3.1. Newtonian fluid spectra with and without porous walls

To investigate the behaviour of viscoelastic fluid in plane Poiseuille flow with porous walls, we begin by analysing the evolution of the eigenspectra with increasing viscoelastic effects. Starting with the Newtonian spectrum ( $ \textit{Wi}=0$ ), we gradually introduced the effects of viscoelasticity by incrementally increasing the $ \textit{Wi}$ from zero while keeping $\beta$ constant and using a high value of the permeability parameter ( $\alpha = 500$ ) to minimize the effect of the porous walls. We then replicated this analysis for a channel with porous walls, characterized by a low value of $\alpha$ (i.e. $\alpha =50$ ), thereby enhancing the effect of fluid exchange with the porous layers. To further isolate the role of permeability parameter, we also examine the variation of the eigenspectrum with decreasing $\alpha$ at fixed $ \textit{Wi}$ and $\beta$ , capturing the transition from nearly impermeable to strongly permeable walls. Throughout our analysis, we maintain a constant porosity ( $\epsilon = 0.6$ see Appendix B) and a zero momentum-transfer coefficient at the interface ( $\tau = 0$ ), ensuring continuity of tangential velocity and normal stress across the fluid–porous interface.

The eigenspectra for Newtonian plane Poiseuille flow with both impermeable walls (high $\alpha$ value) and porous walls at $ \textit{Re} = 800$ and $k = 1.5$ are presented in figure 4. Note that the Oldroyd-B eigenspectrum reduces to the Newtonian eigenspectrum when either $ \textit{Wi}$ = 0 (for any $\beta$ ) or $\beta$ = 1 (for any $ \textit{Wi}$ ). The dimensionless parameters for the case with porous walls are set to $\alpha = 50$ and $\delta = 1$ . The Newtonian eigenspectrum for plane Poiseuille flow generally exhibits a distinct ’Y-shaped’ pattern at sufficiently high Reynolds numbers (indicated by green circles in figure 4). While the Newtonian plane Poiseuille flow with the impermeable surface is stable, when the Reynolds number exceeds 5772, an instability arises from a WM known as the TS instability (Schmid et al. Reference Schmid, Henningson and Jankowski2002), where the eigenfunctions predominantly are very near the walls. Note that the flow is stable under the parameters examined in the figure. In the case of a Newtonian fluid in channel with porous walls, permeability introduces two new WMs, labelled 3 and 4 (blue triangles in figure 4), along with a group of new eigenvalues labelled 7, referred to in this study as the ’porous mode’ (Tilton & Cortelezzi Reference Tilton and Cortelezzi2008). As described in Tilton & Cortelezzi (Reference Tilton and Cortelezzi2008), the porous modes consist of a series of repeated pairs of eigenvalues generated by the influence of permeability, approaching the real axis with small but non-zero phase speed ( $c_r$ ). These modes remain stable regardless of the permeability’s magnitude. However, increasing permeability (decreasing $\alpha$ ) while maintaining a constant Reynolds number can destabilize the two Orr–Sommerfeld WMs (Tilton & Cortelezzi Reference Tilton and Cortelezzi2008), labelled 1 and 2 (blue triangles in figure 4). Wall modes 1 and 2 become unstable at a critical permeability as reported in (Tilton & Cortelezzi Reference Tilton and Cortelezzi2008). In this figure, only WM 1 is observed to be destabilized under the parameters analysed.

Figure 4.

Eigenspectra for Newtonian plane Poiseuille flow at $ \textit{Re}=800$ and $k=1.5$ overlying porous walls for $\alpha = 500$ (representing an impermeable channel) and $\alpha = 50$ (representing channel with permeable wall).

Figure 5.

Eigenspectra for plane Poiseuille flow of an Oldroyd-B fluid at $ \textit{Re}=800$ , $k=1.5$ and $\beta = 0.8$ , for different $ \textit{Wi}$ numbers: (a) $ \textit{Wi}=0.08$ , (b) $ \textit{Wi}=0.8$ , (c) $ \textit{Wi}=4$ , (d) $ \textit{Wi}=7.2$ , (e) $ \textit{Wi}=40$ , (f) $ \textit{Wi}=80$ . For all plots $\alpha = 500$ , representing an impermeable channel. Also, $N$ and $N_{m}$ are considered $400$ .

3.2. Oldroyd-B viscoelastic fluid spectra with and without porous walls

We next examine the effect of viscoelasticity on the eigenspectrum of an Oldroyd-B fluid in an impermeable channel by gradually increasing the $ \textit{Wi}$ from 0.08 to 80 while keeping the viscosity ratio fixed at $\beta =0.8$ , $ \textit{Re}=800$ and $k=1.5$ . These parameter values are chosen to match those reported by Khalid et al. (Reference Khalid, Chaudhary, Garg, Shankar and Subramanian2021a ), enabling direct comparison with their benchmark results for Oldroyd-B fluids in impermeable channels. In all cases, to ensure minimal interference from wall permeability, a high permeability parameter ( $\alpha =500$ ) is used. As shown in figure 5(a), the eigenspectrum at low Wiessenberg number ( $ \textit{Wi}=0.08$ ) remains qualitatively similar to the classical Newtonian Y-shaped spectrum, with only minor perturbations. However, additional discrete modes emerge due to the presence of polymer stresses, signalling the onset of viscoelastic effects even at modest $ \textit{Wi}$ . For comparison, the Newtonian eigenspectrum for the same configuration (green circles) is included, highlighting the deviation introduced by viscoelasticity. As previously reported by Graham (Reference Graham1998), Wilson, Renardy & Renardy (Reference Wilson, Renardy and Renardy1999), Chaudhary et al. (Reference Chaudhary, Garg, Shankar and Subramanian2019) and Khalid et al. (Reference Khalid, Chaudhary, Garg, Shankar and Subramanian2021a ), the eigenspectrum of the Oldroyd-B fluid is characterized by a pair of continuous spectra (denoted CS1 and CS2), which arise due to the local nature of the polymeric stress constitutive model. These continuous spectra appear as balloon-like structures in the complex phase speed plane, around which a ring of discrete modes is typically observed. The positions of the continuous spectra can be analytically determined by identifying the locations where the coefficient of the highest-order derivative in the linear stability equations vanishes. This leads to two horizontal lines in the $c_r$ – $c_i$ plane, corresponding to $c_i = -1/(\textit{kWi})$ for CS1 and $c_i = -1/(\beta \textit{kWi})$ for CS2, where $0 \leqslant c_r \leqslant 1$ (Wilson et al. Reference Wilson, Renardy and Renardy1999; Chokshi & Kumaran Reference Chokshi and Kumaran2009). In our analysis, we also observe an inverted Y-shaped structure emerging just above CS1 (see inset B in figure 5 a), consistent with the findings of Khalid et al. (Reference Khalid, Chaudhary, Garg, Shankar and Subramanian2021a ). Note that despite the high permeability parameter ( $\alpha = 500$ ) employed to emulate impermeable channel conditions, subtle deviations from the benchmark impermeable-wall results remain. Specifically, a faint vertical branch appears between CS1 and CS2, slightly tilted and approaching the real axis with a small but non-zero phase speed. This branch reflects the residual influence of wall permeability, highlighting the sensitivity of the viscoelastic eigenspectrum even to weak permeability.

Consistent with the observations of Khalid et al. (Reference Khalid, Chaudhary, Garg, Shankar and Subramanian2021a ), we find that increasing $ \textit{Wi}$ significantly alters the structure of the eigenspectrum in an impermeable channel. At low to moderate $ \textit{Wi}$ , the discrete ‘ring’ of eigenvalues surrounding the continuous spectrum (CS1 and CS2) begins to collapse inward, progressively concentrating around the two CS branches. With further increase in $ \textit{Wi}$ , this ring-like structure disintegrates and wraps around the CS modes, with only a small segment remaining near $c_{r} \approx 1$ (figure 5 b). At higher $ \textit{Wi}$ (figure 5 c), the ring fully collapses onto the CS curves. Concurrently, the vertical position of the continuous spectra shifts upward towards $c_{i}=0$ , indicating a weakening of the decay rates. This collapse of discrete modes into the continuous spectrum is accompanied by a merging of the elastically modified Newtonian modes with the CS. In particular, the Newtonian centre modes (NCMs), and the WMs, namely the least stable TS mode (TSM) and a higher-order WM (WM-2), gradually coalesce with the CS curves as $ \textit{Wi}$ increases (figure 5 c). At $ \textit{Wi}=7.2$ (figure 5 d), the least stable eigenvalue above the CS remains the elastically modified TSM. However, as first identified by Khalid et al. (Reference Khalid, Chaudhary, Garg, Shankar and Subramanian2021a ), a new type of eigenmode, termed the elasto-inertial centre mode (ECM-1), emerges above CS1, lacking a Newtonian counterpart. Our results reproduce this behaviour: ECM-1 appears clearly above CS1, while additional ECMs (ECM-2, ECM-3 and ECM-4) form below CS1 (inset of figure 5 d). With increasing $ \textit{Wi}$ , ECM-1 migrates upward and ultimately becomes unstable (at $ \textit{Wi} = 80$ ), as illustrated in figure 5(f). This instability, first reported by Garg et al. (Reference Garg, Chaudhary, Khalid, Shankar and Subramanian2018) for pipe flows and subsequently examined by Khalid et al. (Reference Khalid, Chaudhary, Garg, Shankar and Subramanian2021a ) for planar channel flows, signifies a fundamental departure from Newtonian dynamics and highlights the rich modal structure introduced by viscoelasticity. Our findings for viscoelastic flow in a channel with impermeable walls are consistent with the results reported by Khalid et al. (Reference Khalid, Chaudhary, Garg, Shankar and Subramanian2021a ).

It is important to emphasize that in our study, the linearized system governing the coupled free-flow and porous regions cannot be reduced to the smooth-wall Oldroyd–B formulation by taking the porous-layer height $\delta \to 0$ . This is because the two domains are governed by fundamentally different models: the free-flow region satisfies the Navier–Stokes–Oldroyd–B equations, while the porous layers are described by the modified Darcy–Brinkman–Oldroyd–B model, which includes additional terms (see (2.5)). These terms remain present regardless of the value of $\delta$ , preventing the system from collapsing algebraically to the smooth-wall equations used by Khalid et al. (Reference Khalid, Shankar and Subramanian2021b ). Moreover, because $\delta$ appears only in the boundary and interface conditions – and not in the governing equations themselves – the limit $\delta \to 0$ alters the geometry but does not transform the porous-region model into the classical channel-flow formulation.

To examine the impact of porous walls on flow instability, we consider a channel configuration with a relatively high permeability ( $\alpha =50$ ). All other parameters ( $ \textit{Re}$ , $k$ and $\beta$ ) are kept consistent with those used in the impermeable-wall case (figure 5) to enable direct comparisons. We systematically increased the $ \textit{Wi}$ from 0.08 to 59 and tracked the corresponding evolution of the eigenspectra. We compared the results with the Newtonian case ( $ \textit{Wi} = 0$ ) for a channel with porous walls ( $\alpha = 50$ ), as shown by the purple circles in figure 6. Figure 6(a) reveals that even at small $ \textit{Wi}$ values, such as $ \textit{Wi}=0.08$ , the eigenspectra exhibits significant deviations from the Newtonian case in a channel with porous walls. In particular, the structure of the spectrum differs noticeably from the viscoelastic flow with impermeable walls case, indicating a heightened sensitivity to viscoelastic effects when permeability is present. Compared with figure 5, the eigenspectrum exhibits a modified Y-shaped structure, along with two CS modes and a surrounding ring of discrete modes (hereafter referred to as ‘ring1’), features previously reported by Khalid et al. (Reference Khalid, Chaudhary, Garg, Shankar and Subramanian2021a ). However, the position of the Y-structure and the spatial extent of ‘ring1’ are both shifted in the porous-wall case, suggesting that permeability subtly alters the distribution and interaction of eigenmodes even at low $ \textit{Wi}$ .

Several key features arise in the eigenspectra due to the presence of porous walls. Most importantly, the modified TSM ( $1'$ ) becomes unstable (inset A in figure 6 a), and two new WMs (labelled as $3'$ and $4'$ ) appear alongside a vertically oriented porous mode ( $7'$ ), which is consistent with the mode previously identified by Tilton & Cortelezzi (Reference Tilton and Cortelezzi2008). Two notable differences from figure 5(a) include: first, the formation of a continuum balloon-like structure at the end of the CS-1 line, where it intersects the vertical porous mode line (inset B in figure 6 a); second, a secondary ring of discrete modes (hereafter ‘ring2’) emerges, enveloping the original ring structure (‘ring1’), the two CS branches and the vertical porous mode line. This arrangement creates a distinctive $\varphi$ -shaped spectral topology. In the Newtonian limit ( $ \textit{Wi} = 0$ ), the porous mode (mode 7) extends almost vertically towards lower $c_{i}$ values (figure 6, purple circles). When viscoelasticity is introduced at a small Weissenberg number ( $ \textit{Wi} = 0.08$ ), the corresponding porous mode ( $7'$ ) initially traces the Newtonian branch at higher $c_{i}$ , as seen in inset A of figure 6(a). However, at lower $c_{i}$ values, it deviates from the Newtonian path and contributes to the formation of ‘ring2.’ The lower portion of ‘ring2’ connects from the opposite side to the vertical line bridging CS1 and CS2, thereby confirming the existence of the vertical linkage between the two CS modes previously observed in figure 5. This finding underscores the influence of wall permeability, even in regimes where it might otherwise appear negligible.

With increasing $ \textit{Wi}$ , both CS modes and the associated ring structures are displaced upward in the complex phase velocity plane. Similar to the contraction observed in ‘ring1’, the outer ring structure (‘ring2’) also contracts with increasing $ \textit{Wi}$ (figure 6 b). While the overall $\varphi$ -shaped configuration of ‘ring2’ remains recognizable, its symmetry relative to ‘ring1’ is progressively lost. As ‘ring2’ collapses inward, it begins to merge with the vertical line connecting the CS1 and CS2 branches. In particular, in figure 6(b), the two modified WMs introduced by the porous walls ( $3'$ and $4'$ ) are observed to shift rightward in comparison with their Newtonian counterparts (modes 3 and 4). Simultaneously, the modified porous mode ( $7'$ ), initially distinct and located above ‘ring2’ in figure 6(a), gradually descends and merges into the upper portion of ’ring2.’ By the time $ \textit{Wi}$ reaches the value shown in figure 6(b), mode $7'$ is no longer distinguishable as a separate eigenmode. Nonetheless, its spectral signature persists in the lower section of ‘ring2’, suggesting the porous mode’s continued, albeit integrated, influence on the eigenspectrum dynamics.

Figure 6.

Eigenspectra for the channel with porous walls ( $\alpha = 50$ ) of an Oldroyd-B fluid at $ \textit{Re}=800$ , $k=1.5$ and $\beta = 0.8$ , for different $ \textit{Wi}$ numbers: (a) $ \textit{Wi}=0.08$ , (b) $ \textit{Wi}=0.4$ , (c) $ \textit{Wi}=0.8$ , (d) $ \textit{Wi}=2$ , (e) $ \textit{Wi}=4$ , (f) $ \textit{Wi}=10$ (g) $ \textit{Wi}=58$ , (h) $ \textit{Wi}=59$ . Here, $N$ and $N_{m}$ are considered $400$ .

As $ \textit{Wi}$ increases further, the ‘ring1’ structure collapses onto the two CS modes (figure 6 c), while ‘ring2’ shrinks until it only encircles the vertical connecting line between the two CS modes, maintaining its characteristic $\varphi$ shape. With a continued increase in $ \textit{Wi}$ , ‘ring2’ also collapses onto this vertical line. As the CS modes shift upward with increasing $ \textit{Wi}$ , the modes $3'$ and $4'$ merge into the CS modes and later reappear on the opposite side. At a higher value of $ \textit{Wi} = 2$ (figure 6 d), the reemerged $3'$ and $4'$ modes are positioned above the Newtonian counterparts (modes 3 and 4). At this stage, ‘ring2’ has completely collapsed, leaving only the vertical line of a disconnected porous mode connecting CS1 and CS2, along with a vertical line near zero phase speed ( $7'$ ), resembling the Newtonian case (7). Moreover, a continuum balloon shape forms at the end of the CS2 line, similar to the one observed for CS1 in inset B of figure 6(a). A few discrete modes appear above these small balloon structures but they remain stable within the examined parameter range. As $ \textit{Wi}$ increases further and the CS modes continue their upward shift, the NCMs and WMs begin merging into CS1 (figure 6 e).

A significant result is that, as $ \textit{Wi}$ increases more, the modified TSM ( $1'$ ), initially located above the $c_{i}=0$ line, moves downward as $ \textit{Wi}$ increases. At $ \textit{Wi}=10$ , it crosses below the $c_{i}=0$ line, rendering the flow stable (inset B in figure 6 f). Simultaneously, the ECM-1 mode emerges to the right of the CS1 mode (inset A in figure 6 f), consistent with previous findings reported by Khalid et al. (Reference Khalid, Chaudhary, Garg, Shankar and Subramanian2021a ). With a further increase in $ \textit{Wi}$ , at around $ \textit{Wi}=15$ , the modified TSM ( $1'$ ) merges into the CS1 mode (not shown here). When we increased $ \textit{Wi}$ to higher values, and at $ \textit{Wi}=58$ , we observed the emergence of a new mode from the left-hand branch above the CS1 mode, which we refer to as elasto-inertial WM 1 (EWM-1) (inset in figure 6 g). These modes, absent in Newtonian flows and viscoelastic flows with smooth walls, shift upward as $ \textit{Wi}$ increases. Notably, the flow remains stable between $ \textit{Wi}=10$ and $ \textit{Wi}=58$ . However, at $ \textit{Wi}=59$ , EWM-1 becomes unstable for the first time (inset B in figure 6 h). This significant result reveals that adding porous walls to a viscoelastic channel induces instability through a new WM (EWM-1), in contrast to viscoelastic flows in a channel with smooth walls, where instability arises from the ECM-1 mode from the right-hand branch, as reported by Khalid et al. (Reference Khalid, Chaudhary, Garg, Shankar and Subramanian2021a ).

Figures 7(a) and 7(b) illustrate the wall-normal velocity eigenfunction, $\hat {u}_y$ , for the least unstable mode identified in figure 6 at $ \textit{Wi}=0$ and $ \textit{Wi}=59$ , respectively. The unstable mode corresponds to the TSM ( $1$ ) for the Newtonian case ( $ \textit{Wi}=0$ ). As previously discussed by Tilton & Cortelezzi (Reference Tilton and Cortelezzi2008), the perturbation velocity amplitude is influenced by the porous walls, with its magnitude at the fluid–porous interface and monotonically decreasing within the porous layers until it vanishes at the solid walls. In contrast, at $ \textit{Wi}=59$ , the eigenfunction corresponds to the EWM-1, which exhibits a distinct structure characterized by sharp oscillations in the perturbation velocity near the fluid–porous interface.

Figure 8 presents the contour plots of the eigenfunctions for wall-normal velocity ( $\hat {u}_y$ ), streamwise velocity ( $\hat {u}_x$ ) and the streamwise component of total stress ( $\hat {\tau }_{xx}$ ) at these two elasticity levels. Specifically, figures 8(a), 8(c) and 8(e) correspond to $ \textit{Wi}=0$ , while figures 8(b), 8(d) and 8(f) correspond to $ \textit{Wi}=59$ . These contour plots emphasize the stark contrast between the two cases, particularly highlighting the pronounced oscillations near the interface for the EWM-1 mode compared with the smoother structure observed for the TSM.

Figures 9(a) and 9(b) illustrate the variation of the growth rate ( $c_i$ ) and phase speed ( $c_r$ ) with respect to $ \textit{Wi}$ for three distinct modes, the new EWM-1, the ECM-1 and the modifed TSM ( $1'$ ), at $ \textit{Re}=800$ , $k=1.5$ , $\beta =0.8$ and $\alpha =50$ . As shown in figure 9(a), in the near-Newtonian limit ( $ \textit{Wi} \rightarrow 0$ ), the modified TSM is the only unstable mode and it will be the dominant unstable mode until $ \textit{Wi} \approx 8$ . With a slight increase in $ \textit{Wi}$ ( $ \textit{Wi} \approx 8$ ), ECM-1 emerges from the CS1 mode and continues to grow in amplitude. The growth rate of modified TSM initially increases but then decreases until it stabilizes around $ \textit{Wi}=10$ . By approximately $ \textit{Wi}=16$ , the modified TSM completely disappears into the CS1 mode, as previously observed in figure 6(f). As $ \textit{Wi}$ increases further, EWM-1 emerges from CS1 at $ \textit{Wi} = 58$ and becomes unstable for the first time at approximately $ \textit{Wi} = 60$ , crossing the threshold $c_i = 0$ . With a further increase in $ \textit{Wi}$ , around $ \textit{Wi}=80$ , ECM-1 also destabilizes, consistent with the findings of Khalid et al. (Reference Khalid, Chaudhary, Garg, Shankar and Subramanian2021a ). The corresponding phase speeds of these modes, shown in figure 9(b), reveal a slight increase in the phase speed of modified TSM, while ECM-1 remains nearly constant at values close to unity. In contrast, the phase speed of EWM-1 remains steady at $c_r = 0.16$ across the entire range of $ \textit{Wi}$ considered. Note that the dominant unstable mode depends on the Reynolds number value. We found that for lower Reynolds numbers and specific viscosity ratio, the new EWM-1 will be the dominant mode (Mirbod & Taheri Reference Mirbod and Taheri2025).

Figure 7.

Wall normal velocity eigenfunction, $|\hat {u}_y|$ (red line), $\hat {u}_{y_i}$ (blue line) and $\hat {u}_{y_r}$ (green line) corresponding to the least unstable eigenmodes: (a) TSM at $ \textit{Wi}=0$ ( $c$ = $0.419537$ + $0.007281$ i) and (b) the new EWM (EWM-1) at $ \textit{Wi}=59$ ( $c$ = $0.160793$ + $0.000581$ i). The parameters used are $ \textit{Re}=800$ , $k=1.5$ , $\beta =0.8$ and $\alpha =50$ .

Figure 8.

Contour plots of leading mode for (a,b) wall-normal velocity $\hat {u}_y$ , (c,d) streamwise velocity $\hat {u}_x$ and (e,f) streamwise component of the total stress $\hat {\tau }_{xx}$ . Results are shown for the TSM at $ \textit{Wi}=0$ (a,c,e) and EWM (EWM-1) at $ \textit{Wi}=59$ (b,d,f). The parameters used are $ \textit{Re}=800$ , $k=1.5$ , $\beta = 0.8$ and $\alpha =50$ .

Figure 9.

Relative stability of the first three least stable eigenmodes, EWM-1, ECM-1 and TSM, at $ \textit{Re}=800$ , $k=1.5$ , $\beta =0.8$ and $\alpha =50$ . (a) Variation of growth rate ( $c_i$ ) with $ \textit{Wi}$ . (b) Variation of phase speed ( $c_r$ ) with $ \textit{Wi}$ .

Figure 10.

(a) Variation of the growth rate ( $c_i$ ) with respect to $\beta$ for EWM-1. (b) Corresponding variation of $c_r$ with $\beta$ . Results are shown for $ \textit{Wi}=59$ , $ \textit{Re}=800$ , $k=1.5$ and $\alpha =50$ .

To further investigate the effect of the viscosity ratio ( $\beta$ ) on the new EWM (EWM-1), we varied $\beta$ from 0.1 to 0.8 while keeping $ \textit{Wi} = 59$ , $ \textit{Re} = 800$ , $k = 1.5$ and $\alpha = 50$ fixed. Figure 10(a) shows the growth rate ( $c_i$ ) as a function of $\beta$ . Since EWM-1 emerges at high $ \textit{Wi}$ , the location of CS1 (given by $-1/(kWi)$ ) is very close to $c_i = 0$ . As $\beta$ decreases from the Newtonian limit ( $\beta \to 1$ ), EWM-1 emerges from CS1 and rapidly becomes unstable. Further decreasing $\beta$ towards the Maxwell limit ( $\beta \to 0$ ) leads to a further increase in the growth rate ( $c_i$ ) of EWM-1, indicating enhanced instability. In addition, figure 10(b) illustrates the variation of the phase speed ( $c_r$ ) with $\beta$ , showing a clear increasing trend in $c_r$ as $\beta$ decreases, suggesting that EWM-1 propagates faster in more elastic-dominated flows.

We then analyse the effect of increasing permeability (by decreasing $\alpha$ from $300$ to $50$ ) on the eigenspectra while keeping $ \textit{Wi}$ constant and setting $\beta = 0.5$ at a high Reynolds number ( $ \textit{Re}$ ). We selected these values of $ \textit{Re}$ and $k$ to align with one of the figures in Sureshkumar & Beris (Reference Sureshkumar and Beris1995), enabling a direct comparison with their results. Figure 11(a–d) show the eigenspectra for $ \textit{Wi} = 0.5$ , $2.31$ , $5$ and $25$ , respectively, with $ \textit{Re} = 2310$ and $k = 1.31$ . Consistent with our earlier findings (see figure 6), the eigenspectra in these figures display a vertical line corresponding to the porous modes, which have a phase speed close to zero. Also, they feature a ring-shaped structure, ‘ring2’, which arises due to the effect of porous walls in viscoelastic fluids. A comparison of these figures reveals that as permeability increases (i.e. $\alpha$ decreases), ‘ring2’ expands in size. However, for all values of $\alpha$ , the size of ‘ring2’ decreases as $ \textit{Wi}$ increases. At $ \textit{Wi} = 25$ , most of the ring structure collapses into the vertical porous mode line, indicating a significant structural transformation in the eigenspectrum. Furthermore, features associated with the Oldroyd-B fluid, including CS1, CS2 and the gradually collapsing ‘ring1’, are visible across all cases. The insets within the figures provide a magnified view of two modified WMs (labelled $1'$ and $2'$ ), highlighting their evolution with permeability. It should be noted that modified WM $1'$ is the modified TSM that we referred to in previous figures. As reported by Tilton & Cortelezzi (Reference Tilton and Cortelezzi2008), in a channel with two porous walls, permeability can destabilize up to two Orr–Sommerfeld WMs.

For comparison with previous studies, figure 11(b) presents the eigenspectra for a viscoelastic fluid at $ \textit{Wi}=2.31$ , a case previously examined by Sureshkumar & Beris (Reference Sureshkumar and Beris1995) for a smooth-walled channel. In the presence of porous walls, we observe a distinct impact of permeability on the eigenspectrum. For higher values of $\alpha$ ( $\alpha =300$ , $200$ and $150$ ), the characteristic ring structure (‘ring2’) is absent. However, as permeability increases ( $\alpha$ decreases), the formation of the ring structure begins at $\alpha =125$ . With further increases in permeability, the ring structure becomes more pronounced and expands in size. In addition, permeability significantly influences the two modified WMs (labelled $1'$ and $2'$ ) within the Y-shaped structure. Flow destabilization is first observed at $\alpha =200$ , where the modified WM $1'$ (modified TSM) becomes unstable. As permeability continues to increase ( $\alpha$ decreases from 200 to 75), the modified WM $1'$ remains unstable. At $\alpha =50$ , both modified WMs $1'$ and $2'$ become unstable, as highlighted in the figure’s inset.

To examine the variation of the modified WMs shown in figures 11, 12(a) and 12(b) illustrates the growth rate $c_{i}$ as a function of $\alpha$ for modes $1'$ (modified TSM) and $2'$ , respectively. The non-monotonic behaviour of $ \textit{Wi}$ , previously observed by Sureshkumar & Beris (Reference Sureshkumar and Beris1995), is evident in both WMs across all values of $\alpha$ . This effect is particularly pronounced for larger values of $\alpha$ (specifically $\alpha = 75$ and above) in WM $1'$ (figure 12 a). As $ \textit{Wi}$ increases from 5 to 25, the growth rate $c_{i}$ decreases and eventually stabilizes for $\alpha$ values greater than $100$ when $ \textit{Wi} = 25$ . It is important to note that WMs $1'$ and $2'$ merge with CS1 at $\alpha = 150$ and $\alpha = 75$ , respectively.

Figure 11.

Eigenspectra of an Oldroyd-B fluid in a channel with porous walls at $ \textit{Re}=2310$ , $k=1.31$ and $\beta = 0.5$ for different permeability parameter ( $\alpha$ ) values: (a) $ \textit{Wi}=0.5$ , (b) $ \textit{Wi}=2.31$ , (c) $ \textit{Wi}=5$ and (d) $ \textit{Wi}=25$ .

Figure 12.

Relative stability of the elastically modified WMs (shown in 11) for permeability parameter ( $\alpha$ ) and different $ \textit{Wi}$ at $ \textit{Re}=2310$ , $k=1.31$ and $\beta =0.5$ : (a) modified WM $1'$ , (b) modified WM $2'$ .

3.3. Marginal stability curves and the effect of varying $ \textit{Wi}$ and $\alpha$ at fixed $\beta$

We examine how viscoelasticity and wall permeability influence the critical Reynolds number $ \textit{Re}_{\textit{cr}}$ , critical wavenumber $k_{\textit{cr}}$ and the nature of the dominant instability modes in a fluid–porous channel system. Figures 13(a) and 13(b) show the neutral stability curves for Newtonian flows ( $ \textit{Wi}=0$ ) and viscoelastic Oldroyd-B fluids with increasing values of $ \textit{Wi}$ from 0.1 to 10, for a permeability parameter $\alpha = 50$ for two representative viscosity ratios, $\beta = 0.5$ and $\beta = 0.8$ , respectively. The neutral curves reveal the transition boundaries separating stable and unstable flow regimes, plotted in terms of Reynolds number and wavenumber.

Throughout the range of $ \textit{Wi}$ , the dominant instability corresponds to a modified TSM. For this configuration ( $\delta$ = 1), the neutral curves exhibit a characteristic bimodal structure, with two distinct lobes: a long-wave branch (left) and a short-wave branch (right). This bimodal behaviour is consistent with earlier observations in porous channels with moderate to high permeability (see, e.g. Chang et al. Reference Chang, Chen and Straughan2006; Camporeale, Mantelli & Manes Reference Camporeale, Mantelli and Manes2013; Wu & Mirbod Reference Wu and Mirbod2019; Samanta Reference Samanta2020). The long-wave branch, known as the porous mode, is associated with instabilities located within the porous layer, whereas the short-wave branch, called the fluid mode, is driven primarily by flow dynamics in the fluid domain. In all cases examined at $\alpha = 50$ , the fluid mode instability spectrum dominates, regardless of $ \textit{Wi}$ or $\beta$ . This indicates that within this moderate range of elasticity, viscoelastic effects reinforce instabilities already present in the Newtonian case, without introducing new dominant modes. We then explore the regime of significantly higher elasticity to investigate whether qualitatively different instability mechanisms emerge beyond the TS-like observed here. Figures 13(a) and 13(b) show how increasing elasticity affects the transition between instability modes for two different viscosity ratio, i.e. $\beta =0.5$ and $\beta =0.8$ , respectively. For $\beta =0.5$ , increasing $ \textit{Wi}$ shifts the transition point between the porous and fluid modes towards higher wavenumbers ( $k$ ). This shift significantly attenuated for $\beta =0.8$ , indicating that a higher solvent viscosity fraction suppresses the influence of elasticity on the modal structure. For both viscosity ratios, the critical Reynolds number ( $ \textit{Re}_{\textit{cr}}$ ), defined as the minimum value of Re on the neutral curve, exhibits a non-monotonic dependence on $ \textit{Wi}$ . Initially, increasing $ \textit{Wi}$ destabilizes the flow, leading to a decrease in $ \textit{Re}_{\textit{cr}}$ . Beyond a certain threshold, however, further increases in $ \textit{Wi}$ lead to flow stabilization and a rise in $ \textit{Re}_{\textit{cr}}$ . This non-monotonic behaviour is consistent with previous findings in viscoelastic Poiseuille flows with impermeable boundaries (Sureshkumar & Beris Reference Sureshkumar and Beris1995; Sadanandan & Sureshkumar Reference Sadanandan and Sureshkumar2002; Khalid et al. Reference Khalid, Chaudhary, Garg, Shankar and Subramanian2021a ).

Figure 13.

Marginal stability curves in the Re–k plane for $\alpha =50$ and different $ \textit{Wi}$ for (a) $\beta =0.5$ , and (b) $\beta =0.8$ .

Figure 14.

(a) Marginal stability curves in the Re–k plane for $\alpha =50$ , $\beta =0.8$ and high $ \textit{Wi}$ values, (b) Eigenspectra for $ \textit{Wi} = 30$ , $k = 1.5$ and $ \textit{Re} = 1500$ and $100$ shows the instability from modified TSM and EWM, respectively. (c,d) Wall normal velocity eigenfunction for $ \textit{Wi} = 30$ , $k = 1.5$ , $|\hat {u}_y|$ (red line), $\hat {u}_{y_i}$ (blue line) and $\hat {u}_{y_r}$ (green line) corresponding to the least unstable eigenmodes: (c) modified TSM (at $ \textit{Re} = 1500$ ), (d) EWM (at $ \textit{Re} = 100$ ).

As the elasticity of the fluid increases, the structure of the neutral stability curve becomes increasingly complex, revealing new features that are absent in the lower $ \textit{Wi}$ . Figure 14(a) superimposes the neutral curves for $ \textit{Wi} = 20$ and $ \textit{Wi} = 30$ on those previously presented for lower values of $ \textit{Wi}$ ( $0 \leqslant Wi \leqslant 10$ ) in figure 13(b). Although the general two-lobe structure associated with the upper branch, comprising fluid and porous instability modes, persists at higher $ \textit{Wi}$ , a new feature emerges: an additional, lower branch of instability appears at significantly reduced Reynolds numbers over a narrow wavenumber interval. This newly formed branch is not present at lower $ \textit{Wi}$ and signifies the onset of a fundamentally different instability mechanism. As the lower branch appears at $ \textit{Re} \to 0$ indicates the presence of an instability in the inertialess limit. This behaviour is known as purely elastic instability, often associated with the onset of elastic turbulence, where the flow becomes unstable at negligible inertia due to strong elastic stresses Khalid et al. (Reference Khalid, Shankar and Subramanian2021b ). For $ \textit{Wi} = 20$ , the lower branch appears as a modest extension below the main neutral curve, confined to a narrow range of wavenumbers. However, at $ \textit{Wi} = 30$ , the lower branch expands markedly, both in Reynolds number and in wavenumber extent. It initiates at a smaller $k$ and persists over a broader interval compared with its counterpart at $ \textit{Wi} = 20$ , indicating that the associated instability is amplified with increasing elasticity. This behaviour suggests that the system is susceptible to a qualitatively new mode of elastic instability that is activated only beyond a certain threshold in elasticity.

Closer inspection of the neutral curves confirms that this additional lower branch corresponds to an elastic instability that behaves similarly to a distinct wall-localized family of modes, which we identified as EWM-1, associated with elasto-inertial instability. This mode dominates the flow instability landscape at low Reynolds numbers in the high- $ \textit{Wi}$ regime. The eigenvalue spectrum and the structure of the corresponding eigenfunctions, presented in later sections, confirm that here also EWM is characterized by strong localization near the fluid–porous interface, with perturbations concentrated in regions adjacent to the walls. Its spatial confinement and sharp gradients distinguish it from the modified TSM that governs the upper branch. Importantly, the EWM mode governs the lower branch only within the low- $ \textit{Re}$ regime which corresponds to elastic instability. As $ \textit{Re}$ increases, the flow transitions back to being dominated by the TS-type instability along the upper branch.

The persistence of the upper branch structure, even at high $ \textit{Wi}$ , highlights the continuing influence of viscoelastic modifications to the TSM. In particular, the non-monotonic relationship between critical Reynolds number ( $ \textit{Re}_{\textit{cr}}$ ) and elasticity ( $ \textit{Wi}$ ), previously observed at lower $ \textit{Wi}$ for the upper branch, remains present in this high-elasticity regime. This trend, wherein $ \textit{Re}_{\textit{cr}}$ initially decreases with increasing $ \textit{Wi}$ before rising again, indicates that the modified TS instability is sensitive to both elastic and viscous effects, and its stability properties are shaped by a delicate balance between these competing influences. A magnified view of the transition between the first and second lobes of the upper branch is shown in the inset of figure 14(a), revealing subtle shifts in the neutral curve topology that accompany increasing $ \textit{Wi}$ .

To elucidate the distinct nature of the two instability regions observed in the neutral stability diagram at high elasticity, we examine the eigenspectra and associated eigenfunctions corresponding to the lower and upper branches at $ \textit{Wi} = 30$ . Figure 14(b) presents the eigenvalue spectra for fixed wavenumber $k = 1.5$ at two representative Reynolds numbers: $ \textit{Re} = 100$ , which lies below the lower branch, and $ \textit{Re} = 1500$ , which lies above the upper branch. These values are chosen to illustrate the dominant modes responsible for each of the two instability branches. At the lower Reynolds number, $ \textit{Re} = 100$ , the dominant eigenvalue is associated with EWM-1, a mode that emerges in the high- $ \textit{Wi}$ regime and is localized near the fluid–porous interface. This eigenvalue resides closest to the imaginary axis, indicating that EWM-1 is the least stable (and therefore most influential) mode in this parameter regime. It is also worth noting that in the limit $ \textit{Re} \to 0$ , we observe the same instability discussed earlier, indicating that this branch persists as an elastic instability in the inertialess regime. In contrast, at the higher Reynolds number, $ \textit{Re} = 1500$ , the leading eigenvalue corresponds to a viscoelastic modification of the classical TSM. This mode governs the upper branch and is distributed more broadly across the channel, consistent with its inertial origin. The corresponding wall-normal velocity eigenfunctions, $\hat {u}_y$ , are plotted in figures 14(c) and 14(d) for the dominant modes at $ \textit{Re} = 100$ and $ \textit{Re} = 1500$ , respectively. The spatial structure of these eigenfunctions reveals striking differences. For the EWM-1 mode (figure 14 c), the perturbation is sharply localized near the walls, with amplitude rapidly decaying away from the interface, indicating strong coupling between elasticity and wall permeability. This spatial confinement is characteristic of WMs and aligns with the structural features described earlier in figure 7. In contrast, the eigenfunction associated with the TS-like mode at $ \textit{Re} = 1500$ (figure 14 d) exhibits a more extended profile, with perturbation energy distributed across the channel height, reflecting the broader influence of inertia and its less localized nature.

Figure 15(a) illustrates the variation of the $ \textit{Re}_{\textit{cr}}$ with $ \textit{Wi}$ over the range of 0.1–10 for three viscosity ratios: $\beta$ = 0.5, 0.8, 0.95, in a channel with porous walls characterized by $\alpha =50$ . A distinct non-monotonic trend is observed for $\beta =0.5$ and $0.8$ : as $ \textit{Wi}$ increases, $ \textit{Re}_{\textit{cr}}$ initially decreases, indicating destabilization, before increasing again at higher $ \textit{Wi}$ , reflecting a stabilizing effect. This behaviour becomes progressively less pronounced with increasing $\beta$ . For $\beta = 0.95$ , the non-uniform trend vanishes entirely, and $ \textit{Re}_{\textit{cr}}$ gradually increases with increasing $ \textit{Wi}$ , approaching a Newtonian-like trend as $\beta \rightarrow 1$ , where the influence of elasticity is negligible. Another notable feature is the presence of a threshold $ \textit{Wi}$ for each $\beta$ , beyond which the stabilizing effect of elasticity dominates. Initially, increasing $ \textit{Wi}$ destabilizes the flow, causing $ \textit{Re}_{\textit{cr}}$ to decrease. However, $ \textit{Re}_{\textit{cr}}$ starts increasing beyond this threshold. This threshold shifts to lower $ \textit{Wi}$ values as $\beta$ increases. Simultaneously, the minimum value of $ \textit{Re}_{\textit{cr}}$ rises (i.e. the range of $ \textit{Wi}$ where $ \textit{Re}_{\textit{cr}}$ decreases) narrows with increasing $\beta$ . These trends suggest that a higher solvent viscosity fraction suppresses the wall-mode-driven instability induced by porous walls. This behaviour contrasts with the findings of Khalid et al. (Reference Khalid, Chaudhary, Garg, Shankar and Subramanian2021a ) for plane Poiseuille flow with smooth, impermeable walls, where at high elasticity values a centre-mode instability was shown to intensify with increasing $\beta$ , lowering $ \textit{Re}_{\textit{cr}}$ , and expanding the unstable region. In the present case, the presence of permeable walls changes the dominant instability mechanism. Wall permeability amplifies the destabilizing influence of elasticity, while an increase in $\beta$ (i.e. greater solvent contribution) counteracts this effect and stabilizes the flow. Consequently, for wall-mode instabilities in porous channels, increasing $\beta$ shifts the minimum of the $ \textit{Re}_{\textit{cr}}-Wi$ towards higher $ \textit{Re}_{\textit{cr}}$ , unlike the behaviour seen in centre-mode-dominated flows. It is worth noting that this trend is consistent with the findings of Sureshkumar & Beris (Reference Sureshkumar and Beris1995), who showed that in channels with smooth walls, the WM instability remains dominant at low to moderate elasticity, and $ \textit{Re}_{\textit{cr}}$ increases with increasing $\beta$ . The variation of the critical wavenumber ( $k_{\textit{cr}}$ ) with $ \textit{Wi}$ is depicted in figure 15(b). Like $ \textit{Re}_{\textit{cr}}$ , $k_{\textit{cr}}$ exhibits a curve with a well-defined peak aligning closely with the threshold $ \textit{Wi}$ identified in figure 15(a). For all $ \textit{Wi}$ , increasing $\beta$ leads to a decrease in $k_{\textit{cr}}$ , indicating that higher solvent viscosity dampens shorter-wavelength (i.e. smaller-scale) instabilities.

Figure 15.

Critical parameters as a function of $ \textit{Wi}$ for $\alpha = 50$ and different viscosity ratio ( $\beta$ ): (a) critical Reynolds number $ \textit{Re}_{\textit{cr}}$ , and (b) critical wavenumber $k_{\textit{cr}}$ .

The effect of wall permeability on the critical parameters is examined in figure 16. Figure 16(a) shows the variation of $ \textit{Re}_{\textit{cr}}$ with $ \textit{Wi}$ for $\beta =0.5$ and three values of the permeability parameter: $\alpha = 500$ , $100$ and $50$ . As permeability increases (i.e. $\alpha$ decreases from $500$ to $50$ ), $ \textit{Re}_{\textit{cr}}$ decreases significantly across the range of $ \textit{Wi}$ , demonstrating the strong destabilizing effect of porous walls. This highlights the role of wall permeability in enhancing instability. Moreover, for the most permeable case ( $\alpha = 50$ ), the threshold $ \textit{Wi}$ defined as the value at which $ \textit{Re}_{\textit{cr}}$ reaches the minimum, changes to lower values. This suggests that elastic effects become significant earlier in highly permeable channels. The corresponding neutral stability curves in the $ \textit{Re}_{\textit{cr}}-Wi$ plane further highlight how permeability influences the stabilizing regime (i.e. where $ \textit{Re}_{\textit{cr}}$ increases with $ \textit{Wi}$ ). In particular, the slope of the stabilizing branch is steeper for $\alpha = 50$ than for $\alpha = 100$ and $500$ , indicating that in more permeable channels, elasticity more effectively restabilizes the flow after initial destabilization. Figure 16(b) presents the corresponding variation of the critical wavenumber $k_{\textit{cr}}$ with $ \textit{Wi}$ . In contrast to the behaviour of $ \textit{Re}_{\textit{cr}}$ , increasing permeability (decreasing $\alpha$ ) leads to an increase in $k_{\textit{cr}}$ , indicating that more permeable walls favour the onset of shorter-wavelength instabilities. This trend underscores the role of permeability in altering the characteristic scales of the dominant unstable modes in viscoelastic flows. In the study by Khalid et al. (Reference Khalid, Chaudhary, Garg, Shankar and Subramanian2021a ), which focused on channels with smooth, impermeable walls, the instability was primarily attributed to the centre mode for the range of $\beta$ and $ \textit{Wi}$ they considered. By contrast, in the present study, even for the least permeable case ( $\alpha =500$ ), the dominant instability corresponds to a modified WM. While $\alpha =500$ represents a near-impermeable limit, comparisons between our results in figure 16(a) and those of Khalid et al. (Reference Khalid, Chaudhary, Garg, Shankar and Subramanian2021a ) show that the left-hand branch of the $ \textit{Re}_{\textit{cr}}-Wi$ curve approaches the Newtonian (Poiseuille) limit in both cases. It is important to note that Khalid et al. (Reference Khalid, Chaudhary, Garg, Shankar and Subramanian2021a ) reported critical Reynolds number ( $ \textit{Re}_{\textit{cr}}$ ) in terms of $E(1-\beta )$ , with their minimum $\beta$ values being 0.65. Moreover, their analysis covered much higher values of the elasticity number $E$ than the $ \textit{Wi}$ range considered in the current study. These differences in parameter regimes and wall properties further underscore the significant effect of wall permeability on the nature and structure of the instability.

Figure 16.

The critical parameters versus $ \textit{Wi}$ for $\beta =0.5$ and different permeability parameters ( $\alpha$ ): (a) critical Reynolds number, $ \textit{Re}_{\textit{cr}}$ , (b) critical wavenumber, $k_{\textit{cr}}$ .

Figure 17 explores the impact of a wider range of permeability parameters on flow stability. In figure 17(a), the eigenspectra are shown for $\alpha = 50, 100, 200, 300, 400$ and $500$ at fixed $ \textit{Wi} = 10$ , $\beta = 0.5$ , $ \textit{Re} = 2310$ and $k = 1.31$ , matching the Reynolds number and wavenumber used in figure 11 for direct comparison. These spectra reveal that for higher permeability (i.e. smaller $\alpha$ ), a distinct ring-like structure (referred to as ‘ring2’) associated with porous wall effects emerges. At $\alpha = 50$ , both modified WMs ( $1'$ and $2'$ ) become unstable, and ‘ring2’ structure fully develops, displaying a vertical alignment reminiscent of a $\varphi$ -shape. The instability in this regime originates from the modified TSM in the viscoelastic flow. As permeability decreases (i.e. $\alpha$ increases), the WMs ( $1'$ and $2'$ ) shift downward in the eigenspectrum and stabilize.

Figure 17(b) illustrates the wall-normal velocity eigenfunction magnitude, $|\hat {u}_y|$ , for the dominant mode in each spectrum, normalized by the peak streamwise velocity perturbation for comparison across cases. In all cases, the dominant mode corresponds to the modified TSM $1'$ . The wall-normal perturbation decays monotonically into the porous region, but its magnitude at the fluid–porous interface increases as permeability increases (i.e. as $\alpha$ decreases). This leads to larger centreline amplitudes, indicating that the destabilizing effect of permeability arises primarily from the increased wall-normal perturbation at the interface. This trend is most pronounced for $\alpha = 100$ and $50$ , consistent with the simultaneous instability of both WMs ( $1'$ and $2'$ ) observed in figure 17(a).

Figure 17.

Effect of permeability parameter $\alpha$ on the stability of an Oldroyd-B fluid with $\beta =0.5$ : (a) eigenspectra when $ \textit{Re}=2310$ , and $k=1.31$ , (b) wall normal velocity eigenfunction, $|\hat {u}_y|$ , (c) marginal stability curves for $ \textit{Wi}=10$ at different $\alpha$ values, (d) critical Reynolds number ( $ \textit{Re}_{\textit{cr}}$ ) versus $\alpha$ for both Newtonian and Oldroyd-B fluids, with results shown for $ \textit{Wi}=2$ , $4$ and $10$ .

The effect of permeability on the neutral stability curves is shown in figure 17(c) for $\alpha$ ranging from $50$ to $500$ at $\beta =0.5$ and $ \textit{Wi}=10$ . As permeability increases (i.e. $\alpha$ decreases), the unstable regions expand and flatten, leading to a broader range of unstable wavenumbers ( $k$ ) and a reduced in $ \textit{Re}_{\textit{cr}}$ . Notably, for $\alpha = 50$ , the neutral curve exhibits a bimodal structure, reflecting the emergence of two distinct instability branches. The right-hand lobe corresponds to the fluid mode, while the left-hand lobe, induced by high permeability, represents the porous mode. This dual-branch structure signifies the growing influence of wall permeability in shaping the instability landscape, even though the instability still originates from the fluid side.

Figure 17(d) presents $ \textit{Re}_{\textit{cr}}$ with the permeability parameter ( $\alpha$ ) for several values of $ \textit{Wi} = 0$ (Newtonian), 2, 4 and 10, with $\beta =0.5$ . As $\alpha$ increases (i.e. permeability decreases), $ \textit{Re}_{\textit{cr}}$ rises monotonically across all cases, indicating increasing flow stability. Comparing the viscoelastic cases with the Newtonian case reveals that for $\alpha$ $\gt$ $100$ , increasing $ \textit{Wi}$ from $0$ to $4$ destabilizes the flow, resulting in a decrease in $ \textit{Re}_{\textit{cr}}$ . However, for higher elasticity ( $ \textit{Wi} = 10$ ), the trend reverses; elasticity now exerts a stabilizing effect. Interestingly, for highly permeable walls ( $\alpha$ $\lt$ $100$ ), the critical Reynolds number for $ \textit{Wi}=10$ becomes higher than that of the Newtonian case, suggesting a transition in the dominant instability mechanism. This transition is highlighted in the zoomed-in region of figure 17(d), which shows that at high $ \textit{Wi}$ , walls with strong permeability (e.g. $\alpha = 50$ ) actually stabilize the flow, contrary to the behaviour seen at lower $ \textit{Wi}$ . Finally, for the Newtonian case ( $ \textit{Wi}=0$ ), as permeability decreases (i.e. as $\alpha$ $\rightarrow$ $\infty$ ), $ \textit{Re}_{\textit{cr}}$ approaches the classical Poiseuille value of 5772, consistent with the threshold for plane Poiseuille flow with smooth walls (Schmid et al. Reference Schmid, Henningson and Jankowski2002). It is worth noting that Tilton & Cortelezzi (Reference Tilton and Cortelezzi2008) studied channels with two porous walls using a much higher permeability ( $\alpha =5000$ ), where the critical Reynolds number $ \textit{Re}_{\textit{cr}}$ differed by only $0.3\,\%$ from the classical Poiseuille value. At lower permeability values ( $\alpha =50$ and 100), our computed $ \textit{Re}_{\textit{cr}}$ values in figure 17(d) are slightly higher than those in Tilton & Cortelezzi (Reference Tilton and Cortelezzi2008), primarily due to differences in porous layer thickness: they used a depth ratio $\delta =2$ , whereas we use $\delta =1$ . For example, at $\alpha =50$ , they reported $ \textit{Re}_{\textit{cr}}=624$ , while our result is 664.

3.4. Neutral stability curves and the effect of varying $\beta$ at fixed $ \textit{Wi}$ and $\alpha$

In the Oldroyd-B model, the viscosity ratio $\beta$ ranges from $0$ to $1$ , with $\beta = 0$ corresponding to a UCM fluid and $\beta = 1$ representing the Newtonian limit. Figure 18 shows the variation of the eigenspectra for different values of $\beta$ for two wall conditions: (a) a nearly smooth channel ( $\alpha = 500$ ) and (b) a highly permeable channel with porous walls ( $\alpha = 50$ ), under fixed parameters $ \textit{Wi} = 0.2$ , $ \textit{Re} = 800$ and $k = 1.5$ .

In the smooth/impermeable channel (figure 18 a), a viscoelastic ring structure, referred to ‘ring1’, is evident in the eigenspectrum. As $\beta$ increases from 0.5 to 0.95, the size of ‘ring1’ progressively diminishes, in agreement with earlier observations by Khalid et al. (Reference Khalid, Chaudhary, Garg, Shankar and Subramanian2021a ). Notably, as $\beta$ increases from the UCM limit ( $\beta = 0$ ), the previously dominant high-frequency generalized least stable branch gradually curves into a closed loop, giving rise to the formation of ‘ring1’, consistent with the findings of Chaudhary et al. (Reference Chaudhary, Garg, Shankar and Subramanian2019) and Khalid et al. (Reference Khalid, Chaudhary, Garg, Shankar and Subramanian2021a ). When porous walls are introduced ( $\alpha = 50$ ), the eigenspectrum undergoes a significant transformation (figure 18 b). In addition to the inner viscoelastic ring (‘ring1’), a larger outer ring structure, called ‘ring2’, emerges, forming a distinctive $\varphi$ -shaped feature. Similar to ‘ring1’, the size of ‘ring2’ decreases with increasing $\beta$ , gradually vanishing as the fluid approaches the Newtonian regime. Furthermore, a separate porous mode branch appears near $c_r \approx 0$ , which was absent in the smooth-wall case, underscoring the coupling between wall-normal perturbations and the dynamics of the porous boundary.

Figure 18.

Eigenspectra of an Oldroyd-B fluid in a channel with porous walls at $ \textit{Wi}=0.2$ , $ \textit{Re}=800$ and $k=1.5$ at different values of $\beta$ : (a) $\alpha = 500$ , (b) $\alpha = 50$ .

Figure 19.

Neutral stability curves illustrating the effect viscosity ratio $\beta$ on the stability of an Oldroyd-B fluid with $ \textit{Wi}=5$ , in a channel with porous walls for different permeability parameters: (a) $\alpha = 500$ , and (b) $\alpha = 50$ .

Figures 19(a) and 19(b) shows the marginal stability curves at a fixed $ \textit{Wi}=5$ for various values of $\beta$ , ranging from $0.2$ to $1$ , in channels with different permeability levels. Specifically, figure 19(a) corresponds to a nearly impermeable channel ( $\alpha = 500$ ), while figure 19(b) depicts results for a highly permeable configuration ( $\alpha = 50$ ). Here, $\alpha$ is inversely proportional to the intrinsic permeability $\kappa$ , with lower values of $\alpha$ indicating higher wall permeability. For the low-permeability case (figure 19 a), decreasing $\beta$ from its Newtonian limit ( $\beta = 1$ ) leads to a systematic expansion of the unstable region in the $(Re, k)$ plane, along with a reduction in the critical Reynolds number $ \textit{Re}_{\textit{cr}}$ . This trend signifies enhanced instability as the fluid becomes more strongly viscoelastic, consistent with the known destabilizing influence of elasticity in confined shear flows. In contrast, the behaviour is more nuanced in the high-permeability case (figure 19 b). When $\beta$ is reduced below approximately 0.8, $ \textit{Re}_{\textit{cr}}$ continues to decrease, indicating increased instability. However, for $\beta \gt 0.8$ , the marginal curves begin to converge, and the rate of change in $ \textit{Re}_{\textit{cr}}$ with $\beta$ diminishes, deviating from the trend observed at lower $\beta$ values. This non-monotonic behaviour suggests a more intricate interplay between viscoelastic effects and wall permeability.

Moreover, in the highly permeable case, the neutral stability curves exhibit a two-lobed structure corresponding to distinct instability mechanisms: a left-hand lobe associated with the porous mode and a right-hand lobe linked to the fluid mode. As $\beta$ decreases from the Newtonian limit, the transition between these modes shifts to higher wavenumbers. This shift indicates that increasing viscoelasticity modifies the relative prominence of the two instability branches, with the porous mode becoming more dominant at intermediate-to-high wavenumbers before eventually yielding to the fluid mode.

Further information on the impact of the viscosity ratio $\beta$ on flow stability is provided in figure 20, which shows the variation of the critical Reynolds number $ \textit{Re}_{\textit{cr}}$ as a function of $\beta$ for two levels of wall permeability: $\alpha = 500$ (figure 20 a) and $\alpha = 50$ (figure 20 b) when $ \textit{Wi} = 5$ and $10$ . In the highly permeable case ( $\alpha = 50$ ), the relationship between $ \textit{Re}_{\textit{cr}}$ and $\beta$ is distinctly non-monotonic, and this behaviour becomes more pronounced with increasing elasticity (i.e. higher $ \textit{Wi}$ ). At $ \textit{Wi} = 10$ , the stability curve assumes a parabolic form: the flow initially stabilizes as $\beta$ increases from low values, reaching a maximum $ \textit{Re}_{\textit{cr}}$ near $\beta \approx 0.5$ , beyond which further increases in $\beta$ reduce $ \textit{Re}_{\textit{cr}}$ , indicating a re-emergence of instability. This reversal reflects a shift in the dominant instability mechanism due to the competing influences of elasticity and solvent viscosity. At a lower Weissenberg number ( $ \textit{Wi} = 5$ ), a similar trend is observed, but the turning point occurs at a higher value of $\beta$ (around $\beta \approx 0.9$ ), suggesting that stronger elastic effects enhance the non-monotonicity of the stability response.

For comparison, the nearly impermeable configuration ( $\alpha = 500$ ; figure 20 a) exhibits a more regular behaviour. At $ \textit{Wi} = 5$ , $ \textit{Re}_{\textit{cr}}$ increases monotonically with $\beta$ , indicating a consistent stabilizing influence of increasing solvent viscosity. However, for $ \textit{Wi} = 10$ , this trend weakens, and beyond $\beta \approx 0.8$ , the curve begins to decline, suggesting a modest destabilization at high $\beta$ under strong elastic forcing. These results demonstrate the significant role of wall permeability in modulating the effect of $\beta$ on flow stability. The presence of a porous boundary amplifies the non-monotonic response of $ \textit{Re}_{\textit{cr}}$ to variations in $\beta$ , particularly at higher $ \textit{Wi}$ . This behaviour underscores the intricate interplay between viscoelasticity, solvent-to-total viscosity ratio and wall permeability in determining the onset of instability in viscoelastic channel flows. Understanding this coupling is essential for predicting and controlling flow transitions in porous-walled systems.

3.5. Comparison with experiments

To evaluate the predictive capability of our theoretical model, we compare its results with two limiting cases from experimental studies: (i) Newtonian planar Poiseuille flow in a channel with porous walls and (ii) viscoelastic planar Poiseuille flow in a channel with smooth walls. These comparisons, summarized in table 2, correspond to the experimental investigations of Sparrow et al. (Reference Sparrow, Beavers, Chen and Lloyd1973) and Srinivas & Kumaran (Reference Srinivas and Kumaran2017), respectively.

Table 2.

Comparison of the present theoretical predictions with experimental results from two distinct studies. (a) Newtonian fluid over a porous-walled channel, based on the study by Sparrow et al. (Reference Sparrow, Beavers, Chen and Lloyd1973). In this study, the lower wall of the channel is porous, with permeability parameter $\alpha =100$ , $\epsilon =0.95$ and $\tau =0.878$ . (b) Viscoelastic fluid flow in a channel with impermeable walls, as investigated by Srinivas & Kumaran (Reference Srinivas and Kumaran2017). This case considers an Oldroyd-B fluid characterized by an elasticity number $E=0.22$ , $\beta =0.92$ and $C_{p}(ppm)=30$ .

Figure 20.

Variation of the critical Reynolds number $ \textit{Re}_{\textit{cr}}$ as a function of the viscosity ratio $\beta$ for an Oldroyd-B fluid with $ \textit{Wi}=5$ , and $ \textit{Wi}=10$ , in a channel at (a) $\alpha = 500$ , and (b) $\alpha = 50$ .

In the first case, Sparrow et al. (Reference Sparrow, Beavers, Chen and Lloyd1973) studied Newtonian flow through a channel with a porous bottom wall and an impermeable top wall. They reported critical Reynolds numbers across a range of non-dimensional permeabilities by varying the channel’s half-height, using Foametal as the porous medium. Following Tilton & Cortelezzi (Reference Tilton and Cortelezzi2008), who numerically analysed this configuration, we adopted interface parameters $\epsilon = 0.95$ and $\tau = 0.878$ to closely approximate the effective slip coefficient of Beavers and Joseph (Beavers & Joseph Reference Beavers and Joseph1967) used in that study. We selected a representative case with a non-dimensional permeability corresponding to $\alpha = 100$ . As shown in table 2, the critical Reynolds number predicted by our model is substantially higher than the experimental value reported by Sparrow et al. (Reference Sparrow, Beavers, Chen and Lloyd1973), consistent with the trend observed by Tilton & Cortelezzi (Reference Tilton and Cortelezzi2008). They noted that even in channels with impermeable walls, linear stability theory tends to overpredict the transition threshold relative to experiments. In addition, this discrepancy can be attributed to the neglect of inertial effects within the porous region and the potential influence of surface roughness at the porous interface, both of which can promote earlier transition to turbulence in experiments.

In the second case, Srinivas & Kumaran (Reference Srinivas and Kumaran2017) examined the viscoelastic flow of polyacrylamide solutions in a rectangular microchannel. The onset of instability was identified based on the rise in the standard deviation of velocity fluctuations obtained from particle image velocimetry measurements. To compare with this study, we followed the methodology of Khalid et al. (Reference Khalid, Chaudhary, Garg, Shankar and Subramanian2021a ), who aligned theoretical predictions with experimental observations by matching the elasticity number $E$ and viscosity ratio $\beta$ . We selected a case corresponding to a 30 p.p.m. polyacrylamide solution with $E = 0.22$ and $\beta = 0.95$ . Our model used a large value of $\alpha$ to approximate a smooth-walled channel. As reported in table 2, the predicted critical Reynolds number shows good agreement with experimental results in this configuration. Compared with the porous-walled Newtonian case, the improved correspondence here may be attributed to the absence of wall permeability effects and the influence of the polymer solution, which modifies the flow behaviour and likely suppresses secondary disturbances not captured in the linear framework.