3.1. Multiplicity of arrowhead states

We begin by examining arrowhead branches continued in ${\textit{Wi}}$ . Figure 2 shows that the branches exhibit distinct bifurcation structures and topological features at different streamwise lengths $L$ . At $L=2\pi$ , the arrowhead branch forms an isola where the travelling wave exists only at finite amplitude thus does not bifurcate from the one-dimensional laminar state (2.1). This also implies bistability, with two distinct stable solutions coexisting in parameter space. For larger $L$ , the branches reconnect to the laminar state at ${{\textit{Wi}}} \sim 30$ , consistent with the emergence of a centre-mode instability (Lewy & Kerswell Reference Lewy and Kerswell2025b ). At $L=4\pi$ , multiple branches coexist. Around ${{\textit{Wi}}} =40$ , these branches are linked by two saddle–node bifurcations, yielding three AHs at the same parameters – two stable states separated by one unstable state. To verify this, the unstable state III at ${{\textit{Wi}}}=40$ was disturbed by a pair of small perturbations proportional to the difference between states II and III,

(3.1) \begin{equation} \boldsymbol{\phi} ^{\pm }= \boldsymbol{\phi} _{\textit{III}}\pm 0.01(\boldsymbol{\phi} _{\textit{II}}-\boldsymbol{\phi} _{\textit{III}}), \end{equation}

where $\boldsymbol{\phi} =(\boldsymbol{u},\boldsymbol{\alpha })$ , and integrated forward in time by DNS. As expected, the final state was either state II or state IV, depending on the sign of the perturbation applied (the two time series $\boldsymbol{\phi} ^{\pm }(t)$ are projected onto the $\langle v^{2}\rangle$ continuation branch in figure 2 a).

Figure 2.

Branches of arrowhead solutions continued in ${\textit{Wi}}$ : (a) cross-shear velocity $\langle v^2 \rangle$ (where $\langle {\cdot }\rangle$ denotes spatial averaging); (b) distance between the stagnation points $D_{{sp}}$ . Thick lines indicate relative centreline velocity $u_c-c\geqslant 0$ throughout, corresponding to at most one stagnation point ( $n_{{sp}}\leqslant 1$ ). Arrow lines: projection of time series of perturbed III states (see (3.1)). Inset: $u_{c}-c$ versus $x$ . Dashed lines indicate $u_{c}-c=0$ , and dotted lines mark the arch location. Flow fields of the six marked cases are shown in figure 3.

At the large- ${\textit{Wi}}$ limit, the $L=3\pi$ and $L=4\pi$ branches close and reconnect to the laminar state at finite ${\textit{Wi}}$ . On the contrary, the $L=6\pi$ branch persists to arbitrarily large ${\textit{Wi}}$ , and its amplitude asymptotes to zero as ${{\textit{Wi}}}\to \infty$ . Comparable asymptotics occur for $L\gt 6\pi$ (not shown), implying that the solution is effectively insensitive to $L$ . Interestingly, the $L=3\pi$ and $L=6\pi$ solution curves connect in a modulational bifurcation shown by a blue star marker at ${{\textit{Wi}}} \approx 10$ . In this, two wavelengths of a $L=3\pi$ arrowhead solution suffer a modulational instability to a $6\pi$ wavelength.

For a typical arrowhead, the centreline velocity $u_c$ decreases as the flow approaches the arch, and increases afterwards. In a frame moving with the arrowhead, the relative centreline velocity $u_{c}-c$ can drop below zero, creating two stagnation points (see the inset of figure 2 b). Strong extensional flow develops around these stagnation points, which is implicated in the formation of AHs (Goffin, Dubief & Terrapon Reference Goffin, Dubief and Terrapon2025; Morozov et al. Reference Morozov, Lellep, Capocci and Linkmann2025). However, this behaviour is not universal. Figure 2(b) shows the distance $D_{{sp}}$ between the two stagnation points in an arrowhead solution. In some cases, $u_{c}-c\geqslant 0$ is satisfied everywhere, which corresponds to $D_{{sp}}=0$ . Such cases typically occur on the lower branch near the laminar state, where the structure is weak. Their persistence indicates that stagnation points may not play an essential role in sustaining AHs. Apart from these cases, $D_{{sp}}$ varies continuously, with substantial diversity across streamwise lengths $L$ . For $L=2\pi$ , $3\pi$ and $4\pi$ , $D_{{sp}}$ shows no clear asymptotic trend with ${\textit{Wi}}$ . By contrast, for $L=6\pi$ , once past the left saddle–node, $D_{{sp}}$ increases asymptotically with ${\textit{Wi}}$ .

Figure 3.

Typical structures of arrowheads: (a) I, near-onset arrowhead solution resembles the eigenfunction of centre-mode instability; (b) II, lower branch with attached strand; (c) III, intermediate unstable solution with detached strand; (d) IV, upper branch arrowhead with detached weak strand; (e) V, modulated solution at $L=6\pi$ ; (f) VI, upper asymptote solution. Contours denote the normalised trace of polymer conformation $\mathrm{tr}\,\boldsymbol{\alpha }/\max (\mathrm{tr}\,\boldsymbol{\alpha })$ . Lines indicate the streamlines in the phase-speed frame; asymmetry arises from unconstrained seeding.

In figure 3, we show representative structures of AHs. Case I lies close to the laminar state and closely resembles the eigenfunction of the centre-mode instability (see Lewy & Kerswell Reference Lewy and Kerswell2025b ), consistent with the fact that arrowhead is the nonlinear saturation of the centre-mode instability. In the moving frame of the AHs, AHs consist of two legs joined by an arched head that blocks throughflow and traps a downstream counter-rotating vortex pair. Figures 3(b–d) display three solutions straddling a fold (saddle–node) bifurcation: II and IV are stable, whereas III is unstable. In case II, a clear strand is attached to the arch, while such a strand is not distinguishable from the arch in cases III and IV. In fact, a detached, weak strand forms downstream within the counter-rotating vortex pair, likely generated by the intervening extensional corridor, and unrelated to the convex side of the downstream arch. For larger $L$ , the continuation diagram becomes richer, leading to a modulated arrowhead (panel e) and a weak nested arrowhead at extreme ${\textit{Wi}}$ (panel f).

3.2. Persistence of strand

In the literature, the persistence of an upstream ‘strand’ (or ‘spike’) ahead of the arch is often reported and treated as an essential feature of AHs (Morozov Reference Morozov2022; Morozov et al. Reference Morozov, Lellep, Capocci and Linkmann2025), whereas in other studies – e.g. figure 7 in Buza et al. (Reference Buza, Beneitez, Page and Kerswell2022), figure 10 in Lewy & Kerswell (Reference Lewy and Kerswell2025b ), and figure 1c in Nichols, Guy & Thomases (Reference Nichols, Guy and Thomases2025) – such strands are not observed. In figure 3, we have already shown the coexistence of three distinct solutions – one with a clear strand (II) and two without (III, IV). To interrogate this, we perform transient DNS at $(L,{{\textit{Wi}}})=(4\pi ,36)$ , initiating from a ‘no-strand’ solution at ${{\textit{Wi}}}=38$ (near case IV) and evolving to a strand-bearing solution at ${{\textit{Wi}}}=36$ in figure 4. We use $-\partial ^{2}\alpha _{xx}/\partial y^{2}$ as a strand indicator: positive values denote a convex polymer-stretch profile (i.e. a strand). In figures 4(a–c) and supplementary movie 1 available at https://doi.org/10.1017/jfm.2026.11457, a weak strand is initially present between the downstream counter-rotating vortices, detached from the downstream arch. As ${\textit{Wi}}$ decreases, it extends and strengthens, ultimately attaching to the downstream arch and forming a pronounced strand. Figures 4(d–f) show a measure of flow extension, $R=\lVert \boldsymbol{\varOmega} \rVert _{2}/\lVert \boldsymbol{S}\rVert _{2}$ (where $\boldsymbol{\varOmega}$ and $\boldsymbol{S}$ are the rotation and strain-rate tensors). In figure 4(d), extension is weak and only a weak strand forms within the extensional corridor between the counter-rotating vortices. By contrast, in figure 4(f), a strong extensional region develops near the arch, generating the attached strand. The occasional absence of a strand indicates that it is not required for the persistence of the arrowhead state. Rather, it is a by-product of extensional flow, which need not occur upstream of the arch.

Figure 4.

Transitions between detached and attached strands in DNS. (a–c) Instantaneous fields of strand measure $-\partial ^2 \alpha _{xx}/\partial y^2$ (colours) and $\mathrm{tr}\,\boldsymbol{\alpha }$ (lines) at $t=50$ , $1200$ and $2000$ . (d–f) Measures of flow extension $R=|\boldsymbol{\varOmega} |/|\boldsymbol{S}|$ at the same times. (g,h) Space–time $x{-}t$ diagrams of (g) $-\partial ^2 \alpha _{xx}/\partial y^2$ and (h) $u_c-c$ at $y=0$ . Green dashed lines indicate the times shown in (a–f). (i) The $x$ -averaged relative centreline velocity $\langle u_c-c\rangle _x$ versus the arch-strand edge separation $D_{{AS}}$ , measured as the distance between the two zero crossings of $-\partial ^2 \alpha _{xx}/\partial y^2$ shown in (g).

In figures 4(g,h), we present space–time ( $x$ – $t$ ) diagrams of $-\partial ^{2}\alpha _{xx}/\partial y^{2}$ and $u_{c}-c$ at $y=0$ , respectively. The arch position at each time is aligned to isolate structural changes. In figure 4(g), a strong, persistent negative band marks the arch and varies little over time. On the downstream (right-hand) side, a second negative band, representing the strand–arch gap, shrinks gradually and suddenly disappears at $t\approx 1200$ . Thereafter, a positive band (the strand) intensifies and makes direct contact with the arch, indicating strand attachment. Consistently, figure 4(h) shows the evolution of $u_{c}-c$ : the two zero crossings (black lines, stagnation points) approach one another. A subsequent jump to large positive values indicates enhanced flow extension in the attached state. Interestingly, the left-hand stagnation line is vertically fixed, and in fact, as in figure 2(b), coincides with the arch head.

The strand attachment correlates with the relative centreline velocity. In figure 4(i), we show the joint distribution of the arch–strand edge separation $D_{{AS}}$ and the $x$ -averaged relative centreline velocity $\langle u_{c}-c\rangle _x$ for all branches in figure 2. The separation $D_{{AS}}$ is defined as the streamwise distance between the upstream edge of the arch and the strand, measured as the spacing between the two zero crossings of $-\partial ^{2}\alpha _{xx}/\partial y^{2}$ (see figure 4(g)). Many points cluster near a horizontal band at $D_{{AS}}\approx 0.8$ , indicating arrowheads with attached strands. This separation suggests the nearly uniform arch thickness of the attached-strand cases across different ${\textit{Wi}}$ and $L$ . A second cluster appears along a vertical band at $\langle u_{c}-c\rangle _{x}\approx 0.01$ , characteristic of arrowheads with a detached strand. In practice, $\langle u_{c}-c\rangle _{x}$ can serve as a diagnostic: values near $0.01$ indicate a detached strand, whereas larger values suggest attached strands.

3.3. Duct length and localisation

In figure 2, we have shown that travelling-wave branches depend sensitively on the streamwise length $L$ . To further clarify this dependence and identify possible spatial localisation of AHs, we now perform branch continuation in $L$ . We first define a localisation measure $L_{{ah}}$ as the streamwise length of the domain containing $95\,\%$ of the total stretched polymers, quantified by

(3.2) \begin{align} {L_{{ah}}=\int _L \boldsymbol{1}_{[\langle\text{tr}\,\boldsymbol{\alpha }\rangle _{y}\geqslant \tau ^\star ]}\,{\rm d}x, \ \ \text{where} \ \tau ^\star =\mathrm{sup}\left\{\tau : \frac {\int _L\langle \mathrm{tr}\,\boldsymbol{\alpha } \rangle _y\boldsymbol{1}_{[\langle \text{tr}\,\boldsymbol{\alpha } \rangle _y \geqslant \tau ]}\,{\rm d}x}{\int _L\langle \mathrm{tr}\,\boldsymbol{\alpha } \rangle _y\,{\rm d}x}\geqslant 95\,\% \right\},} \end{align}

where $\langle {\cdot }\rangle _y$ denotes averaging over $y$ . This definition eliminates bias from variations in the overall $\mathrm{tr}\,\boldsymbol{\alpha }$ magnitude induced by changes in ${\textit{Wi}}$ , $L$ or branch. Varying the $95\,\%$ threshold shifts the precise $L$ and $L_{{ah}}$ at which localisation onsets, but it does not materially affect the overall localisation trend.

Figure 5.

Influence of streamwise length: (a) localisation (VII, $L=76$ ) and (b) train of arrowheads (VIII, $L=67$ ) in the long channel (colours for $\mathrm{tr}\,\boldsymbol{\alpha }/\max (\mathrm{tr}\,\boldsymbol{\alpha })\in [0,1]$ ), lines for streamlines on the coordinate moving with $c$ ). Branches of AHs projected onto (c) $\langle v^2 \rangle$ versus streamwise length $L$ , and (d) the arrowhead length ( $L_{{ah}}$ ) versus $L$ space, respectively. The black dashed line is $L_{{ah}}=L$ .

In figure 5(c), the upper and lower branches are observed at ${{\textit{Wi}}}=10,20,30$ . The upper branch corresponds to stronger saturated arrowheads (cf. figure 5 a), whereas the lower branch is weaker. Notably, for ${{\textit{Wi}}}=30$ , the lower branch experiences a sequence of bifurcations that produces a train of arrowheads (figure 5 b), visible as vibration in the continuation curve. Phenomenologically, additional arches proliferate as $L$ increases via destabilisation and reconnection of the elongated arrowhead. To our knowledge, this phenomenon has not been reported previously.

In figure 5(d), we investigate localisation by comparing $L_{{ah}}$ with $L$ . When a branch aligns with the diagonal $L_{{ah}}=L$ , the state is effectively non-localised (small deviation is due to the 95 % threshold). As the branch curve bends away from the diagonal and plateau, the arrowhead localises, and further increasing $L$ does not change its length or shape.

For the upper branch, localisation depends on ${\textit{Wi}}$ . For ${{\textit{Wi}}}=10$ , $L_{{ah}}$ plateaus at $\approx 37$ for $L\approx 57$ ; for ${{\textit{Wi}}}=20$ , $L_{{ah}}\approx 88$ at $L\approx 132$ . For ${{\textit{Wi}}}=30$ , no plateau is reached within the explored $L$ , but one is expected at sufficiently large $L$ .

For the lower branch, plateaus occur earlier and at smaller $L_{{ah}}$ . For ${{\textit{Wi}}}=10$ , a plateau appears at $L\approx 25$ with $L_{{ah}}\approx 14$ ; for ${{\textit{Wi}}}=20$ , at $L\approx54$ with $L_{{ah}}\approx 24$ . Owing to the arrowhead train, the lower branch at ${{\textit{Wi}}}=30$ does not localise.

Figure 6.

Continuations of arrowhead branches in ${\textit{Wi}}$ and $L$ : three-dimensional branch manifolds (a) ( ${{\textit{Wi}}},L,\langle v^2\rangle$ ) and (b) ( ${{\textit{Wi}}},L,\langle \mathrm{tr}\,\boldsymbol{\alpha }\rangle$ ), and (c,d) projections of these branches on the ${{\textit{Wi}}}{-}L$ space.

To summarise the solution landscape, figures 6(a,b) present a joint view of branches continued in ${\textit{Wi}}$ and $L$ , plotted against $\langle v^{2}\rangle$ and the trace of the conformation tensor $\langle \mathrm{tr}\,\boldsymbol{\alpha }\rangle$ . Colours represent the arch–strand edge separation $D_{{AS}}$ . A distinct region of large $D_{{AS}}$ appears for ${{\textit{Wi}}}\gt 20$ and $L\lt 6\pi$ , indicating detached strands; a second detached cluster occurs on the lower branch near the laminar state. Velocity fluctuations ( $\langle v^{2}\rangle$ ) are largest at relatively small ${\textit{Wi}}$ and short $L$ , where polymer stretching ( $\langle \mathrm{tr}\,\boldsymbol{\alpha }\rangle$ ) is weak. By contrast, the polymer stretch increases at high ${\textit{Wi}}$ and $L$ , and most dominantly within the detached arrowhead region, underscoring the contrast with attached states.

In figure 6(c), the projection onto the ${\textit{Wi}}$ – $L$ plane delineates the existence boundary of AHs. The minimum ${\textit{Wi}}$ varies little with $L$ , whereas the minimum $L$ changes with ${\textit{Wi}}$ : it decreases from ${{\textit{Wi}}}\approx 10$ to increase approximately linearly for $20\lesssim {{\textit{Wi}}} \lesssim 100$ , i.e. $L_{min }\approx 0.125\,{{\textit{Wi}}} + 1.5$ . The turnover coincides with the attached $\rightarrow$ detached transition (visible as a sharp colour change), after which detached states determine $L_{min }$ . This linear trend cannot persist indefinitely: for detached states, the admissible $L$ gradually approaches $6\pi$ , beyond which only attached AHs remain, as the $L=6\pi$ branch tends to the laminar base as ${{\textit{Wi}}}\to \infty$ . Consequently, we expect $L_{min }\to 6\pi$ at sufficiently large ${\textit{Wi}}$ (estimated as $\approx 140$ by equating $6\pi$ with $0.125\,Wi+1.5$ ). Interestingly, figure 6(d) for the $x$ -averaged centreline relative velocity $\langle u_{c}-c\rangle _x$ shows a similar pattern: the detached arrowhead region coincides with small values of $\langle u_{c}-c\rangle _x$ . This suggests again that $\langle u_{c}-c\rangle _x$ can serve as an effective diagnostic for distinguishing detached/attached AHs.

3.4. Reynolds number dependence

Finally, we examine the Reynolds number dependence of the arrowhead solutions by continuing the branch at fixed $(L,{{\textit{Wi}}})=(4\pi ,40)$ up and down from ${\textit{Re}}=0.5$ . Mathematically, the solution branch can be smoothly continued from positive to negative and then back to positive ${\textit{Re}}$ . This procedure indicates how physical $({\textit{Re}}\gt 0$ ) solutions are related, and gives a strategy of how to find one from the other ( ${\textit{Re}}\lt 0$ solutions are, of course, unphysical). Practically, since (2.2) as written becomes singular when ${\textit{Re}} \to 0$ , a small-continuation-step computation (red line in figure 7 a) can stall near ${\textit{Re}}=0$ , whereas a larger-continuation-step computation (green) can reach across the singular region and proceed beyond it.

Figure 7.

Reynolds number dependence: (a) continuation of arrowhead branch in ${\textit{Re}}$ at $(L,{{\textit{Wi}}})=(4\pi ,40)$ ; structures of arrowhead at (b) IX, ${\textit{Re}}=20$ and (c) X, ${\textit{Re}}=380$ . Purple markers are states II, III and IV from figures 2 and 3.

In figure 7(a), two distinct branches are observed, separated by ${\textit{Re}}=0$ . The upper branch corresponds to arrowhead states with a detached strand (thus states III and IV lie on this branch), whereas the lower branch corresponds to attached-strand arrowheads (where state II is located). On the large- ${\textit{Re}}$ side, the lower branch persists beyond the largest value tested in this study ( ${\textit{Re}} \approx 400$ ), with its velocity fluctuations gradually decreasing. Figures 7(b,c) show the arrowhead structures at ${\textit{Re}}=20$ (state IX) and ${\textit{Re}}=380$ (state X). While state IX remains broadly similar to state II at ${\textit{Re}}=0.5$ , state X exhibits substantial differences: the arrowhead occupies a narrower region in the $y$ -direction, and an upstream ‘leg’ connects to the downstream structure, yielding a more nested topology. The strand becomes less distinct than in the lower- ${\textit{Re}}$ states.

The persistence of both branches as ${\textit{Re}} \to 0$ provides strong evidence that the arrowhead travelling wave is primarily elasticity driven, thereby linking the inertialess limit to a broad range of ${\textit{Re}}$ . Furthermore, the upper branch is associated with substantial velocity fluctuations, as quantified by $\langle v^{2}\rangle$ , suggesting that detached-strand arrowheads may efficiently enhance mixing in the inertialess regime. A systematic assessment over a wider range of $L$ and ${\textit{Wi}}$ would be needed to test this conjecture, but this lies beyond the scope of the present study.