Elastic instability in a family of rectilinear viscoelastic channel flows devoid of centerline symmetry

We analyze the linear stability of viscoelastic channel flows, with velocity profiles that are asymmetric about the channel centerline, belonging to the one-parameter Couette-Poiseuille family (CPF). These flows are driven by a combination of an imposed pressure gradient and (tangential) wall motion. A particular member of this family, corresponding to a zero net volumetric flux, may be experimentally realized in a shallow lid-driven cavity flow configuration, as well as in the narrow-gap limit of the Taylor-Couette geometry with an obstruction at a fixed azimuthal angle (a narrow gap Taylor-Dean flow). Recent work by Khalid et al. [Phys. Rev. Lett. 127, 134502 (2021)] has shown, using the Oldroyd-B model, that plane Poiseuille flow with a symmetric velocity profile becomes unstable, even in the absence of inertia, to an elastic “center mode” with phase speed close to the base-state maximum. In contrast, viscoelastic plane Couette flow is linearly stable. The objective of this study is to determine parameter regimes where viscoelastic CPF, whose members include the two limiting flows above, is unstable in the inertialess limit. The dimensionless groups that govern stability are the Weissenberg number $W=\lambda U_{\text{avg}}/L$, the parameter $\alpha$ characterizing the relative importance of Couette ($\alpha =0$) and Poiseuille flow ($\alpha =1$) components, and the ratio of solvent to solution viscosities $\beta ={\mu }_s/\mu$. Here, $\lambda$ is the polymer relaxation time, $L$ the channel half-width and $U_{\text{avg}}$ the average speed; $\beta \in [0,1]$, and $\alpha \in (-\infty ,\infty )$ with $\alpha \to \pm \infty$ representing the unidirectional flow in a shallow lid-driven cavity. We show that, similar to plane Poiseuille flow, an elastic center-mode instability does indeed exist for the aforesaid family in the limit of ultra-dilute polymer solutions ($\beta \gtrsim 0.99$); the instability relies on the existence of a base-state maximum, implying its absence for CPF members with $\alpha \in (-0.5,0.25)$. Our results point to the potential relevance of the center-mode instability to viscoelastic Taylor-Dean flows and other curvilinear shear flow configurations.

Figure 1

Schematic of the flow geometry. The base-state flow occurs as a result of the applied pressure gradient and the motion of the top plate with velocity $U_w$; the bottom plate remains stationary. The velocity profiles shown correspond to the pressure-driven (black) and plate-driven (blue) components.

Figure 2

The nondimensional base-state velocity profiles for the CPF family for different $\alpha$: $\alpha =0$ and 1 represent plane Couette and plane Poiseuille flows, respectively. For $-0.5\le \alpha <0.25$, the base flow velocity is monotonic; for $\alpha \to \pm \infty$, the flow reduces to nonmonotonic cavity flows that are mirror images or each other. Note that the cavity flow profile has been nondimensionalized using the top-plate velocity, since the average speed is zero.

Figure 3

Unfiltered elastic eigenspectra for CPF of an Oldroyd-B fluid with $\alpha =2$; data shown for $W=900,k=0.4,\beta =0.995$ and $N=450$ and 500. The inset in panel (a) shows a poorly converged spurious mode located near $c_r=-2$, which arises in the spectral method (this mode is nonconvergent in the shooting method.) The magnified view in panel (b) shows a (converged) discrete unstable mode with $c=2.08157+0.00003594i$.

Figure 4

Unfiltered elastic eigenspectra for CPF of an Oldroyd-B fluid with $\alpha =0.75$; data shown for $W=2500,k=0.4,\beta =0.995$, and $N=450$ and 500. Both the main figure and the inset in panel (a) highlight the spurious modes located near $c_r\approx 0$ and 0.5 which remain unconverged in the shooting method. The magnified view in panel (b) shows a (converged) discrete unstable mode with $c=1.3883837+0.00005081i$.

Figure 5

Real, imaginary, and absolute values of the (a) streamwise and (b) wall-normal velocity eigenfunctions for CPF of an Oldroyd-B fluid with $\alpha =1,1.5$, and 2. The data is shown for the neutral eigenmode at $\beta$ = 0.995; the other parameter values are given in Table 2. For $\alpha =1.5$ and 2, only the portion of the eigenfunction from $z=-1$ to $z=z_0$ has been plotted, the latter value being the zero-crossing of the base-state velocity profile; further, this domain has been rescaled to $z^{\prime }\in [-1,1]$ to facilitate comparison.

Figure 6

Real, imaginary, and absolute values of the stress eigenfunctions of the (a) streamwise normal stress ${\tilde{\tau }}_{xx}$ and (b) shear stress ${\tilde{\tau }}_{xz}$, for CPF of an Oldroyd-B fluid with $\alpha =1,1.5$, and 2. The data is shown for the neutral eigenmode at $\beta$ = 0.995; the other parameter values are given in Table 2. For $\alpha =1.5$ and 2, only the portion of the eigenfunction from $z=-1$ to $z=z_0$ has been plotted, the latter value being the zero-crossing of the base-state velocity profile; further, this domain has been rescaled to $z^{\prime }\in [-1,1]$ to facilitate comparison.

Figure 7

Contours of the real part of velocity perturbations in $x-z$ plane: (a) the streamwise velocity (${\widehat{v}}_x$) and (b) wall-normal (${\widehat{v}}_z$), for CPF of an Oldroyd-B fluid with $\alpha =2$ (upper subfigure) and $\alpha =1$ (lower subfigure); data corresponding to parameters shown in Table 2.

Figure 8

Contours of the real part of stress perturbations in $x$–$z$ plane: (a) the streamwise stress (${\widehat{\tau }}_{xx}$) and (b) shear stress (${\widehat{\tau }}_{xz}$) for CPF of an Oldroyd-B fluid with $\alpha =2$ (upper subfigure) and $\alpha =1$ (lower subfigure); data corresponding to parameters shown in Table 2.

Figure 9

The $c_i$ of the center mode, as a function of $W$, for different members of the CPF family ($\alpha =0.25$, 0.5, 1 and 2) and $\beta =0.99$; (a) $k=0.1$ and (b) for $k=1$. The inset in panel (a) shows a magnified view of the region in the vicinity of $c_i=0$. The flow remains linearly stable in all cases.

Figure 10

The variation of $c_i$ for the center mode as a function of (a) $\alpha$ for $\beta =0.99, k=0.05$, 0.1, 0.25, 0.5; (b) $\beta$ for different values of $\alpha$ and $k=0.1$. The inset in panel (a) shows the region near $\alpha =0.25$, where the center mode merges into the CS. While, the inset in panel (b), shows the narrow range over which the mode is unstable; data for $W=3200$.

Figure 11

Neutral stability curves in the $W-k$ plane for CPF at various $\alpha$ ($>0$). (a) Instability is present for $\alpha \in$ (0.67, 2). (b) Instability exists $\forall \alpha \gtrsim 0.25$.

Figure 12

Variation of the (a) critical Weissenberg number ($W_c$) and (b) the critical wave number ($k_c$) for $\alpha =0.8$, 1, and 1.1.

Figure 13

Variation of the critical Weissenberg numbers ($W_c$ and $W_{Pc}$) with $\alpha$ ($\alpha >0$). For $\beta \ge$ 0.996 and $\alpha \ge$ 2.0, $W_c$ decreases as ${\alpha }^{-1}$, while $W_{Pc}$ asymptotes to a plateau. In panel (b), the dashed straight lines are the large-$\alpha$ asymptotes for cavity flow. The $(\beta ,W_{Pc})$ pairs from the cavity-flow limit are $(0.996,3499.8), (0.997,2373.9), (0.998,2875)$, and (0.999,5278.4).

Figure 14

Neutral stability curves in the $W-k$ plane and variation of the critical Weissenberg numbers (${W_P}_c$ and $W_c$) for $\alpha <0$. Data is presented at a fixed value of $\beta =0.996$. (a) Neutral curves shift upwards as $\alpha \to -0.5.$ (b) For $\alpha \le -100$, the value of ${W_P}_c$ is nearly equal to the critical Weissenberg number of cavity flow, which is represented as a black dashed line.

Figure 15

(a) Schematic of the flow in a shallow lid-driven cavity; (b) the velocity profile characterizing the unidirectional flow away from the end walls; this profile is a limiting member of the Couette-Poiseuille family, corresponding to $\alpha \to \infty$.

Figure 16

Panel (a) shows the zoomed-in unfiltered eigenspectra around $c_r$ = 0.333 for cavity flow of an Oldroyd-B fluid with $W_{\infty }=7500, k=1$, and $\beta =0.998$. Spectra are shown at two different collocation points ($N$ = 550 and 600) with a discrete eigenvalue, having $c=0.3332252+0.0000148i$. Panel (b) shows the variation of $c_i$ with scaled Weissenberg number $\left(k{W}_{\infty }\right)$ for cavity flow at various wave numbers and fixed $\beta$ = 0.996. The inset shows an expanded view of the region where $c_i$ is positive.

Figure 17

Neutral stability curves in the $W_{\infty }-k$ plane, for cavity flow of an Oldroyd-B fluid, for different $\beta$.