.

(a) Filtered eigenspectrum for $E=0.005,k=1.3,Re=8000$ (location 2 in figure 17 a) for $N=500$ . Panels (b–d) show the normalized eigenfunctions for $\tilde{v}_{x}$ corresponding to the three unstable modes shown in panel (a).

Figure 21.

Filtered eigenspectrum (a) for $E=0.005,k=1.67,Re=9800$ (location 3 in figure 17 a) for $N=900$ . Panels (b–d) show the normalized eigenfunctions for $\tilde{v}_{x}$ corresponding to the three unstable modes shown in (a).

Figure 22.

Evolution of the unstable modes $A2$ and $A3$ from the $E\rightarrow 0$ limit: panel (a) shows the spectra (symbols) and results from the shooting procedure (continuous line) for the scaled growth rate $kWc_{i}$ versus $c_{r}$ for mode $A2$ . Panel (b) shows the Newtonian spectra (symbols) and shooting (continuous line) results for mode $A3$ at different $E$ . The inset in panel (b) zooms into the region where the mode becomes unstable.

4.2.3 Origin of the new elasto-inertial unstable modes

The neutral stability curves shown, for example, in figure 18, do not contain the information on whether the new unstable modes identified in this study are a continuation, to finite $E$ , of the stable modes in the spectrum for Newtonian channel flow, or whether they are genuinely new modes that do not have any counterparts in the Newtonian spectrum. If the latter scenario holds, then, the growth rates of the modes should diverge to $-\infty$ in the limit $E\rightarrow 0$ . In this case, it is still possible for the modes to emerge either from the continuous spectrum or the HFGL line, since both these features of the viscoelastic spectrum have $c_{i}\propto 1/E$ for $E\rightarrow 0$ . In order to understand this better, we continued the unstable modes $A2$ and $A3$ to small $E$ , and the results are shown in figure 22. The unstable mode $A2$ smoothly continues to an HFGL mode in the limit $E\rightarrow 0$ . In order to illustrate this better, in figure 22(a), we plot $kWc_{i}$ , since $c_{i}=-1/(2kW)$ for the HFGL modes, and the quantity $kWc_{i}=-0.5$ . The continuous line is the result from the shooting code, while the discrete symbols are results from the spectral method for a few representative values of $E$ . In figure 22(b), we show the evolution of the unstable mode $A3$ , which (in contrast to $A2$ ) smoothly continuous (as $E\rightarrow 0$ ) to the third mode in the $A$ -branch of the Newtonian spectrum. Our results for the other elasto-inertial modes (not shown in the figures) show that similar to $A2$ , the modes $A4$ , $S1$ and $S3$ continue to one of the HFGL modes as $E\rightarrow 0$ . However, the mode $S2$ , similar to $A3$ , smoothly continues to a stable mode in the Newtonian spectrum as $E\rightarrow 0$ . Thus, it appears that there is no generic trend with regard to the origin of the new elasto-inertial unstable modes in the Newtonian limit.

Based on the eigenfunctions presented here, all the new unstable modes are qualitatively similar in that the disturbances are confined close to the walls. The $c_{r}$ for these modes are small ( $c_{r}<0.45$ ) and the maximum oscillation amplitude and the reduced spatial frequency occurs in the vicinity of the critical layer, in a manner similar to the least stable modes of the hourglass structure seen earlier (figures 10 d and 13 c). This similarity with the least stable hourglass modes remains true for both the symmetric and antisymmetric modes. In all cases considered, the length scale of the oscillations decreases with increasing $Re$ . More unstable modes could well be present at higher $Re$ and $E$ , but on account of the spatial scales in the eigenfunctions (shown, for example, in figures 19 b, 20 b and 21 b) becoming increasingly small, computation of eigenvalues (via either of the two methods used) becomes difficult. It is necessary to either take closer initial guesses (for the shooting method) or consider a higher number of collocation points (for the spectral method).

Figure 23.

Variation of neutral stability curves with $E$ for eigenmodes (a) $A2$ , and (b) $S1$ .

4.2.4 Neutral stability curves for higher unstable elasto-inertial modes

In this section, we examine the role of fluid elasticity on the neutral stability curves for the new elasto-inertial unstable modes ( $A2$ – $A4$ and $S1$ – $S3$ ) found in this study. A similar discussion for the elastically modified TS mode ( $A1(a)$ ) and the neutral curve that coalesced/bifurcated from $A1(a)$ , viz., $A1(b)$ , was presented in § 4.2.1. Figure 23 shows that the neutral stability curves for both modes $A2$ and $S1$ shift to lower $Re$ as $E$ is increased. Similar behaviour is observed for modes $A3$ and $S2$ (not shown). For the values of $E$ explored, the neutral stability curves for modes $A1(a)$ , $A1(b)$ and $A2$ form closed loops at sufficiently high $E$ . We expect this trend to carry over to other higher modes as well at high enough $E$ , but could not explicitly verify this due to computational difficulties associated with accessing that regime.

Figure 24(a) shows the variation of $Re_{c}$ with $E$ for all the modes identified in this study. As indicated, our results compare well with the (limited) results reported in Sureshkumar & Beris (Reference Sureshkumar and Beris1995b ). For eigenmodes $A1(a)$ , $A1(b)$ and $A2$ the critical Reynolds number $Re_{c}$ first decreases with increasing $E$ , passing through a minimum, before increasing again. This feature could not be explored for the modes higher than $A2$ for the values of $E$ probed ( $E\leqslant 0.01$ ). All three eigenfamilies exist only over a finite range of $E$ , this arising from the shrinking and subsequent disappearance of the corresponding neutral loops, in the $Re$ – $k$ plane, at critical values of $E$ (as discussed earlier in § 4.2.1). For the modes $A1(a)$ and $A1(b)$ , the threshold values are $E\approx 0.003$ and $0.008$ , respectively. For the $A2$ mode, the threshold $E$ might lie just outside the range of $E$ ( ${\leqslant}0.01$ ) examined here. Our expectation that the neutral loop corresponding to mode $A2$ will nevertheless close up is reinforced by the appearance of the non-monotonicity in $Re_{c}$ (figure 24 a). This is because, for modes $A1(a)$ and $A1(b)$ , the non-monotonicity appears just before the values of $E$ for which the neutral loop begins to shrink in size. In general, the critical $Re$ for a given modal family initially decreases with $E$ , and after reaching a minimum, again increases with $E$ . The neutral curve of a given family reduces to a shrinking loop that disappears beyond a critical $E$ . Concomitantly, new (unstable) modal families continue to appear as $E$ is increased, albeit with larger values of $Re_{c}$ . Thus, if $Re$ is fixed, plane Poiseuille flow of a UCM fluid is eventually stable at sufficiently high $E$ .

Of the three eigenmodes above, the lowest $Re_{c}=1210.9$ (at $E=0.0066$ ) occurs for $A1(b)$ . In cases where data are available, for $E\ll 1$ , $Re_{c}\propto E^{-1}$ for eigenmodes $S1$ and $S2$ , whereas $Re_{c}\propto E^{-5/4}$ for eigenmodes $A2$ and $A3$ . The phase speed $c_{r,c}\propto E^{1/2}$ for all the modes except $A1(a)$ (figure 24 b), thus indicating that the unstable elasto-inertial modes are a type of ‘wall modes’ wherein the disturbances are confined to a thin region near the wall (Schmid & Henningson Reference Schmid and Henningson2001). The critical wavenumber $k_{c}\propto E^{-1/2}$ for eigenmodes $S1$ and $S2$ , while it scales as $E^{-1/4}$ for eigenmode $A3$ as shown in figure 24(c). The data for mode $A2$ , for the lowest values of $E$ probed, are suggestive of the scaling $k_{c}\sim E^{-1/4}$ similar to $A3$ .

We also analysed the trend exhibited by the critical frequency corresponding to the minimum of the $Re$ – $k$ neutral curves for various modal families. When the critical frequency, $\unicode[STIX]{x1D714}_{c}^{\ast }$ , is made dimensionless using the relaxation time, $\unicode[STIX]{x1D706}$ , the resulting quantity can be interpreted as a Deborah number $De_{c}=\unicode[STIX]{x1D714}_{c}^{\ast }\unicode[STIX]{x1D706}=Wk_{c}c_{r,c}$ . When the values of $De_{c}$ for various modes are plotted against $E$ (figure 24 d), they seem to approach an $O(1)$ constant as $E\rightarrow 0$ , although the constants are different for different modes. Scaling arguments given below also imply that, for the class of wall modes, $De_{c}\sim O(1)$ for $E\ll 1$ .

Figure 24.

Variation of critical parameters with $E$ for modes $A1(a),A1(b),A2,A3,S1$ and $S2$ for $E\leqslant 0.01$ : (a) $Re_{c}$ versus $E$ , eigenmodes $A2$ and $A3$ scale as $Re_{c}\propto E^{-5/4}$ , and eigenmodes $S1$ and $S2$ scale as $Re_{c}\propto E^{-1}$ , (b) $c_{r}$ versus $E$ , eigenmodes $A2,A3,S1$ and $S2$ scale as $c_{r}\propto E^{1/2}$ , (c) $k_{c}$ versus $E$ , eigenmodes $S1$ and $S2$ scale as $k_{c}\propto E^{-1/2}$ , and eigenmodes $A3$ and $A2$ at the lower $E$ end, scale as $k_{c}\propto E^{-1/4}$ . Present results are also compared with those of Sureshkumar & Beris (Reference Sureshkumar and Beris1995b ) (denoted as $SB1$ and $SB2$ ). The insets in each panel show the variation for mode $A1(a)$ and its approach to the Newtonian TS mode as $E\rightarrow 0$ , and (d) the variation of the dimensionless critical frequency $De_{c}$ with $E$ for modes $A2$ , $A3$ , $S1$ and $S2$ shows that it approaches an $O(1)$ constant as $E\rightarrow 0$ .

The numerically obtained scalings for $Re$ , $c_{r}$ and $k$ in the limit $E\ll 1$ may be rationalized using a boundary layer analysis near the walls. It is worth emphasizing that this scaling analysis does not explain why the modes are unstable, but only serves to organize the elasto-inertial unstable modes identified in this study into a class of viscoelastic wall modes. In the limit $Re\gg 1$ and $E\ll 1$ , there is a thin ‘wall layer’ near the walls of (non-dimensional) thickness $\unicode[STIX]{x1D6FF}\ll 1$ . The scalings for $Re$ , $k$ , $\unicode[STIX]{x1D6FF}$ and $c_{r}$ in terms of $E$ can be derived by rescaling the (4.10)–(4.15) in the region near the wall as follows. The wall-normal coordinate $z$ near the wall at $z=1$ can be expressed as $z=1-\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D709}$ . In the limit $Re\gg 1$ , $E\ll 1$ , the eigenvalue $c$ is asymptotically small, and hence we expand $c=\unicode[STIX]{x1D6FF}c^{(0)}$ . The derivatives near the wall get rescaled as $\text{d}/\text{d}z=-(1/\unicode[STIX]{x1D6FF})(\text{d}/\text{d}\unicode[STIX]{x1D709})\equiv -\unicode[STIX]{x1D6FF}^{-1}{\mathcal{D}}$ , where ${\mathcal{D}}\equiv \text{d}/\text{d}\unicode[STIX]{x1D709}$ . The base-flow profile becomes $U=1-z^{2}\equiv 2\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D709}-\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D709}^{2}$ , which for $\unicode[STIX]{x1D6FF}\ll 1$ implies that $U\sim \unicode[STIX]{x1D6FF}$ near the wall. Using the boundary layer rescaling, equations (3.8)–(3.13) are modified for the region near the wall as

(4.10) $$\begin{eqnarray}\displaystyle & \displaystyle -\frac{1}{\unicode[STIX]{x1D6FF}}{\mathcal{D}}\tilde{v}_{z}+\text{i}k\tilde{v}_{x}=0, & \displaystyle\end{eqnarray}$$

(4.11) $$\begin{eqnarray}\displaystyle & \displaystyle Re\{\text{i}k(2\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D709}-\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D709}^{2}-\unicode[STIX]{x1D6FF}c^{(0)})\tilde{v}_{x}-2(1-\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D709})\tilde{v}_{z}\}=-\text{i}k\tilde{p}+\text{i}k\tilde{\unicode[STIX]{x1D70F}}_{xx}-\unicode[STIX]{x1D6FF}^{-1}{\mathcal{D}}\tilde{\unicode[STIX]{x1D70F}}_{xz}, & \displaystyle\end{eqnarray}$$

(4.12) $$\begin{eqnarray}\displaystyle & \displaystyle Re\text{i}k(2\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D709}-\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D709}^{2}-\unicode[STIX]{x1D6FF}c^{(0)})\tilde{v}_{z}=-\unicode[STIX]{x1D6FF}^{-1}{\mathcal{D}}\tilde{p}+\text{i}k\tilde{\unicode[STIX]{x1D70F}}_{xz}-\unicode[STIX]{x1D6FF}^{-1}{\mathcal{D}}\tilde{\unicode[STIX]{x1D70F}}_{zz}, & \displaystyle\end{eqnarray}$$

(4.13) $$\begin{eqnarray}\displaystyle & & \displaystyle \{1+\text{i}kW(2\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D709}-\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D709}^{2}-\unicode[STIX]{x1D6FF}c^{(0)})\}\tilde{\unicode[STIX]{x1D70F}}_{xx}=\left[\left\{4W(1-\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D709})\unicode[STIX]{x1D6FF}^{-1}{\mathcal{D}}+2\text{i}k\right.\right.\nonumber\\ \displaystyle & & \displaystyle \quad \left.\left.+\,16\text{i}kW^{2}(1-\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D709})^{2}\right\}\tilde{v}_{x}-16W^{2}(1-\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D709})\tilde{v}_{z}\right]-4W(1-\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D709})\tilde{\unicode[STIX]{x1D70F}}_{xz},\end{eqnarray}$$

(4.14) $$\begin{eqnarray}\displaystyle & & \displaystyle \{1+\text{i}kW(2\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D709}-\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D709}^{2}-\unicode[STIX]{x1D6FF}c^{(0)})\}\tilde{\unicode[STIX]{x1D70F}}_{xz}=\left[-\unicode[STIX]{x1D6FF}^{-1}{\mathcal{D}}\tilde{v}_{x}+\left\{\text{i}k\right.\right.\nonumber\\ \displaystyle & & \displaystyle \quad \left.\left.+\,8\text{i}kW^{2}(1-\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D709})^{2}+2W\right\}\tilde{v}_{z}\right]-2W(1-\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D709})\tilde{\unicode[STIX]{x1D70F}}_{zz},\end{eqnarray}$$

(4.15) $$\begin{eqnarray}\displaystyle \{1+\text{i}kW(2\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D709}-\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D709}^{2}-\unicode[STIX]{x1D6FF}c^{(0)})\}\tilde{\unicode[STIX]{x1D70F}}_{zz}=-2\{\unicode[STIX]{x1D6FF}^{-1}{\mathcal{D}}+2\text{i}kW(1-\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D709})\}\tilde{v}_{z}. & & \displaystyle\end{eqnarray}$$

Near the walls, $(U-c)\sim \unicode[STIX]{x1D6FF}$ . For $E\ll 1$ , our numerical results show that $k_{c}$ becomes large (figure 24 c), and so we set $k\sim \unicode[STIX]{x1D6FF}^{-1}$ in the following analysis. The left side of the linearized constitutive equations reveal that in order for the elastic stresses and viscous relaxation to be of the same order, we require $W\sim O(1)$ , or $Re\sim E^{-1}$ . Further, the linearized constitutive relations indicate that $\tilde{\unicode[STIX]{x1D70F}}_{zz}\sim \tilde{\unicode[STIX]{x1D70F}}_{xz}\sim \tilde{\unicode[STIX]{x1D70F}}_{xx}\sim \unicode[STIX]{x1D6FF}^{-1}$ . The linearized $x$ -momentum equation indicates that $Re\sim \unicode[STIX]{x1D6FF}^{-1}\unicode[STIX]{x1D70F}_{xz}$ , which then yields $Re\sim \unicode[STIX]{x1D6FF}^{-2}$ . Using $Re\sim E^{-1}$ we therefore obtain the following scaling relationships in terms of $E$ for $E\ll 1$ :

(4.16a-d ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}\sim E^{1/2},\quad k\sim E^{-1/2},\quad Re\sim E^{-1},\quad c_{r}\sim E^{1/2}. & & \displaystyle\end{eqnarray}$$

From figures 24(a) and 24(c) it is clear that the modes $S1$ and $S2$ follow the above scaling very well. The scalings are not followed exactly for the antisymmetric modes in the range of $E$ explored. It is possible that the numerical and asymptotic scaling exponents will agree for much lower values of $E$ and higher values of $Re$ , a regime that is difficult to access numerically.