3.1 A flexible sheet perpendicular to the flow

A series of measurements were made over a range of Reynolds numbers up to $Re=3.5\times 10^{-4}$ (defined as $Re=\unicode[STIX]{x1D70C}Uw/\unicode[STIX]{x1D702}_{0}$ , where $\unicode[STIX]{x1D70C}$ is the density of the fluid, $U$ is the flow velocity, $w$ is the width of the sheet and $\unicode[STIX]{x1D702}_{0}$ is the zero-shear-rate viscosity). As a result, these experiments were in the Stokes flow regime and the inertial flow effects could be neglected. At zero flow velocity, the sheet was undisturbed and was aligned perpendicular to the flow direction, as seen in figure 2(a). As the cross-flow was applied, the sheet was bent in the flow direction (second and third images in figure 2 a). It should be noted that in addition to the induced curvature along the length of the sheet that can be observed from the front view in figure 2(a), the low flexural rigidity of the sheet resulted in a secondary curvature across the sheet that can be observed from the side view in figure 2(a). The resulting cross-sectional profile was ‘C’ shaped. At low flow velocities, where the Weissenberg number was small, the flow of the wormlike micelle solution remained stable and the static deflection of the sheet grew with increasing flow velocity. As the flow velocity was increased beyond a critical velocity and a corresponding critical Weissenberg number ( $Wi_{crit}=13$ ), the flow of the wormlike micelle solution became unstable, with periodic fluctuations in the velocity and stress fields around the sheet, observed by tracking particle motion and FIB in the fluid. As was the case for the flow of wormlike micelle solutions past circular cylinders and spheres (Chen & Rothstein Reference Chen and Rothstein2004; Moss & Rothstein Reference Moss and Rothstein2010; Mohammadigoushki & Muller Reference Mohammadigoushki and Muller2016), the flow instability originated as a slow growth and fast decay of extensional stress in the wake of the flexible sheet. This can be observed in figure 3 from the time-dependent extension and retraction of the birefringent tail in the wake of the sheet.

Figure 2. The centreline deflection of the flexible sheet versus the flow velocity (a), and the time histories of the centreline deflection of the sheet at (b) $U=4.3$   $\text{mm}~\text{s}^{-1}$ , (c) $U=7.15$   $\text{mm}~\text{s}^{-1}$ and (d) $U=11.44~\text{mm}~\text{s}^{-1}$ . The critical onset condition, $U=1.4~\text{mm}~\text{s}^{-1}$ , corresponds to $Wi_{crit}=13$ .

Figure 3. Time history of the centreline deflection of the sheet for one period of oscillation (a), together with bright-field images of the deformed sheet (left) and the extensional birefringent patterns (right) at $U=4.3~\text{mm}~\text{s}^{-1}$ and for (b) $t_{1}=0$ , (c)  $t_{2}=3.45$  s, (d) $t_{3}=7.2$  s and (e) $t_{4}=7.95$  s. The dashed lines highlight the change in the birefringence patterns and deformations of the sheet between each time interval. The FIB images are viewed perpendicular to the flow direction, while the real bright-field images are taken from an angle to the flow direction in order to show the entire length of the flexible sheet.

The FIB measurements in figure 3 were taken through crossed polarizers to emphasize the intensity of the elastic extensional stress in the fluid. It is difficult to deconvolute these FIB measurements directly into a quantitative value of the viscoelastic stress in the fluid for a number of reasons. Flow-induced birefringence is a line-of-sight technique, which integrates the birefringence contribution from all fluid elements along the light path. Because the flow is three-dimensional, it is not possible to accurately use these FIB measurements to determine the stress. Additionally, at these birefringence levels, the stress-optical rule has been shown to break down for this fluid (Rothstein Reference Rothstein2003), and any stresses calculated using it would dramatically underpredict the true state of stress in the fluid. The FIB measurements are still a valuable tool for qualitatively observing the growth in the viscoelastic stress and its time-dependent fluctuations. As seen in figure 3, a narrow region of micelle deformation, known as a birefringent tail, formed in the strong extensional flow region just downstream of the stagnation point. In the wake, the fluid must accelerate from rest along the trailing edge of the sheet to the maximum flow velocity, $U_{max}$ , over a short distance downstream of the flexible sheet. This extensional flow resulted in strong micelle alignment and deformation, as seen in figure 3(b). As the flow velocity was increased, a stable birefringent tail grew both in length and intensity until the onset of the viscoelastic flow instability. After the onset of the flow instability, the maximum extent of the birefringent tail approached an asymptotic limit. A series of dashed lines have been added to figure 3 to graphically illustrate the magnitude and direction of the changes to the birefringence pattern with time during one oscillation cycle. These lines are placed in the wake of the sheet at a location corresponding to a fixed value of the FIB or elastic stress in the fluid. A similar set of lines have been added to show the deflection of the sheet in the bright-field images to the left of the FIB. At the start of an oscillation (figure 3 b), the wormlike micelle solution already exhibited a significant amount of elastic stress in the wake. The birefringent tail in the fluid grew in length and intensity with time (figure 3 c), resulting in still further stretching of the flexible sheet. At its maximum extent (figure 3 d), the birefringence grew by approximately 30 % beyond its minimum extent (figure 3 b) to nearly 10 $w$ downstream, while the deflection of the sheet increased by roughly 12 % from 8.5 mm to 10 mm. When the displacement of the sheet reached 10 mm, an abrupt breakdown of the wormlike micelles in the high-stress wake was observed, resulting in a rapid loss of elastic stress in the wake. This can be seen in figure 3(e) as a significant reduction in the length and extent of the birefringent tail in the fluid. The next oscillatory cycle begins with the flow of fresh unruptured wormlike micelles from upstream of the flexible sheet into its wake, where with time, elastic stress is once again built up in the fluid in the wake of the sheet as the sheet is stretched and deformed back towards its maximum deformation. The result of the cyclical tensile loading and failure of the viscoelastic fluid is the observed periodic motion of the flexible sheet. Similar time-dependent FIB patterns were observed at all flow velocities where the flow of wormlike micelle solution became unstable.

The presence of this viscoelastic flow instability caused the flexible sheet to oscillate around a mean stretched position. As seen in figure 3, the stress growth/decay and the sheet deflection were directly correlated. For this fluid and sheet, the critical velocity for the onset of oscillations was found to be $U_{crit}=1.4~\text{mm}~\text{s}^{-1}$ (figure 2 a), corresponding to a critical Weissenberg number of $Wi_{crit}=13$ . Unlike several cases of inertia-driven flow-induced instabilities of flexible structures (e.g. vortex induced vibrations (VIV) of a flexible (Bourguet et al. Reference Bourguet, Modarres-Sadeghi, Karniadakis and Triantafyllou2011) or a flexibly mounted structure (Williamson & Govardhan Reference Williamson and Govardhan2004), or the response of a flexible cylinder in axial flow (Païdoussis Reference Païdoussis2004; Modarres-Sadeghi et al. Reference Modarres-Sadeghi, Païdoussis, Semler and Grinevich2011)), the displacement of the sheet over time was not sinusoidal. The saw-toothed pattern of the time histories (figure 2 b–d) shows that these oscillations were highly nonlinear. At a cross-flow velocity of $U=4.3~\text{mm}~\text{s}^{-1}$ (figure 2 b), the structure was found to oscillate with a dominant frequency of $f=0.11$  Hz. Under these flow conditions, the flexible sheet stretched slowly at a rate of $0.17~\text{mm}~\text{s}^{-1}$ until a critical breakdown of the wormlike micelles, and the subsequent loss of the viscoelastic stress in the wake of the sheet caused the flexible sheet to recoil abruptly. The recoil rate was found to be considerably faster at $1.5~\text{mm}~\text{s}^{-1}$ . As the flow velocity was increased to $U=7.15~\text{mm}~\text{s}^{-1}$ (figure 2 c), the frequency of oscillations increased to $f=0.2$   Hz. The frequency of the flexible sheet oscillations increased roughly linearly with increasing flow rate, seen later in figure 6(c). As will be discussed in further detail later, this increase in the overall frequency of oscillations was accompanied by an increase in both the deformation rate of the sheet during the growth phase of the deformation cycle and a slightly slower increase in the recoil rate of the sheet during the decay phase of the deformation cycle. These rates will be presented as a function of flow velocity, alongside a plot of their relative magnitude, later in figure 7. At velocities of $U=11.44~\text{mm}~\text{s}^{-1}$ and beyond, higher harmonics began to emerge in the oscillations of the flexible sheet. These can be observed in figure 2(d) and are quite clear from the fast Fourier transform analysis of the data (not shown).

The maximum and minimum deformations of the flexible sheet versus the flow velocity are shown in figure 2(a). For each time history, a minimum of 10 cycles and 180 s of data were used. Below $U_{crit}=1.4~\text{mm}~\text{s}^{-1}$ , the flow was stable and the structure underwent a static deflection but did not oscillate. Above $U_{crit}$ , a periodic response was observed. Just beyond this point, the maximum displacement of the flexible sheet increased linearly with the flow velocity until a velocity of $U=6~\text{mm}~\text{s}^{-1}$ , beyond which the maximum sheet deflection reached a plateau and remained more or less unchanged for all flow velocities tested. On the other hand, the amplitude of the sheet oscillations increased linearly until it reached a maximum at $U=4~\text{mm}~\text{s}^{-1}$ . Thereafter, with increasing flow velocity, the oscillation amplitude began to decay.

Two observations can be gleamed from figure 2. First, there are two types of sheet deformation: one is the in-flow bending and stretching of the sheet span and the other is the bending of the cross-section of the sheet into a semicircular ‘C’. A simplified diagram of these two types of flexible sheet deformation is shown in figure 4. During the experiments, the flexible sheet deformation remained within the material’s elastic limits. Therefore, the stress needed for the cross-flow bending of the sheet from a straight profile into a ‘C’ profile can be approximated from Hooke’s law, $\unicode[STIX]{x1D70E}_{cross\text{-}flow}=E\unicode[STIX]{x1D716}$ , where $\unicode[STIX]{x1D70E}_{cross\text{-}flow}$ is the bending stress at a layer above/below the neutral axis, $E=101$  kPa is the known elastic modulus of the flexible sheet and $\unicode[STIX]{x1D716}$ is the strain, which is the ratio of the distance from the neutral axis and the radius of curvature to the neutral surface. The result is a maximum bending stress of approximately 8 kPa. The stress needed for the in-line bending of the sheet span length to the maximum deflection seen in figure 2(a) was calculated using beam theory (Beer et al. Reference Beer, Johnston, Dewolf and Mazurek2014). The force acting on the sheet was approximated by using the equation of a simply supported beam, $p=-384EIy_{max}/5L^{4}$ , where $p$ is the force per unit length acting on the sheet, $E$ is the elastic modulus, $I=0.4~\text{mm}^{4}$ is the moment of inertia for the deformed shape as a C-channel, $y_{max}$ is the maximum deflection measured from figure 2(a) and $L$ is the flexible sheet length. The resulting in-line bending stress, $\unicode[STIX]{x1D70E}_{inline}=p/w$ , where $w$ is the sheet’s width, was found to be significantly smaller, in the range of only 10 Pa. In addition to bending, there is a tensile stress which results from the sheet stretching from its initial undeformed state to its deformed elongated contour length at maximum deflection. By calculating the strain, $\unicode[STIX]{x1D716}$ , from the images of the deformed sheets, this stress was found to be approximately $\unicode[STIX]{x1D70E}_{tensile}=E\unicode[STIX]{x1D716}=13~\text{kPa}$ . From filament stretching extensional rheology measurements, the wormlike micelle solution was found to rupture and fail at an extensional stress of 7 kPa. Thus, these calculations appear to confirm that the viscoelastic fluidic stresses needed to deform the sheet are large enough to result in breakdown of viscoelastic wormlike micelles in the wake of the sheet, resulting in a time-dependent flow field, which in turn drives the observed time-dependent oscillations of the sheet.

Second, as mentioned before, the deformation rate of the sheet during the oscillations was found to grow linearly with increasing velocity and change more dramatically with increasing flow velocity than the recoil rate. As a result, at low velocities, when the recoil rate was much faster than the growth in the sheet deflection or the deformation rate, the sheet was able to fully recoil before any significant extensional stress could be rebuilt in the micelle solution in the wake of the sheet. As the flow velocity was increased, the deformation and recoil rate of the sheet became more comparable, moving from a 20 : 1 to a 5 : 1 ratio between the deformation and recoil rates, as discussed later. As a result, the sheet was not able to fully recoil before new elastic stresses began building in its wake. At the same time, the breakdown of micelles transitioned from a global to a local phenomenon. The wormlike micelle solution became unstable at multiple isolated locations in the wake of the sheet rather than failing simultaneously across the entire downstream edge. The result was that a complete recoil of the sheet could not be achieved because all of the fluid did not yield simultaneously. Such an instance of local regions of elastic stress in the fluid in the wake along the length of the flexible sheet is illustrated in the time series of FIB images in figure 5. In these images, a patchwork of bright irregular regions of high elastic stress in the fluid was found to fluctuate in space and time during the sheet oscillations observed at a constant flow velocity of $U=10~\text{mm}~\text{s}^{-1}$ . The consequence of these spatial fluctuations was the appearance and growth of higher harmonics in the data and a reduction in the oscillation amplitude with increasing velocity beyond $U=8~\text{mm}~\text{s}^{-1}$ . At lower velocities, the birefringence was found to be constant along the span of the sheet and vary uniformly across the span with the onset of the viscoelastic flow instability and the resulting sheet oscillations. The spatial fluctuations of the FIB within the high-stress regions (figure 5) and the feather-like structures (figure 3) are reminiscent of the Taylor–Gortler vortex structures observed downstream of a circular cylinder in cross-flow of a viscoelastic polymer solution by McKinley, Armstrong & Brown (Reference McKinley, Armstrong and Brown1993) and Shiang, Özkekin, Lin & Rockwell (Reference Shiang, Özkekin, Lin and Rockwell2000). In their work, three-dimensional stationary roll cells were observed to be spaced periodically along the axis of the cylinder.

Figure 4. A schematic diagram of the in-line and cross-flow deformation of the flexible sheet.

Figure 5. Flow-induced birefringence snapshots corresponding to the sequence of fluid rupture in the wake of the flexible sheet for a velocity of $U=10~\text{mm}~\text{s}^{-1}$ . In (a), an FIB image is shown at the onset of sheet recoil, time $t=0$ ; in (b), $t=0.25$  s, while in (c), $t=0.60$  s, which corresponds to the end of the recoil phase in the oscillation cycle. The bright regions highlight areas of large micelle deformation and large elastic stresses in the fluid, while the darker regions show the locations where the micelles have broken down and the elastic stress in the fluid has been lost.

3.2 Modifying the flexible sheet inclination

An additional set of experiments was conducted over the same range of Reynolds numbers for cases wherein the flexible sheet was placed at $0^{\circ }$ , $20^{\circ }$ and $45^{\circ }$ to the flow direction. The desired inclination of the flexible sheet was achieved through external rotation of the fixtures holding the flexible sheet inside the flow cell. The $0^{\circ }$ orientation did not oscillate for the range of flow rates tested, and, as a result, discussion of the $0^{\circ }$ case will not be included in the following. The centreline deflection of the flexible sheet for the three different inclination angles tested is shown in figure 6(a). For completeness and ease of comparison, the data for the $90^{\circ }$ case have been included in all subsequent figures. The side views of the complex deformation that the flexible sheet underwent for each inclination angle are shown in inset images in figure 6. As the flow began, the $45^{\circ }$ flexible sheet was deformed into the flow direction, with its cross-section forming a ‘C’ shape that was significantly less deformed than that of the $90^{\circ }$ case. Unlike the $90^{\circ }$ case, the ‘C’ shape remained open for all of the flow velocities tested and did not completely bend back onto itself. At a critical flow velocity of $U=2.8~\text{mm}~\text{s}^{-1}$ , the wormlike micelle solution became unstable, and periodic oscillations of the flexible sheet started in the same manner as previously observed for the $90^{\circ }$ case. The centreline deflection of the $45^{\circ }$ flexible sheet increased linearly with flow velocity up to a maximum value that was smaller than that observed for the $90^{\circ }$ case. After reaching this maximum value at $U=5.7~\text{mm}~\text{s}^{-1}$ , the centreline deflection decreased and approached a plateau at the higher flow velocities tested. For the $20^{\circ }$ inclination, the flexible sheet began from a position aligned almost completely with the flow, and the resulting cross-section profile of the sheet had a much smaller curvature (figure 6 a). The centreline deflection of the sheet was much smaller than the $45^{\circ }$ and $90^{\circ }$ cases, with a maximum deflection of $5$ mm compared with nearly $11$ mm for the $90^{\circ }$ case. For flow velocities larger than $U=4.8~\text{mm}~\text{s}^{-1}$ , the $20^{\circ }$ flexible sheet did not continue to hold the stretched deformation seen in figure 6(a), but instead rotated off the centreline, where it remained in an asymmetric position closer to one of the sidewalls for the rest of the flow velocities tested and exhibited no further large-amplitude fluctuations. Data for the $20^{\circ }$ case beyond $U=4.8$ $\text{mm}~\text{s}^{-1}$ are therefore not presented in figure 6 or any subsequent plots. The amplitude and frequency of oscillations from the centreline deflection time histories are mapped out in figures 6(b) and 6(c) respectively. The amplitude of oscillations initially increased roughly linearly with increasing flow velocity for all three inclinations tested and reached a maximum. Beyond this maximum, the amplitudes of all three angles began to decay with increasing flow velocity. The oscillation frequency increased with the flow velocity for all three inclinations.

Figure 6. (a) Centreline displacement of the flexible sheet for $20^{\circ }$ (●), $45^{\circ }$ (▴) and $90^{\circ }$ (▪) inclinations. The filled and hollow symbols are used for the maximum and minimum flexible sheet displacements respectively during oscillations at each flow velocity. The insets contain the side views of the cross-flow deformation of the flexible sheet. (b,c) Amplitude and frequency of oscillations of the flexible sheet over the range of flow velocities tested. The error margin for the amplitude of oscillations is less than 10 % for flow velocities below $4~\text{mm}~\text{s}^{-1}$ and less than 5 % for higher flow velocities. The error margin is less than 5 % for the frequency plots.

Figure 7. (a) The deformation rate (open symbols) and recoil rate (filled symbols) during an oscillation cycle for the flexible sheet aligned at a $20^{\circ }$ (●), $45^{\circ }$ (▴) and $90^{\circ }$ (▪) inclination to the flow direction as a function of the flow velocity. (b) The ratio of deformation and recoil rates of the flexible sheet as a function of the flow velocity for the three flexible sheet inclination angles.

From the sawtooth waveform of the centreline deflection time histories, it was clear that the flexible sheet stretches slowly before recoiling rapidly during each oscillation cycle. In figure 7(a), the deformation and recoil velocities of the flexible sheet during oscillations are presented as a function of the flow velocity for all three inclinations. It can be observed that the deformation velocity curve followed a similar trend for all three inclinations over the range of flow velocities tested. The recoil velocities, on the other hand, were very different for the three inclination angles. The $90^{\circ }$ inclination curve had the largest recoil velocity by almost a factor of three. This can likely be attributed to the flexible sheet recoil velocity being dependent on the elastic stress built up within the sheet, which is clearly at a maximum within the $90^{\circ }$ sheet due to the increased sheet deformation at any given flow velocity. Conversely, the deformation velocity of the sheet is strongly dependent on the flow conditions as it is coupled to the convection of fluid from upstream of the sheet to rebuild elastic stress in the wake of the sheet after a fluid rupture event. As a result, as seen in figure 7(a), the deformation velocity is not strongly coupled to the sheet orientation. A plot of the ratio of deformation and recoil velocities of the flexible sheet over varying flow velocities is shown in figure 7(b). For the $45^{\circ }$ and $90^{\circ }$ inclination angles, the ratio of the deformation rate to the recoil rate was found to start quite small at a value of less than 0.1 and then to increase with the flow velocity to reach a plateau beyond $U=8~\text{mm}~\text{s}^{-1}$ . This plateau of the ratio of deformation and recoil velocity corresponds to the point at which higher harmonics appear in the oscillations and the amplitude begins to decay. For the case of the $90^{\circ }$ sheet, the ratio of deformation rate to recoil rate reached a plateau of just over 0.2. As such, the dynamics of oscillation was dominated by the fast recoil of the sheet after fluid rupture. Conversely, this rate approached 0.6 for the $45^{\circ }$ sheet, resulting in a more symmetric oscillation cycle and a growth in deformation that nearly matched its decay. Although the $20^{\circ }$ sheet did not reach a plateau before twisting from the centreline, breaking symmetry and ceasing to oscillate, its behaviour appears to be closer to that of the $45^{\circ }$ sheet than the $90^{\circ }$ sheet.

Finally, we concentrate on the results presented in the previous sections with the objective of combining the dimensional results from the different flexible sheet inclinations into a cohesive set of results using a single non-dimensional parameter. In order to collapse the data for all three inclination angles, a non-dimensional flow velocity, $U^{\ast }$ , a non-dimensional oscillation amplitude, $A^{\ast }$ , and a non-dimensional frequency, $f^{\ast }$ , were considered. The non-dimensional velocity that best collapsed the data in terms of both the critical onset conditions and the flow velocity corresponding to the maximum oscillation amplitude was found to be $U^{\ast }=U/(w\,\text{sin}(\unicode[STIX]{x1D703})f_{\unicode[STIX]{x1D703}})$ . Here, $U$ is the flow velocity, $w$ is the width of the flexible sheet, $\unicode[STIX]{x1D703}$ is the inclination angle of the sheet, $w\,\text{sin}(\unicode[STIX]{x1D703})$ is the equivalent surface area of the flexible sheet exposed to the flow and $f_{\unicode[STIX]{x1D703}}$ is the frequency of oscillations at the critical flow velocity, which depends on the inclination angle of the flexible sheet. As seen in figure 8(a), using this dimensionless velocity, both the onset condition and the maximum of the oscillation amplitude collapse to values of $U^{\ast }\approx 7$ and $U^{\ast }\approx 42$ respectively. It is expected that the appropriate dimensionless velocity should also contain some information about the viscoelastic fluid properties, specifically the relaxation time of the fluid, $\unicode[STIX]{x1D706}$ . However, because in this study only a single fluid composition was used, this hypothesis could not be fully tested. As seen in the inset in figure 8(b), a renormalization of the critical oscillation frequency with the relaxation time to form a modified Weissenberg number, $Wi_{f}\,=\,\unicode[STIX]{x1D706}f_{\unicode[STIX]{x1D703}}$ , appears to be a promising first step towards incorporating rheological information into our analysis as the value approaches $Wi_{f}\,\sim \,1$ . Future studies will focus on how changes in the fluid rheology affect the observed viscoelastic FSI. The appropriate non-dimensional amplitude and frequency were found to be $A^{\ast }=A/w\,\text{sin}(\unicode[STIX]{x1D703})$ and $f^{\ast }=f/f_{\unicode[STIX]{x1D703}}$ , where $A$ and $f$ are the dimensional amplitude and frequency of flexible sheet oscillations respectively. Using these dimensionless parameters, the data from the three inclination angles are collapsed onto a master curve, as seen in figure 8(a,b). The slight variations in the data are likely the result of the complex deformation of the flexible sheet under flow. The deviation of scaling for higher flow velocities could be due to the higher harmonics that become dominant during oscillations.

Figure 8. The dimensionless (a) amplitude and (b) frequency of oscillations as a function of the dimensionless flow velocity for flexible sheet inclination angles of $20^{\circ }$ (●), $45^{\circ }$ (▴) and $90^{\circ }$ (▪). The inset in (b) contains the dimensionless product, $\unicode[STIX]{x1D706}f_{\unicode[STIX]{x1D703}}$ , as a function of the inclination angle.