Edinburgh Research Explorer “Phase diagram” for viscoelastic Poiseuille flow over a wavy surface

WeexperimentallyexaminethePoiseuilleﬂowofviscoelasticﬂuidsoverwavysurfaces.Fiveprecisionmicrofabricatedﬂowchannelsareutilized,eachofdepth2 d = 400 µ m, spanwise width (cid:52) = 10 d and with a sinusoidal undulation of amplitude A = d /20 on one of the spanwise walls. The undulation wavelength λ is varied between each of the channels, providing dimensionless channel depths α in the range 0.2 π ≤ α = 2 π d / λ ≤ 3.2 π . Nine viscoelastic polymer solutions are formulated, spanning more than four orders in elasticity number El and are tested in the wavy channels over a wide range of Reynolds and Weissenberg numbers. Flow velocimetry is used to observe and measure the resulting ﬂow patterns. Perturbations to the Poiseuille base ﬂow caused by the wavy surfaces are quantiﬁed by the depth of their penetration P into the ﬂow domain. Consistent with theoretical predictions made for wavy plane-Couette ﬂow [J. Page and T. A. Zaki, “Viscoelastic shear ﬂow over a wavy surface,” J. Fluid Mech. 801 , 392–429 (2016)], we observe three distinct ﬂow regimes (“shallow elastic,” “deep elastic” and “transcritical”) that can be assembled into a “phase diagram” spanned by two dimensionless parameters: α and the depth of the theoretically predicted critical layer Σ ∼ √ El . Our results provide the ﬁrst experimental veriﬁcation of this phase diagram and thus constitute strong evidence for the existence of the predicted In the inertio-elastic transcritical a surprising of the are of


I. INTRODUCTION
The addition of even small quantities of a flexible polymer to a Newtonian solvent can introduce a degree of elasticity to the fluid that fundamentally modifies its flow dynamics. Prominent examples include the shifting of the onset conditions and structure of flow instabilities, 1,2 including that of the laminar to turbulent transition, [3][4][5] and the reduction of turbulent drag in pipe flows. 3,[6][7][8] Quite recently, it has also been reported that even low levels of fluid elasticity can have unexpected and strong effects in shear flows over rough or undulating surfaces. 9,10 While for a Newtonian fluid, surface waviness introduces vorticity to the flow that decays monotonically with distance from the surface, 9,11 for a viscoelastic fluid, amplification of the vorticity is predicted in a critical layer located away from the site of vorticity injection. Apart from being fundamentally interesting, this curious phenomenon may have significant consequences for understanding wide ranging applications of viscoelastic fluids. Examples include viscoelastic flows over rough or deformable surfaces, 12,13 and the generation of viscoelastic interfacial instabilities in stratified flows. 14,15 In addition, these effects may play an important and heretofore overlooked role in multiscale turbulent flows of a) Electronic mail: simon.haward@oist.jp viscoelastic fluids. For example, it has been hypothesized that drag reduction by polymer additive may be linked to the onset of a self-sustaining chaotic state termed "elasto-inertial turbulence" (EIT), which is characterized by the development of near-wall spanwise-coherent vortical structures. 3 The mechanisms driving vorticity amplification at viscoelastic critical layers, namely vorticity-wave propagation along the tensioned streamlines and amplification of the polymer torque, 9 are dominant for parameter values associated with the onset of EIT in nonlinear flows, so these effects may play an important role.
For a laminar Newtonian flow in a plane-Couette geometry with a sinusoidal wavy perturbation on the stationary wall [see schematic diagram in Fig. 1(a)], linear analysis has shown that vorticity perturbations induced by the wavy surface can be classified using two non-dimensionalized parameters: (1) the depth of the flow domain α = kd and (2) the viscous length θ = (ηk 2 /ργ) 1/3 . Here, d is the (dimensional) depth of the domain, k is the wavenumber of the surface undulation, η is the fluid viscosity, ρ is the fluid density, andγ is the uniform velocity gradient (shear rate) across the gap. 9,11 Flow configurations with α 1 are considered "shallow," while those with α 1 are considered "deep." It can be easily shown that θ = (α 2 /Re) 1/3 , where Re = ργd 2 /η is the Reynolds number, which describes the ratio of inertial to viscous forces. Using these two dimensionless parameters, the vorticity perturbations caused by the wavy surface to the Couette base flow fall into one of three regimes [as illustrated in Fig. 1(a)] in the α−θ parameter space, each with characteristic depths of penetration into the flow domain P ω . 9,11 A "shallow viscous" regime occurs for α 1, θ > α; in this regime, the channel depth is small compared to the roughness wavelength so the vorticity perturbation fills the flow domain (P ω ≈ α). The viscous layer is deeper than the flow domain, and so the flow is unaffected by inertia.
A "deep viscous" regime occurs for α 1, θ 1; in this regime, the channel depth is large compared to the roughness wavelength, and the vorticity perturbation decays within the flow domain over approximately one wavelength from the surface (P ω ≈ 1). Again, viscosity is dominant and the flow is unaffected by inertia.
Finally, an "inviscid" regime occurs for the conditions α > θ, θ 1. In this case, the viscous layer is confined within the flow domain and inertial effects are appreciable. The perturbation is confined within the viscous layer (P ω ∼ θ) and is tilted forward due to the inertia. 9,11 Using linear analysis applied to the Oldroyd-B viscoelastic constitutive model, Page and Zaki 9 studied the effects of fluid elasticity on the vorticity field in the same plane-Couette configuration [ Fig. 1(b)]. Somewhat in analogy to Newtonian flows, they found that the vorticity perturbations induced by the wavy surface in viscoelastic flows could also be classified using two non-dimensionalized parameters: (1) the depth of the flow domain α = kd and (2) the depth of the "critical layer" Σ = k 2τη p /ρ. Here, τ is the fluid relaxation time and η p is the polymer contribution to the fluid viscosity. The critical layer is predicted to occur at the depth where vorticity waves traveling along tensioned mean-flow streamlines propagate at the same velocity as the base flow. 9 It can be shown that the critical layer depth is related to the fluid elasticity by 2El, where the elasticity number El = (1 − β)Wi/Re. Here, β = η s /η 0 is the solvent-to-total viscosity ratio (where η s is the solvent viscosity and η 0 = η s + η p is the zero shear viscosity of the fluid), and Wi =γτ is the Weissenberg number, which describes the ratio of elastic to viscous forces. Again in analogy with Newtonian flows, three regimes of vorticity perturbations were found [as illustrated in Fig. 1(b)], 9 each with characteristic depths of penetration P ω .
A "shallow elastic" regime was identified for the conditions α 1, Σ > α. In this regime, the channel depth is small compared to the roughness wavelength, so the vorticity perturbation fills the flow domain (P ω ≈ α). 9,11 However, the fluid elasticity is high enough such that the critical layer is located outside of the flow domain and so has little influence on the flow.
A "deep elastic" regime was identified for the conditions α 1, Σ 1. In this regime, the channel depth is large compared to the roughness wavelength, so the vorticity perturbation decays within the flow domain over approximately one wavelength (P ω ≈ 1). 9,11 However, the critical layer is deeper than the depth of the penetration and so again has little impact on the flow.
While the shallow elastic and deep elastic regimes appear essentially similar to their Newtonian counterparts (shallow viscous and deep viscous, respectively), the third regime predicted for wavy viscoelastic flows contrasts significantly with the third (inviscid) regime observed for Newtonian flow. The so-called transcritical regime was identified for the conditions α > Σ, Σ 1. Here, the critical layer is located both within the flow domain and within the region over which the vorticity perturbation naturally decays. Amplification of the vorticity perturbation takes place at the critical layer, leading to a penetration depth that scales as P ω ∼ Σ. The term "transcritical" is used to describe this regime in order to convey that the base flow velocity varies from "subcritical" near the wavy wall to "supercritical" at depths beyond the critical layer. The generation of vorticity at the critical layer is explained in terms of a kinematic amplification mechanism due to the polymer torque. 9 In recent work, we developed experimental methods to test the analytical predictions for the vorticity perturbations found in shear flows over wavy surfaces. 10,16 For experimental convenience, a Poiseuille flow was employed in favor of plane-Couette; see Fig. 1(c). Five wavy-walled channels were fabricated with a range of 0.6 α 10, and the flow velocimetry in the devices was used to quantify the perturbations based on the measurement of transverse velocity components. Two Newtonian fluids of contrasting viscosity were tested over a range of Re such that the dimensionless viscous length was varied in the range 0.1 θ 30. The excellent agreement between the experimental results and linear theory in the same Poiseuille configuration showed that the devices conform to the constraints of linear assumptions and allowed the first experimental verification of the Newtonian flow phase diagram in the α−θ parameter space. 16 Subsequently, using a weakly viscoelastic polymer solution (El ≈ 0.007) in a deep wavy channel (α ≈ 10, Σ ≈ 1) and by comparison with the linear theory for viscoelastic wavy Poiseuille flows, we demonstrated the first experimental evidence supporting the prediction of the critical layer and hence of the vorticity amplification in the transcritical regime. 10 In the current work, we present an extensive set of experiments utilizing all five of our wavy walled channels (spanning a wide range of α) and employing a variety of well-characterized polymer solutions with a range of rheological properties providing 0.001 < El < 43.6 and hence a range of Σ spanning more than two decades in each channel. We thoroughly investigate the three predicted regimes of wavy viscoelastic shear flow and use our results to construct an experimental phase diagram in the α−Σ parameter space that conforms well to the theoretical prediction.

A. Flow geometries
The five wavy-walled flow channels used in the experiments are described in detail in a previous publication. 16 A schematic representation of the channels, indicating the coordinate system and the important dimensions, is provided in Fig. 1(c). All of the channels have a total depth 2d = 400 µm across y, a spanwise width = 2 mm through z (aspect ratio r a = /2d = 5), and total length L = 30 mm along x. The five channels differ in terms of the wavelength of the wavy surface on the bottom spanwise wall, which is varied in the range 0.125 ≤ λ ≤ 2 mm. The amplitude of the wave is A = 10 µm (i.e. A = d/20 ≤ λ/12.5). Due to the small amplitude, the maximum contraction ratio (i.e. between the widest and narrowest sections of channel) is r c = 410/390 ≈ 1.05, and extensional components in the flow field are considered negligible. Indeed, we have shown that A is small enough to be considered as a linear perturbation to the Poiseuille base state, 16 so we consider the flow kinematics to be entirely dominated by simple shear. For convenience, Table I provides the actual parameters of each of the five channels, including the wavenumber of the surface k and the dimensionless depth α. In this Poiseuille configuration, we consider the depth of the flow domain d to be the region of monotonically increasing flow velocity above the wavy wall (i.e. half the total depth of the channel).  addition of 13 wt. % of a low molecular weight poly(ethylene glycol) (PEG, M = 8 kDa). At this concentration and molecular weight, the aqueous PEG solution behaves as a Newtonian fluid with a viscosity around 8 times that of water. 17 Test fluids are prepared at the desired polymer concentrations 0.001 ≤ c ≤ 0.05 wt. % by dilution of the stock solution in the appropriate quantity of the respective solvent.

B. Test fluids
The viscosities of the test fluids (and the two solvents) were measured at 25 • C using a DHR3 stress-controlled rotational rheometer (TA Instruments, Inc.) fitted with a 40 mm diameter 1 • cone-and-plate fixture and the results are shown in Fig. 2(a). The measured viscosities of the two solvents are η s = 0.87 mPa s (DI water) and η s = 7.72 mPa s (13 wt. % PEG). The zero shear viscosities η 0 of the polymer solutions are all quite close to the viscosity of their respective solvent (i.e. the viscosity ratio β is in general close to unity), and the fluids do not exhibit significant shear thinning.
The densities of the polymer solutions are taken to be equal to those of their solvents, which were measured at 25 • C by weighing various volumes of fluid dispensed from calibrated micropipettes, yielding ρ = 996.9 kg m −3 and ρ = 1015.5 kg m −3 for the pure DI water and the 13 wt. % PEG, respectively.
The relaxation times τ of the polymer solutions were determined by measurements made in a capillary breakup extensional rheometer (CaBER 1, Thermo Haake). 19,20 In this method, an initially cylindrical fluid sample is held by surface tension between two axially aligned circular end plates (diameter D 0 = 6 mm, initial separation l 0 = 1 mm). At time t 0 = −50 ms, the end plates are displaced axially at a relative rate of 0.1 m s −1 to reach a final separation of l 0 = 6 mm at time t = 0 s. A laser micrometer located at the midpoint between the endplates in their final position monitors the diameter D(t) of the liquid bridge as it thins due to capillary forces. For Newtonian fluids, the liquid bridge thins linearly with time at a viscosity-dependent rate. 21 However, an elongational flow is set up within the necking liquid bridge, which for polymer solutions causes alignment of polymer molecules and the generation of elastic forces that retard the capillary-driven thinning process. Under these conditions of elasto-capillarity, the filament diameter decays exponentially with time as D ∼ exp(−t/3τ). 19,20 Figure 2(b) shows the filament diameter as a function of time measured for the various polymer solutions, each showing a clear exponentially decaying region from which the relaxation time is extracted. For the most dilute fluids in the DI water solvent, the filament thinning was too rapid to resolve accurately using the standard CaBER setup. For these fluids, we employed the slow retraction method (SRM) to generate the fluid filament, 18 and we used a high speed Phantom Miro camera (Vision Research, Inc.) rather than a laser micrometer in order to monitor the decay of the filament diameter over time. The results from these SRM tests are shown in the inset to Fig. 2(b), again demonstrating clear elasto-capillary regions from which the relaxation time can be extracted. Note that one particular test fluid (0.01 wt. % PEO4 in DI water) was tested using both standard CaBER and the SRM technique, with reasonably good agreement obtained for the relaxation time in each case (τ = 12 ms by CaBER and τ = 14 ms by SRM). Table II provides the salient rheological parameters of all viscoelastic test fluids employed in the flow experiments.

C. Dimensionless flow parameters
Flow of the test fluids through the wavy channels is driven at a controlled volume flow rate Q using a neMESYS low pressure syringe pump (Cetoni GmbH) fitted with a Hamilton Gastight glass syringe connected to the channel using silicone tubing. The average flow velocity in the channel along the xdirection [ Fig. 1(b)] is then U = Q/2wd. Note that based on the vorticity and polymer torque equations used in the linear theory, we do not expect results to differ depending on whether the flow is driven at controlled rate or controlled pressure. 9,10 As a characteristic shear rate, we take the nominal velocity gradient at the wall for Newtonian flowγ w = 3U/d. Since in the Poiseuille configuration, the total channel depth is 2d, the Reynolds number becomes . (1) The Weissenberg number is given as usual by and hence the elasticity number is The dimensionless viscous length in the wavy Poiseuille configuration is given by and the dimensionless critical layer depth becomes Apart from some numerical factors arising from the definitions of Re andγ w , both θ and Σ in the wavy Poiseuille flow are entirely equivalent to those used in the wavy Couette case. A difference between the two configurations arises in the classification of deep and shallow channels. Experiments and linear theory with Newtonian fluids in wavy Poiseuille flow have shown that channels with α π are to be considered shallow and those with α π should be considered deep. 16 As shown in Table II, by virtue of using two solvents of contrasting viscosities and by varying the polymer molecular weight and concentration, the viscoelastic test fluids span a wide range of elasticity 0.001 ≤ El ≤ 43.6. However, the solvent-to-total viscosity ratio is kept high ( β ≥ 0.69), thereby avoiding complications associated with shear thinning (i.e. rate dependence of τ and η, hence Re, Wi, θ and Σ, and rate dependent base flow profiles in the channels). For a given fluid and elasticity number, the value of Σ depends on the wavelength

Solvent
Polymer of the surface undulation of the channel. Table II provides the values of Σ for each fluid at each value of α. It can be seen by inspection of Eq. (5) that El ≤ 0.75 for transcritical phenomena to be observed (since it is required that Σ/α ≤ 1). In Table II, Σ-α combinations that correspond to expectations for transcritical behavior in wavy Poiseuille flow are identified by bold text. The remaining Σ-α combinations are listed in either bold-italic or italic text in order to identify expected shallow elastic (α < π) or deep elastic (α ≥ π) behavior, respectively.

D. Micro-particle image velocimetry (µ-PIV)
Perturbations to the Poiseuille base flow state caused by the wavy surfaces are characterized by measuring velocity vector fields within the wavy channels using a micro-particle image velocimetry (µ-PIV) system (TSI, Inc.). The fluids are seeded with 1 µm diameter fluorescent microparticles (nile red FluoSpheres, Life Technologies) with excitation and emission wavelengths of 535 and 575 nm, respectively. The flow channel is placed on the imaging stage of an inverted microscope (Nikon Eclipse Ti), and the mid-plane of the device (z = /2) is brought into focus. The measurement section of the test channel commences at a distance of ∆x ≈ 18 mm (∆x/2d ≈ 45) downstream of the inlet, allowing adequate distance for the flow to become fully developed. Depending on the particular flow channel being studied, objective lenses are selected in order to maximize the spatial resolution of the measurement while simultaneously allowing at least one full wavelength of the undulation, and the full depth of the flow domain, to be visualized. Accordingly, for Devices 1 and 2 (with the longest wavelengths), a 4× NA = 0.13 objective is required; for Device 3 a 5× NA = 0.15 objective is used; and for Devices 4 and 5 (with the shortest wavelengths), a 10× NA = 0.3 objective is employed. The corresponding measurement width, over which microparticles contribute to the determination of velocity vectors, is δz m ≈ 142, 109 and 31 µm for the 4×, 5× and 10× lenses, respectively. 22 Even in the worst case of the 4× objective lens, δz m is only ≈7% of the channel width, so there should not be a significant measurement error given the high channel aspect ratios and therefore the expected uniform flow along the z-direction close to the mid-plane. The fluid is illuminated through the microscope objective by a dual-pulsed Nd:YLF laser at 527 nm, exciting the fluorescent particles. The light emitted by the particles is imaged through the same objective lens, passed through a G2-A epifluorescent filter, and projected onto the sensor array of a high speed Phantom Miro camera (Vision Research, Inc.), operating in the frame-straddling mode. Images are captured in pairs, in synchronicity with the pairs of laser pulses. The time between laser pulses ∆t is chosen such that the maximum particle displacement between images in each pair is around 8 pixels, optimal for the standard cross-correlation PIV algorithm used to obtain velocity vectors based on a square grid of 32 × 32 pixel interrogation areas. Since in this work we are only interested in examining steady flows, at each flow rate imposed in each device, vector fields are ensemble-averaged over the data from up to 50 image pairs. In cases where more than one wave of the surface is visualized, the fully developed nature of the flow over the measurement section is clearly evident from the cyclic variation in the velocity field from wave to wave. In such cases, in order to further reduce noise, the data are also averaged over successive waves.
Velocity vector fields measured in the x-y plane at z = /2 have components u and v in the xand y-directions, respectively. Since for the Poiseuille base flow (i.e. if there were no wavy surface) v Pois ≡ 0, our measurement of v allows a simple and direct determination of the perturbation component v = v − v Pois ≡ v. As we have shown, this v component is sufficient to characterize the perturbation, without the need to resort to any assumptions or approximations that are necessary in order to extract the u perturbation component from the data and thus to evaluate the full vorticity perturbation ω = ∂v /∂x − ∂u /∂y. 16 Hence, in this work (as in our previous work), we use v rather than ω to evaluate an experimental measure of the penetration depth of the disturbance. For this measure, which we denote P v , we employ a criterion similar to that proposed by Page and Zaki: 9,10 where |v (y)| is averaged over the full range of x in the field of view.

III. RESULTS
As predicted by linear theory for viscoelastic wavy plane-Couette flow, 9 three broad regimes of viscoelastic flow can also be observed in the flow patterns measured experimentally in the wavy-walled channels. These three regimes, with parameters α and Σ corresponding to the expectations of shallow elastic, deep elastic and transcritical type behavior, are illustrated in Figs. 3(a), 3(b), and 3(c), respectively. As will be shown in detail subsequently, the flow patterns in the shallow and deep elastic regimes are essentially Newtonian-like in appearance; the perturbation extends straight along the ydirection and penetrates all the way through the flow domain in the shallow channel [Device 2, Fig. 3(a)], but decays over a dimensionless depth of ≈3 in the deep channel [Device 4, Fig. 3(b)]. In the transcritical regime [ Fig. 3(c), also shown for Device 4], the perturbation is clearly intensified compared with that in Fig. 3(b), is markedly tilted forward and also penetrates more deeply into the channel.
In Fig. 4 we present the penetration depth measurements [evaluated according to Eq. (6)] made in all five wavy wall test channels (0.2π ≤ α ≤ 3.2π) with all the test fluids (i.e. for a range of Σ). Following the analysis performed by Page and Zaki, 9 the data are plotted for a relatively high value of the Weissenberg number Wi ≈ 40. The reason for this particular choice of Wi is that it is the highest value that could be achieved with all of the fluids in all of the channels. For fluids with short relaxation times, obtaining higher Wi was not feasible due to the high flow velocities required and constraints on the µ-PIV imaging rate. On the other hand, for more elastic fluids with longer relaxation times, higher Wi flows resulted in elasticity-driven recirculations in the troughs of the wavy surface (particularly in channels with shorter wavelengths). These recirculations can be seen by observing the motion of individual µ-PIV tracer particles at different times, as illustrated for one particular case in Fig. 5 (multimedia view). Figure 4(a) shows P v as a function of Σ. Here it can be seen that for the higher values of Σ in each channel (i.e. towards the elastic limit), a region of approximately constant P v is reached. This high Σ plateau appears to reach an asymptotic value P v ≈ 3 (or π, marked by the horizontal dashed line) for α ≥ 5, thus indicating deep elastic behavior. Figure 4(b) shows P v /α as a function of Σ/α. In this plot, it can be seen that the plateau of Σ/α asymptotes to a value of approximately 1 (i.e. Σ = α, marked by the horizontal dashed line) for α ≤ 1, thus indicating shallow elastic behavior. These two regimes are highly analogous to deep viscous behavior and shallow viscous behavior reported previously for Newtonian fluids. 16 For deep channels only (i.e. α ≥ π), the plots in Fig. 4 show a change in behavior for Σ π, Σ/α 1, marking the expected boundary between the transcritical and deep elastic regimes (indicated by the vertical dotted lines in the two plots). At the change in regime between deep elastic and transcritical behavior, we note several similarities between our experimental data and the theoretical plots presented by Page and Zaki (in their case for Wi = 60). 9 In particular, in the deepest channel (α = 3.2π), there is a significant overshoot in P v as Σ increases and the flow regime transitions into the deep elastic. Also, in the shallower (α = 1.6π) channel, the plateau region of P v extends further into the transcritical regime (i.e. to lower Σ). However, the channel with α = π exhibits contrasting behavior, and no clear or general scaling between Σ and P v is evident for the three deep channels in the transcritical region. We note here that we do not expect to see evidence of transcritical behavior in the shallow channels (α < π) since the perturbation spans the entire flow domain under all imposed conditions. For the same reason, we also could not observe inviscid type behavior in shallow channels in our experiments with Newtonian fluids. 16 Focusing only on deep channels (α ≥ π), we can obtain some insight into the absence of a transcritical scaling for P v with Σ in Fig. 4 by examining how the penetration depth for the viscoelastic fluids depends on the imposed flow conditions. In Figs. 6(a)-6(c) we present the penetration depth as a function of the viscous length θ made for all the viscoelastic test fluids and shown in comparison with the results obtained for Newtonian flow. The Newtonian behavior is indicated by the dashed black curves, while the colored curves show the responses of the various polymer solutions. The solid colored data points mark the conditions where Wi = 40 (i.e. the data shown previously in Fig. 4). These plots clarify several key points. At low flow rates (i.e. low Wi, low Re and hence high θ), the polymer solutions all appear Newtonian-like, with the penetration depth lying on the high-θ plateau. The polymer solution responses closely follow the Newtonian curves as θ is decreased (i.e. as Wi and Re increase) until a sudden departure from Newtonian behavior is seen at a Σ-dependent value of θ, where P v dramatically increases. The most significant excursions from Newtonian behavior are seen for the polymer solutions that follow the Newtonian curves into the sloping (inviscid) region, which are in general the same fluids expected to show transcritical behavior. This indicates the importance of the combination of both inertia and elasticity for the clear manifestation of the transcritical regime.
It is evident from Figs. 6(a)-6(c) that by selecting penetration depth measurements from all of the test fluids and channels at a single fixed value of the Weissenberg number (as in Fig. 4 at Wi = 40), the different regions of the phase diagram are not being compared under equivalent conditions in all of the channels. It is clear in Fig. 6(a) that at Wi = 40 in the α = π channel, the response is close to Newtonian for all Σ. On the other hand, at Wi = 40 in the α = 1.6π and α = 3.2π channels [Figs. 6(b) and 6(c), respectively], a significantly non-Newtonian response is recorded. These observations explain the absence of a clear and general scaling between P v and Σ observed within the transcritical regime for the data shown in Fig. 4.

A. Shallow elastic and deep elastic regimes
In this section, we provide additional details of the appearance of the v perturbation patterns observed within the shallow The top panel of Fig. 7 shows v perturbation patterns observed in the shallow elastic regime in Device 1 (α = 0.2π). In Fig. 7(a) we show the result obtained for a Newtonian fluid (13 wt. % PEG) in the shallow viscous regime (α < π, θ > α).

B. Transcritical regime
In this section, we provide additional details on the observations of flow patterns within the transcritical flow regime (α > Σ, Σ < π) in the deep wavy channels (α ≥ π).
The top panel of Fig. 8 shows details of v perturbation patterns observed in the transcritical regime in Device 3 (α = π), as the Weissenberg and Reynolds numbers are progressively increased for a fixed Σ = 0.94. At Wi = 40 [ Fig. 8(a)], the response is essentially that of a Newtonian fluid in the inviscid regime [as established in Fig. 6(a)]. Due to the small viscous length (θ = 0.42), the perturbation is localized close to the wavy wall and is notably tilted forward by the shear. As the Weissenberg number is increased to Wi = 71.4 and Wi = 127.0 in Figs. 8(b) and 8(c), respectively, the perturbation remains tilted forward but also increases substantially in magnitude and penetrates progressively more deeply into the channel (even though the viscous length θ actually decreases).
The middle panel of Fig. 8 shows details of v perturbation patterns observed in the transcritical regime in Device 4 (α = 1.6π) at a fixed Σ = 1.51. As the Weissenberg and Reynolds numbers are progressively increased through Figs. 8(d)-8(f), similar observations can be made as described above for Device 3.
The bottom panel of Fig. 8 shows details of v perturbation patterns observed in the transcritical regime in Device 5 (α = 3.2π). Here, the Weissenberg number is held fixed at Wi ≈ 40 as the value of Σ is progressively increased through Figs. 8(g)-8(i). At low Σ = 0.39 [ Fig. 8(g)], the perturbation is quite intense, tilted forward and localized close to the wavy surface due to a combination of both the small viscous length (θ = 0.55) and the low depth of the critical layer (Σ = 0.39). In Fig. 8(h), the viscous length remains quite small (θ = 0.80), but the increased depth of the critical layer (Σ = 1.01) results in the perturbation penetrating more deeply into the channel while remaining tilted forward. Finally, in Fig. 8(i), θ = 0.92 and Σ = 3.02; now the inertia is becoming less significant and the critical layer is located near the deep limit. The flow is transitioning from the transcritical into the deep elastic regime; the perturbation consequently decreases in intensity and becomes straighter and more aligned along the y-direction.

C. Construction of the "phase diagram"
Using the results of our experiments in five different wavy microchannels (providing a range of 0.2π ≤ α ≤ 3.2π) using nine distinct viscoelastic fluids (providing a range of 0.001 ≤ El ≤ 43.6, hence a range of critical layer depths Σ), we can construct an experimental phase diagram in the α−Σ parameter space delineating the flow regimes observed for viscoelastic wavy Poiseuille flow. This phase diagram is shown in Fig. 9 and is in broad agreement with our stated a priori expectations based on the predictions from linear theory using the Oldroyd-B model in a wavy plane-Couette flow 9 and our earlier experimental work with Newtonian fluids in wavy Poiseuille flow. 16 The theory, applied to plane-Couette flow, predicts transition between the shallow and deep regimes at α = 1 and between the deep elastic and transcritical regimes at Σ = 1. The experiments, conducted in Poiseuille flow, find these two boundaries at α = π and Σ = π, respectively. Given the assumptions involved in the theory (i.e. Oldroyd-B model, linear perturbation modelled by wall slip), the relative complexity of the experiments (i.e. real viscoelastic fluids with polydisperse polymer samples, real wavy walls, and measurement uncertainty) and the contrasting flow configurations, the level of agreement is remarkable.
In the shallow elastic and deep elastic regimes, the penetration is exactly as expected (P v ≈ α and P v ≈ π, respectively). In the transcritical regime, linear theory predicts the FIG. 9. Experimental phase diagram in the α−Σ parameter space, delineating the three regimes observed for viscoelastic wavy Poiseuille flow. penetration depth (based on the full vorticity perturbation) to scale as P ω ∼ Σ. 9 In our case, using a penetration depth based on v , we divide the transcritical regime into two subregimes (as we also subdivided the inviscid regime for Newtonian flows 16 ). In the shallow transcritical subregime P v ≈ α (effectively indistinguishable from the shallow elastic behavior). The transition between the deep elastic regime and the deep transcritical subregime is clearly apparent in the observed flow patterns through a marked increase in intensity, a forward tilt of the v perturbation and an increase in the penetration depth. However, in this work, we find no clear scaling for P v with Σ, so we are unable to provide an experimental verification for the theoretical scaling.

IV. DISCUSSION AND CONCLUSIONS
In this work we used a series of five wavy-walled channels and nine polymer solutions of distinct rheology in order to perform an experimental investigation of viscoelastic shear flows over wavy surfaces. The five channels are all similar except for having distinct ratios of depth to undulation wavelength, α. The channels span the shallow α < π to deep α ≥ π regimes and have been validated against linear theory in previous work using Newtonian fluids. The polymeric fluids have been formulated to span a wide range of elasticity numbers 0.001 ≤ El ≤ 43.6, hence a range of dimensionless critical-layer depths Σ ∼ √ El spanning two orders of magnitude in each channel. The flow patterns observed in our experiments show clear evidence of the three flow regimes (shallow elastic, deep elastic, and transcritical) predicted by linear theory, which fit into the predicted phase diagram in the α−Σ parameter space. This constitutes the first experimental verification of the predicted phase diagram. This provides strong evidence for the existence of the predicted "critical layer" where amplification of the perturbation occurs away from the site of vorticity injection (i.e. away from the wavy surface). In the transcritical regime (which is characterized by a combination of inertia and elasticity), this amplification results in a non-Newtonian increase in the penetration depth above that observed for Newtonian fluids under comparable inertial conditions.
The wavy-wall model studied here is a canonical surrogate to a number of important flows, e.g. the motion of a viscoelastic fluid near a deformable substrate or two-fluid interface, the onset and structure of inertial instabilities, or the flow dynamics in the vicinity of differently sized eddies in elasto-inertial and drag-reduced turbulent flows.