2.1. Periodically constricted tube

Figure 1(a) demonstrates a two-dimensional (2-D) cross-section of the flow setup used for the experiments. The flow consists of several stages. Each stage is detailed starting from the farthest upstream location on the left-hand side of figure 1(a) and moving downstream or to the right. The entrance region was of radius $R_o = 1.07$ mm, and was 68$R_o$ in length – a sufficient length to ensure a fully-developed Poiseuille flow entered the sections to follow. Farther downstream of the entrance, the flow entered the PCT, where the radius of the tube wall, $R_w$, varied sinusoidally along the streamwise direction, x, according to

(2.1)\begin{equation} R_w = R_{o} + \epsilon \left(\cos \left( \frac{2 {\rm \pi}x}{\lambda} \right) -1 \right), \end{equation}

where the sinusoidal amplitude of wall radius was $\epsilon = 0.14$ mm, and the wavelength, $\lambda$, was 4.7 mm. The maximum radius of the PCT was $R_o$, the minimum radius $R_i$ was 0.79 mm and the average radius $R$ was 0.93 mm. The length of the PCT section was $7\lambda$. Figure 1(b) demonstrates a magnified depiction of the PCT portion of the test section. The cylindrical coordinate system is shown for reference in figure 1(b). The streamwise, radial and azimuthal directions are denoted as x, r and $\theta$, respectively. The radius of the tube downstream of the PCT returned to $R_o$ for a length of $28R_o$. The radius then gradually increased to 5.5 mm via a 3-degree axisymmetric conical expansion farther downstream from the PCT.

Figure 1. Two-dimensional schematic of the (a) complete acrylic test section and (b) the periodically constricted tube.

The three-dimensional (3-D) axisymmetric tube was built from two halves of 12.7 mm thick acrylic. The radial profile shown in figure 1 was cut into the two acrylic halves using a computer numerical control router with a precision ball nose end mill. The scallop height – the height of the surface imperfections caused by the curvature and step length of the ball nose tool – was less than 1 $\mathrm {\mu }$m or 0.1 % of $R_{i}$. The two halves were pressed together to form the 3-D axisymmetric tube without using any adhesive. Custom milled steel flanges with lag bolts and nuts were used to apply sufficient compression to the two halves, such that fluid did not expel out the sides of the test section.

Fluid entered the test section from a straight, 1.2 m long stainless-steel tube with an inner radius of $R_o$ that was face-sealed to the left-hand side of the test section shown in figure 1(a). Fluid that exited the test section entered a 0.3 m long stainless-steel tube with an inner radius of 5.5 mm that was joined to the downstream portion of the test section. Fluid temperature was monitored using a K-type thermocouple and a data logger (HH506, Omega Engineering). The average fluid temperature of all experiments was $20.1\,^{\circ }\mathrm {C} \pm 0.2\,^{\circ }{\rm C}$. A syringe pump (Legacy 200, KD Scientific Inc.) with an accuracy of $\pm$1 % was used to propel fluid through the flow facility. Glass syringes (Micro-Mate, Popper & Sons Inc.) with 10 and 30 ml volumes were equipped in the syringe pump; the choice in the syringe volume depended on the required volumetric flow rate $Q$. Flexible PVC tube with an inner radius of 3.18 mm connected the syringe to the 1.2 m long stainless-steel tube. Five flow rates were considered for each Newtonian and non-Newtonian fluid: 1, 3, 6, 9 and 12 ml min$^{-1}$.

The Reynolds number was defined based on $Re = 2 \langle U_0 \rangle R/\nu _w$, where $\nu _w = \eta _w/\rho$ is the kinematic wall viscosity, $\eta _w$ is the dynamic wall viscosity and $\rho$ is the density. This definition of $Re$ is similar to that used by Ahrens, Yoo & Joseph (Reference Ahrens, Yoo and Joseph1987), where the flow of viscoelastic fluids was simulated through a wavy-walled tube. Within the PCT, the centreline velocity $U_0$ oscillates with respect to $x$. As such, the average centreline velocity $\langle U_0 \rangle$ along $x$ was determined from flow measurements in the PCT. Here, angle brackets $\langle \cdots \rangle$ are used to denote spatial averaging along the $x$ direction. Within the straight-walled development region, where $R_w = R_o$, $U_0$ does not vary along $x$ and the Reynolds number is defined as $Re_d = 2 U_0 R_o / \nu _w$. For the flow of water, $Re_d$ was between 13 and 184, which was laminar. The wall shear rate within the straight-walled development region can be derived from a differentiation of the parabolic Poiseuille profile for Newtonian fluids, $\dot {\gamma }_w = 2 U_0/ R_o$. Within the PCT, the fluid is subjected to a combination of shear and extensional deformation. A characteristic near-wall shear rate within the PCT was defined similar to the straight-walled section as $\dot {\gamma }_w = 2\langle U_0 \rangle /R$. A characteristic extensional strain rate $\dot {\varepsilon }$ was defined as the range in $U_0$ (maximum subtracted by minimum) divided by $\lambda /2$. In the present investigation, $\dot {\gamma }_{w}$ was between 13 and 300 s$^{-1}$ and $\dot {\varepsilon }$ was between 2 and 58 s$^{-1}$ depending on the fluid and $Re$. The dynamic wall viscosity $\eta _w$ was derived from shear rheograms for non-Newtonian fluids and using $\dot {\gamma }_w$, as detailed in § 2.3. For water, $\eta _w = \eta _s$, where $\eta _s$ is the viscosity of the solvent and was considered to be 1.00 mPa s according to Cheng (Reference Cheng2008).

2.2. Test fluids

Three non-Newtonian additives, with drag-reducing capabilities (Warwaruk & Ghaemi Reference Warwaruk and Ghaemi2021), were chosen for the experiments: a flexible polymer, a rigid biopolymer and a cationic surfactant. Additives in their solid powder form were weighed using a digital scale (Explorer Analytical, OHAUS Corporation) with a 1 mg resolution. Solid powders were then gradually added to 15 l of distilled water and agitated for 8 h using a stand mixer equipped with a 100 mm diameter impeller (Model 1750, Arrow Engineering Mixing Products). After mixing, the aqueous non-Newtonian solutions were left to rest for 16 h. Fluid samples were then collected for rheology measurements and experiments in the PCT. Experiments in the PCT for the non-Newtonian fluids were conducted at the same flow rates as water, $Q$ between 1 and 12 ml min$^{-1}$. For some of the fluids, their viscosities were different than water, therefore, a similar $Q$ did not constitute a similar $Re$.

The flexible polymer used in the present experiment was polyacrylamide (PAM) from a sample batch contributed by SNF Floerger (6030S, molecular weight of 30–35 Mg mol$^{-1}$). The rigid biopolymer was xanthan gum (XG) (43708, MilliporeSigma). Both polymers, PAM and XG, have been readily used in various experimental investigations involving rheology and turbulent drag reduction (Escudier et al. Reference Escudier, Presti and Smith1999; Mohammadtabar et al. Reference Mohammadtabar, Sanders and Ghaemi2020; Warwaruk & Ghaemi Reference Warwaruk and Ghaemi2021). Cationic surfactants are quarternary ammonium salts of the form C$_n$H$_{2n+1}$N$^+$(CH$_3$)$^3$Cl, where $n$ is an integer, generally between 12 and 18 (Qi & Zakin Reference Qi and Zakin2002). When paired with a counterion, such as sodium salicylate (NaSal), the molecules combine to form complex micellar structures (Bewersdorff & Ohlendorf Reference Bewersdorff and Ohlendorf1988; Zhang et al. Reference Zhang, Schmidt, Talmon and Zakin2005). We found in previous experimental campaigns that trimethyltetradecylammonium chloride ($n = 14$) (T0926, Tokyo Chemical Industry Co., Ltd.) combined with NaSal (71945, MilliporeSigma) at a molar ratio of 1:2, produced considerable amounts of drag reduction (Warwaruk & Ghaemi Reference Warwaruk and Ghaemi2021). Therefore, the same additives and molar ratios were used in the present investigation. Much like other investigations, we abbreviate and refer to the surfactant additive as TTAC for the remainder of the manuscript (Roelants & De Schryver Reference Roelants and De Schryver1987). A parametric sweep of five concentrations were considered for each additive (i.e. PAM, XG and TTAC). The concentrations, $c$, were the same for all additives: 0.01 %, 0.02 %, 0.03 %, 0.04 % and 0.05 %.

2.3. Fluid rheology

Steady and dynamic shear rheology measurements were performed using a controlled-stress single-head torsional rheometer (HR-2, TA Instruments). A single-gap concentric cylinder was used for all viscosity measurements, both steady and dynamic. The radius of the inner rotating cylinder $R_{min}$ was 14 mm, while the radius of the outer fixed cylinder $R_{max}$ was 15.2 mm. The immersion height $L$ was 42.04 mm. Steady shear viscosity measurements involved a logarithmic sweep in the shear rate $\dot {\gamma }$ from 0.1 to 1000 s$^{-1}$ with 10 data points per decade, and the corresponding stress $\tau$ was monitored. Note that the rotational velocity in rad s$^{-1}$, $\varOmega$, can be converted to $\dot {\gamma }$ using $\dot {\gamma } = F_{\gamma } \varOmega$, where $F_{\gamma } = R_{max}/(R_{max}-R_{min})$ is the strain coefficient derived for Taylor–Couette flow (Barnes, Hutton & Walters Reference Barnes, Hutton and Walters1989; Ewoldt, Johnston & Caretta Reference Ewoldt, Johnston and Caretta2015). Similarly, the torque measurements, $T$, can be converted to stress, $\tau$, using $\tau = F_{\tau } T$, where $F_{\tau } = (2 {\rm \pi}R_o^2 L)^{-1}$ is the stress coefficient. The shear viscosity was derived based on $\eta = \tau / \dot {\gamma } = (F_{\tau }/F_{\gamma })(T/\varOmega )$ (Barnes et al. Reference Barnes, Hutton and Walters1989; Ewoldt et al. Reference Ewoldt, Johnston and Caretta2015). The maximum shear rate limit of the steady shear viscosity measurements was determined based on the Taylor number limitation, $Ta = \rho ^2 \varOmega ^2(R_{max}-R_{min})^3R_{min} / \eta ^2 < 1700$ (Ewoldt et al. Reference Ewoldt, Johnston and Caretta2015). Usually, the minimum shear rate limit can be determined from the lower torque limit prescribed by the manufacturer of the rheometer. The lower limit in $T$ provided by TA instrument was 10 nN m, or $\tau = 0.2$ mPa. In practice, we found that the lower limit for steady shear viscosity measurements was higher, $T = 100$ nN m, or $\tau =2$ mPa. Lastly, a power-law model was fit to shear rheograms for fluids that exhibited shear thinning tendencies. The power law was of the form:

(2.2)\begin{equation} \eta = K \dot{\gamma}^{n-1}, \end{equation}

where $K$ is called the consistency and $n$ is the flow index. Fits were performed on profiles of $\eta (\dot {\gamma })$ with $\tau >2$ mPa and $Ta < 1700$ using nonlinear least square regression. The values of $K$ and $n$ for the solutions that were shear-thinning are reported in Appendix A. Recall from § 2.1 that $Re = \rho{2}\langle U_0 \rangle R/\eta _w$, where $\eta _w$ is the viscosity evaluated at $\dot {\gamma }_w = 2\langle U_0 \rangle /R$. In other words, using (2.2) produces $\eta _w = K \dot {\gamma }_w^{n-1} = K 2^{n-1}\langle U_0 \rangle ^{n-1}R^{1-n}$, and the Reynolds number in the PCT flow can be equally represented as $Re = \rho 2^{2-n} \langle U_0 \rangle ^{2-n} R^n / K$ for shear-thinning fluids.

Dynamic shear viscosity measurements were also performed on select PAM and XG solutions. Limitations of the rheometer made performing these measurements only possible for high-concentration solutions. The details and results of the dynamic shear viscosity measurements are provided in Appendix B.

The extensional rheology of the solutions was evaluated using a custom dripping-onto-substrate (DoS) apparatus, depicted in figure 2(a). In this measurement technique, a small droplet was discharged from a blunt-end nozzle with a diameter $D_0$ of 1.27 mm. A syringe pump (Legacy 200, KD Scientific Inc.) was used to expel the droplet from the nozzle at a rate of 0.02 ml min$^{-1}$. Pumping was terminated once the droplet made contact with a glass substrate that was situated $3D_0$ or 3.81 mm below the blunt-end of the nozzle outlet. An apparatus with similar features was used by Dinic et al. (Reference Dinic, Zhang, Jimenez and Sharma2015), Dinic, Jimenez & Sharma (Reference Dinic, Jimenez and Sharma2017) and Zhang & Calabrese (Reference Zhang and Calabrese2022). After the droplet made contact with the substrate, a liquid bridge was formed between the nozzle outlet and the substrate. The diameter of the liquid bridge $D_{min}$ decayed rapidly due to capillary forces. Images of the liquid bridge were collected using a high-speed camera (v611, Vision Research) and back-lit illumination from a light-emitting diode (LED). Figure 2(b) shows a sample image of the liquid bridge for the PAM solution with $c = 0.05\,\%$. The camera had a $1280\times 800$ pixel complementary metal-oxide semiconductor sensor with pixels that were $20\times 20$ $\mathrm {\mu }$m$^2$ in size and had a bit-depth of 12 bit. A zoom lens was used to achieve a magnification of 3.8 and a scale of 5.16 $\mathrm {\mu }$m pixel$^{-1}$. Images were collected at an acquisition rate of 2 kHz. The minimum diameter $D_{min}$ of the liquid bridge was determined using a script developed in MATLAB (Mathworks Inc.).

Figure 2. (a) Isometric view of a 3-D model depicting the DoS setup. (b) A sample image taken for PAM with $c=0.05\,\%$ in elastocapillary thinning.

The pinch-off dynamics of the liquid bridge in the DoS apparatus depends on forces attributed to inertia, surface tension, viscosity and elasticity (Dinic et al. Reference Dinic, Jimenez and Sharma2017). The Ohnesorge number, $Oh = t_v/t_R$, relates the time scale associated with viscous forces to the Rayleigh time $t_R$, which pertains to surface tension and inertial forces. Here, $t_v = \eta D_0 / 2\sigma$ is the characteristic time scale of viscocapillary thinning, $t_R = (\rho D_0^3/8\sigma )^{1/2}$, and $\sigma$ is the surface tension. Low-viscosity fluids typically have $Oh < 1$ and a necking process dominated by inertial and capillary forces. In this regime, inertiocapillary (IC) thinning is described by a 2/3 power law:

(2.3)\begin{equation} \frac{D_{min}(t)}{D_0} = \alpha \left( \frac{t_b-t}{t_R} \right)^{2/3}, \end{equation}

where $t_b$ is the filament break-up time, and $\alpha$ is a multiplicative pre-factor between 0.4 and 1 (Zhang & Calabrese Reference Zhang and Calabrese2022). If $Oh>1$, viscous forces are significant, and the evolution of $D_{min}$ is described by viscocapillary thinning, $D_{min}(t)/D_0 = 0.0709(t_b-t)/t_v$ (McKinley & Tripathi Reference McKinley and Tripathi2000). For elastic fluids, the Deborah number, $De = t_e / t_R$, describes the ratio of the extensional relaxation time $t_e$ and the Rayleigh time (Tirtaatmadja, McKinley & Cooper-White Reference Tirtaatmadja, McKinley and Cooper-White2006). If $De > 1$, the necking process is dominated by elastic and capillary forces. This elastocapillary (EC) regime is described by

(2.4)\begin{equation} \frac{D_{min}(t)}{D_0} = A \exp \left(-\frac{t}{3 t_e} \right), \end{equation}

where $A = (GD_0/2\sigma )^{1/3}$. The fluids in the present investigation exhibit thinning in an IC ($Oh<1$) or EC regime ($De>1$). Nonlinear least-square regression was used to establish $t_e$ for fluids that exhibited EC thinning using measurements of $D_{min}$ and (2.4). Values of $t_e$ are listed in Appendix A.

2.4. Particle shadow velocimetry

Particle image velocimetry (PIV) with backlight illumination, denoted as particle shadow velocimetry (PSV), was used to measure the velocity of the fluid within the test section. In PSV, the thickness of the measurement domain is driven largely by the depth of focus (DOF) of the imaging system. Provided a sufficient magnification and lens aperture, images can be acquired with a thin focal plan that enables 2-D planar flow measurements along a select and narrow region of interest (Santiago et al. Reference Santiago, Wereley, Meinhart, Beebe and Adrian1998; Estevadeordal & Goss Reference Estevadeordal and Goss2006; Khodaparast et al. Reference Khodaparast, Borhani, Tagliabue and Thome2013).

The PSV system consisted of a digital camera (Imager Pro X, LaVision GmbH) with a $2048 \times 2048$ pixel charged-coupled device sensor. Each pixel was $7.4 \times 7.4$ $\mathrm {\mu }$m$^2$ in size and had a 14-bit digital resolution. A Nikon lens with a focal length of $f=105$ mm was equipped to the camera with an aperture diameter of $f/2.8$. The camera focus was adjusted such that images were focused on the radial mid-span of the test section. Two fields of view (FOVs) were considered, as shown in figure 3(a). The first FOV, i.e. FOV1, considered the entrance or development region immediately upstream of the PCT, as demonstrated in the left-hand side of figure 3(a). The FOV1 captured the complete tube radius, $R_o$, and approximately 3$\lambda$ along the $x$ direction and immediately upstream of the first oscillation in the PCT. Only the Newtonian flow of water was considered in FOV1. The objective was to determine if the flow entering the PCT was a fully-developed laminar Poiseuille flow. Experimental results for FOV1 are presented separately in Appendix C. The second field of view, FOV2, measured the velocity between the second and fifth oscillation of the PCT, that is, from $x \approx 2\lambda$ to $5\lambda$. For FOV2, flows of the three non-Newtonian fluids through the PCT were measured. Both FOVs were approximately the same size, $(\Delta x, \Delta r) = 3.24 \times 14.1$ mm$^2$, with a scale of 6.88 $\mathrm {\mu }$m pixel$^{-1}$ after the sensor was cropped to remove unnecessary data for $r/R_o>1$. The magnification was 1.07 and the DOF was 87 $\mathrm {\mu }$m, which was approximately 10 % the minimum radius in the PCT, $R_{i}$.

Figure 3. (a) A two-dimensional schematic showing the different PSV fields of view. (b) Sample PSV image (TTAC at $c = 0.04\,\%$) for FOV2. (c) An enhanced version of the sample image in panel (b) for TTAC at $c=0.04\,\%$ and FOV2.

Backlight illumination of the PIV recordings was achieved using a 15 mJ pulse$^{-1}$ Nd:YAG laser (Solo I-15, New Wave Research Inc.) equipped with a diffuser. A diffuser expanded the laser beam, made the incident light incoherent and changed the wavelength to 610 nm using fluorescent disks. A programmable timing unit (PTU-9, LaVision GmbH) and DaVis 8.4 software (LaVision GmbH) were used to synchronize the camera and laser. Silver coated hollow glass spheres, with diameter $d_p = 10$ $\mathrm {\mu }$m, were used as tracer particles in the flow (S-HGS-10, Dantec Dynamics). These particles were opaque, which was ideal for projecting a shadow on the camera in backlight illumination. The density of the particles, $\rho _p$, was 1400 kg m$^{-3}$. As a result the particle response time, $t_p = \rho _p d_p^2/18 \eta _s$, and particle settling velocity, $u_p = (\rho _p - \rho )d_p^2g/18\eta _s$, could be established. Here, $g$ is the gravitational acceleration. The particle response time, $t_p$, was 7.8 $\mathrm {\mu }$s and the particle settling velocity, $u_p$, was 21.8 $\mathrm {\mu }$m s$^{-1}$. We estimated the Stokes number to be $St = t_p \dot {\gamma }_w$, and the Froude number to be $Fr = 2u_p/U_0$. The largest $St$ was 0.003 and the largest $Fr$ was 0.005, depending on $Q$. Both the Stokes and Froude numbers are small (less than 0.1) and errors attributed to particle inertia and particle settling are negligible.

For FOV1, five sets of measurements were performed for water, each for the different values of $Q$ listed in § 2.1. The results for FOV1 are presented in Appendix C. The measurements of velocity within the entrance region show good agreement with the theoretical expectations for all values of $Q$, providing good confidence in PSV to produce reasonable measurements. For FOV2, measurements were performed for three different non-Newtonian fluids, each having five different concentrations, and five flow rates $Q$ (75 datasets in total). Additionally, five measurements were performed for distilled water at FOV2 for each value of $Q$. Each dataset consisted of 600 pairs of double-frame images recorded at an acquisition frequency of 7.3 Hz. A sample image of the first frame for TTAC at a mass concentration of 0.04 % is shown in figure 3(b). The time delay, $\Delta t$, between image frames was between 500 and 7000 $\mathrm {\mu }$s depending on the value of $Q$, such that the maximum particle displacement between the image frames was no greater than 15 pixel.

Image processing was performed using DaVis 8.4 software (LaVision Gmbh). First, the images were inverted; the intensity signal at each pixel was subtracted from a constant intensity value. Next, the minimum intensity within each pixel and along the complete image ensemble was determined and subtracted from all images in each dataset. Third, the intensity signals at each pixel were normalized by the average intensity of the ensemble. A sample image (TTAC at a mass concentration of 0.04 %) after performing the previously detailed processing steps can be seen in figure 3(c). Compared to the native image, seen in figure 3(b), the processed image has more clearly defined bright particles for all values of $r$.

Vector fields were established using the ensemble-of-correlation method with an initial interrogation window (IW) size of $64 \times 64$ pixel ($0.44 \times 0.44$ mm$^2$ or $0.41R \times 0.41R$) and a final IW size of $16 \times 16$ pixel ($0.11 \times 0.11$ mm$^2$ or $0.10R \times 0.10R$) with 75 % overlap between neighbouring IWs (Meinhart, Wereley & Santiago Reference Meinhart, Wereley and Santiago2000). The velocity vector was denoted as $\boldsymbol {u}$, with components in cylindrical coordinates being $u_r, u_{\theta },u_x$, corresponding to the velocity along the $r, \theta, x$ directions, respectively. The flow is laminar and steady, with presumably no swirl, i.e. $u_{\theta } = 0$, given the geometric dimensions of the PCT and the Reynolds numbers of the flows in the present investigation (Deiber & Schowalter Reference Deiber and Schowalter1979). We also did not observe evidence of secondary flow re-circulations, turbulence or swirl.

Sources of uncertainty in the PSV measurements were assumed to include: (i) errors due to subpixel interpolation of the correlation function; (ii) the finite DOF; and (iii) optical distortion near the walls of the tube from radial curvature and differences in the refractive index. Each source of uncertainty was conservatively estimated, the details for which are listed below.

1. (i) Errors from subpixel interpolation are conservatively estimated to be 0.1 pixels according to Raffel et al. (Reference Raffel, Willert, Scarano, Kähler, Wereley and Kompenhans2018). A 0.1 pixel error in displacement translates to an error in velocity of 0.1–1.4 mm s$^{-1}$ depending on $\Delta t$. If this error is normalized by the average centreline velocity, $\langle U_0 \rangle$, the largest velocity error among all flow conditions was 0.012$\langle U_0 \rangle$.

2. (ii) Quantifying the uncertainties attributed to radial distortion and differences in the refractive index was challenging and would require ray tracing analysis (Minor, Oshkai & Djilali Reference Minor, Oshkai and Djilali2007). Instead, errors from radial distortion were conservatively estimated based on how well the velocity within FOV1 could match the theoretical Poiseuille profile, as discussed in Appendix C. The results of the analysis in Appendix C demonstrated that the largest deviation from the parabolic velocity profile was $0.04\langle U_0\rangle$.

3. (iii) Slower moving particles within the DOF but outside the centre plane of the tube will bias velocity vectors to lower values. If we consider a parabolic velocity profile when the wall radius $R_w$ is equal to $R_i$, a DOF that is $0.1R_i$ in thickness would produce a relative error in $u_x$ of approximately $0.003\langle U_0 \rangle$ near the centreline of the PCT and $0.1\langle U_0 \rangle$ near the wall of the PCT. These errors reduce when considering regions of the PCT with a larger wall radius.

The total uncertainty in measurements of $\boldsymbol {u}$ from PSV was estimated to be the root sum squared value of the three previously listed sources of uncertainty. This was approximately $0.042\langle U_0 \rangle$ near the PCT centreline and $0.108\langle U_0 \rangle$ near the PCT walls, when considering regions of the PCT where $R_w = R_i$. In subsequent plots of velocity within the PCT, error bars are used to display the uncertainty in the velocity measurements from PSV.

2.5. Flow field analysis

The steady flow of complex and Newtonian fluids in the PCT are governed by the following equations for mass and momentum conservation:

(2.5)\begin{equation} \left. \begin{array}{c} \displaystyle \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u} = 0,\\ \displaystyle \rho \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u} ={-}\boldsymbol{\nabla} p + \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{\tau}, \end{array}\right\} \end{equation}

where $p$ is the indeterminate component of the Cauchy stress tensor and $\boldsymbol {\tau }$ is the deviatoric stress tensor. The velocity gradient tensor, $\boldsymbol{\mathsf{L}} = \boldsymbol {\nabla } \boldsymbol {u}$, can be decomposed into the symmetric rate of deformation tensor, $\boldsymbol{\mathsf{D}} = (\boldsymbol{\mathsf{L}} + \boldsymbol{\mathsf{L}}^{{\dagger} })/2$, and anti-symmetric rate of rotation tensor, $\boldsymbol{\mathsf{W}} = (\boldsymbol{\mathsf{L}} - \boldsymbol{\mathsf{L}}^{{\dagger} })/2$, where ${\dagger}$ represents the matrix transpose. The components of $\boldsymbol{\mathsf{D}}$ and $\boldsymbol{\mathsf{W}}$ are listed:

(2.6)\begin{gather} \left. \begin{array}{c} \displaystyle {\mathsf{D}}_{rr} = \dfrac{\partial u_r}{\partial r},\quad {\mathsf{D}}_{\theta \theta} = \dfrac{u_r}{r},\quad {\mathsf{D}}_{xx} = \dfrac{\partial u_x}{\partial x},\\ \displaystyle {\mathsf{D}}_{r x} = {\mathsf{D}}_{x r} = \dfrac{1}{2}\left( \dfrac{\partial u_r}{\partial x} + \dfrac{\partial u_x}{\partial r} \right), \end{array}\right\} \end{gather}

(2.7)\begin{gather}\omega_{\theta} ={-}2{\mathsf{W}}_{xr} = \frac{\partial u_r}{\partial x} - \frac{\partial u_x}{\partial r}, \end{gather}

where $\boldsymbol {\omega }$ is the vorticity vector, whose only non-zero component is $\omega _{\theta }$. Undisclosed components of $\boldsymbol{\mathsf{D}}$ and $\boldsymbol{\mathsf{W}}$ are zero. Equation (2.5) reduces to the Navier–Stokes equation for Newtonian fluids when the deviatoric stress tensor is represented by the constitutive equation $\boldsymbol {\tau } = 2\eta _s \boldsymbol{\mathsf{D}}$. For non-Newtonian fluids, the constitutive relation is much more complex and can be a partial differential equation with nonlinear terms (e.g. Phan–Thien–Tanner and Giesekus models). For most non-Newtonian constitutive models, it is common to segregate the deviatoric stress tensor into a solvent and non-Newtonian stress, i.e. $\boldsymbol {\tau } = \boldsymbol {\tau }_{\boldsymbol {s}} + \boldsymbol {\tau }_{\boldsymbol {nn}}$ (Alves, Oliveira & Pinho Reference Alves, Oliveira and Pinho2020). Here, $\boldsymbol {\tau }_{\boldsymbol {s}} = 2\eta _s \boldsymbol{\mathsf{D}}$ is the solvent stress and $\boldsymbol {\tau }_{\boldsymbol {nn}}$ is the non-Newtonian stress introduced from the polymers or micelles. Note that if $\boldsymbol {\tau }_{\boldsymbol {nn}} = 0$, then $\boldsymbol {\tau } = \boldsymbol {\tau _{s}}$ and the constitutive equation is Newtonian. When substituted into (2.5), the divergence of the non-Newtonian stress, $\boldsymbol {\nabla } \boldsymbol {\cdot } \boldsymbol {\tau }_{\boldsymbol {nn}}$, acts as an additional forcing term and for polymeric flows is often referred to as a ‘polymer force’ (Kim et al. Reference Kim, Li, Sureshkumar, Balachandar and Adrian2007).

Equations (2.6) and (2.7) can be explicitly evaluated using the measured $u_x$ and $u_r$. To circumvent the need for pressure, $p$, we considered the vorticity transport equation obtained from taking the curl of the momentum transport equation, shown in (2.5). The only non-zero component of the vorticity in the PCT flow is $\omega _{\theta }$; therefore, we consider the vorticity transport equation along the azimuthal direction alone:

(2.8)\begin{equation} \underbrace{u_r \frac{\partial \omega_{\theta}}{\partial r} + u_x \frac{\partial \omega_{\theta}}{\partial x} - \frac{u_r \omega_{\theta}}{r}}_{VA} = \underbrace{\nu_s \left( \frac{\partial^2 \omega_{\theta}}{\partial r^2} + \frac{1}{r} \frac{\partial \omega_{\theta}}{\partial r} + \frac{\partial^2 \omega_{\theta}}{\partial x^2} - \frac{\omega_{\theta}}{r^2} \right)}_{VSD} + \hspace{0.3em} T_{\theta}. \end{equation}

The additional term on the right-hand side of (2.8), is the azimuthal component of the non-Newtonian torque, $\boldsymbol {T} = (\boldsymbol {\nabla } \times \boldsymbol {\nabla } \boldsymbol {\cdot } \boldsymbol {\tau }_{\boldsymbol {nn}})/\rho$. The non-Newtonian torque is a vector, whose only non-zero component in the PCT is $T_{\theta }$. Previous numerical investigations have denoted $\boldsymbol {T}$ the ‘polymer torque’ as it can be represented as the curl of the polymer force (Kim et al. Reference Kim, Li, Sureshkumar, Balachandar and Adrian2007, Reference Kim, Adrian, Balachandar and Sureshkumar2008; Kim & Sureshkumar Reference Kim and Sureshkumar2013; Page & Zaki Reference Page and Zaki2015, Reference Page and Zaki2016; Biancofiore, Brandt & Zaki Reference Biancofiore, Brandt and Zaki2017; Lee & Zaki Reference Lee and Zaki2017). Its simplified units are s$^{-2}$ – when multiplied by moment of inertia, the units are force times unit distance, consistent with the true torque definition. The under-braces shown in (2.8) isolate the different combinations of terms within the vorticity transport equation. On the left-hand side of (2.8), $VA$ denotes the azimuthal vorticity advection. The first term on the right-hand side of (2.8), $VSD$, represents vorticity solvent diffusion, where $\nu _s = \eta _s/\rho$. For each flow, the azimuthal non-Newtonian torque was calculated based on the deficit between $VA$ and $VSD$, i.e. $T_{\theta } = VA - VSD$.

To establish the first-order spatial gradients of velocity, a moving second-order polynomial surface was fit on profiles of $u_x$ and $u_r$. The size of the second-order polynomial filter was $20 \times 20$ pixels, $138\times 138$ $\mathrm {\mu } \textrm {m}^2$ or $0.15R \times 0.15R$. Coefficients of the polynomial surface were used to establish first-order spatial derivatives of $u_x$ and $u_r$. Azimuthal vorticity, $\omega _{\theta }$, was then established using (2.7). To determine the higher order spatial gradients in the flow, a moving third-order polynomial surface was fit on profiles of $u_x$ and $u_r$. The size of the cubic polynomial filter was $76 \times 76$ pixels, $522\times 522$ $\mathrm {\mu } \textrm {m}^2$, or $0.56R \times 0.56R$. Coefficients of the third-order polynomial were used to the determine the second- and third-order spatial derivatives of $u_x$ and $u_r$. Three orders of differentiation in $\boldsymbol {u}$ are required due to the $VSD$ term in (2.8). These higher-order derivatives were then used to calculate the azimuthal non-Newtonian torque $T_{\theta }$ using (2.8). Polynomial filters that overlapped with the PCT wall were neglected, and results of $\omega _{\theta }$ and $T_{\theta }$ were not considered close to the wall.

All parameters including $\boldsymbol {u}$, $\omega _{\theta }$ and $T_{\theta }$ exhibited symmetry about $r = 0$. Therefore, $\boldsymbol {u}$, $\omega _{\theta }$ and $T_{\theta }$ on the lower half of the domain ($r<0$) were averaged with the upper half ($r>0$). When comparing $\boldsymbol {u}$, $\omega _{\theta }$ and $T_{\theta }$ in one oscillation to prior or subsequent oscillations, the parameters are not dramatically different for all flow conditions and fluids. Therefore, $\boldsymbol {u}$, $\omega _{\theta }$ and $T_{\theta }$ were periodically averaged over three oscillations, i.e. for $x$-coordinates that share the same wall radius, $R_w$.

The volumetric flow rate, $Q$ can be determined from flow measurements based on a volume integration of $u_x$, i.e. $Q = 2 {\rm \pi}\int _{0}^{R_w} u_x r \,\textrm {d}r$. The bulk velocity can be defined according to $U = Q/({\rm \pi} R_w^2)$. Because of mass conservation and the variation of $R_w$ along $x$, the bulk velocity $U$ changes along the streamwise $x$ direction. Therefore, an average value of $U$ along $x$ was determined, and angle brackets $\langle \cdots \rangle$ were used to denote the spatial averaging along the $x$ direction, i.e. $\langle U \rangle$. Recall from § 2.1 that the same angle brackets were used to define the average centreline velocity along $x$, $\langle U_0 \rangle$. An average shape factor can be determined from the ratio of centreline to bulk velocity, $SF = \langle U_0 \rangle / \langle U \rangle$. For Poiseuille flow in a straight-walled tube, $SF=2$. Lastly, distributions of $\boldsymbol {u}$, $\omega _{\theta }$ and $T_{\theta }$ for flows within the PCT (i.e. FOV2) were normalized by $\langle U_0 \rangle$, $\dot {\gamma }_w$ and $\dot {\gamma }_w^2$, respectively. Spatial variables $x$ and $r$ were normalized by $\lambda$ and $R$, respectively. Normalization of the parameters was denoted using the superscript $+$.

3.1. Shear rheology

Figure 4 displays measurements of $\eta$ as a function of $\dot {\gamma }$ for the Newtonian and non-Newtonian fluids. Shear viscosity distributions of distilled water are shown in figure 4(a). Measurements of $\eta$ for water are constant with respect to $\dot {\gamma }$ provided $\tau > 2$ mPa and $Ta < 1700$. For $\tau < 2$ mPa, measurements of $\eta$ for water are noisy and scattered. When $Ta>1700$, measurements of $\eta$ for water increase abruptly and are no longer constant with respect to $\dot {\gamma }$; Taylor vortices have corrupted the measurements of $\eta$. The average viscosity of water for $\tau > 2$ mPa and $Ta<1700$ is 0.97 mPa s. This is 3 % lower than the theoretical shear viscosity of water at $20.1\,^{\circ }\mathrm {C}$, 1.00 mPa s.

Figure 4. Steady shear viscosity distributions for (a) the baseline Newtonian fluids, (b) flexible polymer solution PAM, (c) rigid biopolymer solution XG and (d) cationic surfactant solution TTAC. The horizontal black solid line is $\eta$ for water determined from the empirical correlation of Cheng (Reference Cheng2008). Dashed black lines indicate the lower torque limit ($\tau < 2$ mPa) and the onset of Taylor vortices ($Ta>1700$). In panels (b,c), solid coloured lines represent the power-law fits for shear-thinning fluids given by (2.2) and with values provided in Appendix A.

Figure 4(b) shows profiles of $\eta$ for the five PAM solutions. All five concentrations of PAM exhibit larger values of $\eta$ than water. They also exhibit shear-thinning, where $\eta$ decreases monotonically with increasing $\dot {\gamma }$. At the higher values of $\dot {\gamma }$, $\eta$ appears to increase sharply for $\dot {\gamma }$ with a $Ta$ less than 1700. Nonetheless, the trend by which $\eta$ reduces with respect to $\dot {\gamma }$ is well represented by the power law model (2.2) for measurements with $\dot {\gamma }>0.1$ s$^{-1}$ and $\dot {\gamma } < 100$ s$^{-1}$ – sufficiently below the shear rate that $\eta$ increases abruptly. Values of the consistency, $K$, and flow index, $n$, for PAM are provided in Appendix A.

Figure 4(c) demonstrates profiles of $\eta$ for the five XG solutions. Similar to PAM, all concentrations of XG exhibit larger values of $\eta$ than water and prevalent shear-thinning. Unlike PAM, $\eta$ appears to increase sharply for $\dot {\gamma }$ with a $Ta$ equal to 1700. The shear-thinning trend is well represented by the power law model (2.2) for measurements with $\dot {\gamma } > 0.1$ s$^{-1}$ and $Ta < 1700$. Values of the consistency, $K$, and flow index, $n$, for XG are provided in Appendix A.

Lastly, figure 4(d) demonstrates shear rheograms for the five TTAC solutions. All five solutions have values of $\eta$ similar to water (approximately 1.00 mPa s) and independent of $\dot {\gamma }$. A similar water-like shear rheogram was observed by Warwaruk & Ghaemi (Reference Warwaruk and Ghaemi2021) for a 0.015 % and 0.02 % TTAC solution, despite the solutions being able to induce upwards of a 70 % reduction in skin friction drag and attenuate velocity fluctuations in a high-Reynolds-number, turbulent channel flow. Although the solutions have a proclivity to induce complex dynamics in other canonical flows, the TTAC solution does not reflect non-Newtonian features in a steady Couette flow with $\dot {\gamma }$ between 2 and 100 s$^{-1}$.

3.2. Extensional rheology

Measurements of $D_{min}/D_0$ from the DoS apparatus are shown in figure 5 for the PAM solutions – the only solutions that demonstrated EC thinning. For $t$ less than the inertial break-up time $t_b$, the evolution of $D_{min}$ is in an IC regime and well described by (2.3). The inertial break-up time $t_b$ was not significantly different among the PAM solutions of different $c$ and was approximately $7.2\ \textrm {ms} \pm 0.6\ \textrm {ms}$. Measurements of $D_{min}/D_0$ for PAM with $c = 0.01\,\%$ are shown in the inset axes of figure 5. The IC thinning represented by (2.3), and shown by the black solid line in the inset axes of figure 5, has $\alpha = 0.5$ and $t_R = 1.9$ ms. This value of $\alpha$ is between 0.4 and 1, which is within the margin of experimental expectations (Zhang & Calabrese Reference Zhang and Calabrese2022). The theoretical Rayleigh time $t_R = (\rho D_0^3/8\sigma )^{1/2}$ should be 1.89 ms (assuming $\sigma \approx 72$ mN m$^{-1}$) – not significantly different than $t_R$ derived from fitting (2.3) onto measurements of $D_{\textrm {min}}/D_0$ in the IC regime. Recall that the Ohnesorge number is defined as $Oh = t_v / t_R$, where $t_v = \eta D_0 / 2\sigma$. If $\eta$ in the equation for $t_v$ is taken to be the largest measured viscosity in figure 4 (approximately 0.1 mPa s for PAM with $c = 0.05\,\%$), then $t_v \approx 0.9$ ms and the largest $Oh$ is approximately 0.5.

Figure 5. Normalized minimum filament diameter with respect to time for the PAM solutions, as determined from the DoS system. The inset figure demonstrates a zoomed in distribution along time for PAM with $c = 0.01\,\%$. Coloured solid lines indicate the fits of the EC regime using (2.4). The solid black line in the inset denotes the fit of the IC regime using (2.3).

For $t>t_b$, all PAM solutions demonstrate EC thinning, well represented by (2.4) and the coloured lines shown in figure 5. As the concentration grows, the extensional relaxation time $t_e$ increases. Values of $t_e$ are provided in Appendix A. If we assumed that $t_R = 1.89$ ms for all PAM solutions, $De$ was between 1.2 and 25.7 depending on $c$. For the high concentration PAM solutions, the 2 kHz image acquisition rate coupled with the spatial resolution of the camera results in repetitive measurements of $D_{min}/D_0$ over several time instances (i.e. the small horizontal lines).

Solutions of TTAC and XG do not demonstrate EC thinning and therefore, $De < 1$. A lack of EC thinning is either a result of a low $t_e$ or a large $t_R$, by definition of the Deborah number $De = t_e / t_R$. It is well known that surfactant solutions have a much lower $\sigma$ than the solvent and hence, a large $t_R$ (Zhang et al. Reference Zhang, Schmidt, Talmon and Zakin2005). It is possible that the lack of EC thinning in TTAC could be attributed to low surface tension. Surface tension $\sigma$ is generally 40 % lower for large concentration solutions of cationic surfactants relative to water (i.e. 35–45 mN m$^{-1}$). This would mean that $t_R$ could be approximately 30 % larger for surfactants. We suspect that a 30 % increase in $t_R$ is not sufficient to explain the lack of EC thinning for TTAC. The present investigation does not measure $\sigma$, and hence no definitive conclusion can be made in this regard. However, we suspect that the lack of EC thinning is attributed to low $t_e$ at the conditions imposed by the DoS rheometer. This is another example of how difficult it is to measure $t_e$ using capillary-driven extensional rheometers for drag-reducing surfactant solutions, as previously seen by Warwaruk & Ghaemi (Reference Warwaruk and Ghaemi2021) and Fukushima et al. (Reference Fukushima, Kishi, Suzuki and Hidema2022). It also highlights a need to develop other techniques for measuring extensional features of non-Newtonian solutions, as done by Wunderlich & James (Reference Wunderlich and James1987). Ultimately, PAM solutions have relatively large $t_e$ that could be derived from the DoS rheometer, while TTAC and XG solutions have $t_e$ that could not measured using the DoS apparatus.

4.1. Newtonian flow

Figure 6 demonstrates contours of the velocity magnitude $\|\boldsymbol {u}\|^+ = (u_x^2+u_r^2)^{0.5}/ \langle U_0 \rangle$ along with streamlines for the flow of water at five different $Re$ within the PCT. All flows with unique $Re$ have a centreline velocity $U_0$ that attains a maximum value around $x^+ = 0.5$. When $x^+ = 0.5$, the wall radius of the PCT $R_w$ is at its smallest value, $R_w = R_i$. For the lowest $Re$ flow (i.e. $Re = 15.7$), the centreline velocity at $x^+= 0.5$ attains 1.25$\langle U_0 \rangle$. The lowest magnitude in $U_0$ occurs when $x^+ = 0$ and $1$, and is approximately equal to $0.7\langle U_0 \rangle$ for $Re = 15.7$. At larger $Re$, the centreline velocity at $x^+= 0.5$ is smaller in magnitude – approximately 1.1$\langle U_0 \rangle$. Values of $U_0$ are also slightly larger for the high-$Re$ cases when $x^+ = 0$ and $1$ compared to the case with $Re = 15.7$ – approximately equal to $0.8\langle U_0 \rangle$. Therefore, when $Re$ increases, the normalized centreline velocity decreases. In all flow conditions, streamlines at large $r^+$ tend to follow the sinusoidal profile of the wall. Near the core, streamlines are more parallel with respect to the streamwise $x$ direction.

Figure 6. Velocity magnitude normalized by the average centreline velocity $\langle U_0 \rangle$ for different $Re$ of water. Solid black lines overlaid on filled contours are streamlines. The solid black line at the limit of the filled contour is the sinusoidal wall profile.

Profiles of $u_x^+$ with respect to $r^+$ at different points of $x^+$ are shown in figure 7(a) for water at $Re = 15.7$, $106$ and $203$. Sample error bars are shown for the flow condition with $Re$ of 15.7 and at $x^+ = 0.5$. Relative errors were conservatively estimated to be $0.042\langle U_0 \rangle$ near the centreline of PCT and $0.108\langle U_0 \rangle$ near the wall at $x^+ = 0.5$, as discussed in § 2.4. As noted in the discussion pertaining to figure 6, the low-$Re$ flow of $15.7$ has a large variation in $U_0$. When $x^+ = 0$, $U_0$ becomes $0.75 \langle U_0 \rangle$ and when $x^+ = 0.5,$ $U_0$ equals $1.25 \langle U_0 \rangle$. Newtonian flows with larger $Re$ of 106 and 203 have a centreline velocity of approximately $0.81\langle U_0 \rangle$ when $x^+ = 0$ and $1.1\langle U_0 \rangle$ when $x^+ = 0.5$. Within the PCT contractions and expansions (i.e. $x^+ = 0.25$ and 0.75, respectively), radial profiles of $u_x^+$ are approximately the same. In other words, the velocity is symmetric about $x^+ = 0.5$. Figure 7(b) demonstrates that the streamlines of the Newtonian flows also depend on $Re$. When $Re$ is low, streamlines are more curved and their radial position is closer to the PCT centreline at $x^+ = 0.5$.

Figure 7. (a) Velocity profiles of water at $Re = 15.7$, 106 and $203$ at different $x$ locations along the PCT. Down sampled error bars are shown for the flow of water at $Re = 15.7$ and $x^+ = 0.5$. (b) Overlaid streamlines of the water flows at different $Re$. The black line in panel (b) indicates the wall profile $R_w$. Symbol colours in panel (a) correspond to the different $Re$ as indicated in panel (b).

As noted with regards to figure 7(a), the velocity in the PCT for water demonstrates a dependence on $Re$ most notable by the differences in the amplitude of the centreline velocity $U_0$. The standard deviation in the centreline velocity $\mathcal {R}(U_0)$ was computed for each $Re$ and normalized by their respective average centreline velocities $\langle U_0 \rangle$. The normalized standard deviation in $U_0$ is denoted as $\mathcal {R}(U_0)^+ = \mathcal {R}(U_0)/\langle U_0 \rangle$. Values of $\langle U_0 \rangle$ and $\mathcal {R}(U_0)^+$ are listed in table 1 for the different water flows. The inverse proportionality between the amplitude of $U_0$ and $Re$ is clearly demonstrated by the decreasing trend in $\mathcal {R}(U_0)^+$ as $Re$ grows. In addition to centreline velocity, table 1 also lists the average bulk velocity $\langle U \rangle$ and shape factor $SF = \langle U_0 \rangle / \langle U \rangle$ for the flows of water in the PCT. For Newtonian Poiseuille flow in a straight-walled pipe, $SF$ equals 2. Although the PCT is not straight-walled, values of $SF$ for all water flows are approximately 2.1 and not too different from the theoretical $SF$ for straight-walled Poiseuille pipe flow.

Table 1. Bulk and centreline velocity statistics for the flow of water within the PCT at different $Re$.

Contours of azimuthal vorticity, $\omega _{\theta }^+$, are shown in figure 8 for water within the PCT. Near the centreline, $\omega _{\theta }^+$ is approximately equal to zero. For all radial and streamwise coordinates, $\omega _{\theta }$ is positive. The maximum $\omega _{\theta }^+$ is situated near the wall and at $x^+ = 0.5$ for all $Re$. Recall from §2.5 that measurements of $\omega _{\theta }^+$ within close proximity (15 % of $R_w$) of the wall were not calculated. This is because the differentiation filter conflicted with the wall.

Figure 8. Vorticity normalized by average wall shear rate $\dot {\gamma }_w$ for different $Re$ of water. The solid black line is the sinusoidal wall profile.

4.2. Xanthan gum solutions

Velocity contours and streamlines are shown in figure 9 for XG solutions at different $Re$. For brevity, only the results of two concentrations, $c = 0.02\,\%$ and $0.05\,\%$, are shown. Similar to water, both of the XG solutions with $c = 0.02\,\%$ and $0.05\,\%$ have magnitudes of $\|\boldsymbol {u}\|^+$ that are lowest when $x^+ = 0$ and $1$, and largest when $x^+ =0.5$. Compared to Newtonian water flows seen in figure 6, the zone with larger values of $\|\boldsymbol {u}\|^+$ is extended farther towards the tube wall. As $c$ increases from 0.02 % to 0.05 %, $\|\boldsymbol {u}\|^+$ also increases at $r^+ > 0$ locations. Streamlines at large $r^+$ take on a similar sinusoidal profile as the wall pattern. Similar to the water flows, the streamlines for XG at $c = 0.02\,\%$ and $0.05\,\%$ are approximately symmetric with respect to $x^+ = 0.5$.

Figure 9. Velocity magnitude normalized by the average centreline velocity $\langle U_0 \rangle$ for different $c$ and $Re$ of XG. Solid black lines overlaid on filled contours are streamlines. The solid black line at the limit of the filled contour is the sinusoidal wall profile.

Streamwise velocity profiles $u_x^+$ at different $x^+$ coordinates are shown in figure 10(a) for XG with $c = 0.05\,\%$ and at $Re = 10.2$. For comparison, the profiles of water at a similar $Re$ are presented alongside those of XG. Relative to water at the same $x^+$ coordinates, XG has larger $u_x^+$ values. The distributions of $u_x^+$ are more flat in the PCT centre; a blunted profile that is common in shear-thinning fluids (Bird, Stewart & Lightfoot Reference Bird, Stewart and Lightfoot2007). Despite the differently shaped velocity profile, the range of $U_0$ appears to be similar among water and XG. For the XG flow, $U_0 = 0.75\langle U_0 \rangle$ at $x^+ = 0$ and $U_0 = 1.25\langle U_0 \rangle$ at $x^+ = 0.5$ – the same as water. Lastly, figure 10(b) compares the streamlines of the same flows of XG and water seen in figure 10(a). Despite having different velocity profiles with respect to $r^+$, the streamlines for water and XG are approximately the same.

Figure 10. (a) Velocity profiles along different $x$ locations for XG with $c = 0.05\,\%$ at $Re = 10.2$ and water at $Re = 15.7$. Down sampled error bars are shown for the flow of water at $Re = 15.7$ and $x^+ = 0.5$. (b) Overlaid streamlines of XG and water. The black line in panel (b) indicates the wall profile $R_w$. Red symbols in panle (a) correspond to the water flow with $Re = 15.7$, while blue symbols represent the XG flow with $c = 0.05\,\%$ and $Re = 10.2$.

Based on figure 10(a), it was shown that the shape of $u_x^+$ profiles with respect to $r^+$ were different, but the relative variations in $U_0$ were the same among the flows of water and XG at similar $Re$. Figure 11(a) demonstrates the standard deviation in $U_0$ normalized by the average centreline velocity, $\mathcal {R}(U_0)^+$, for XG and water. Low-concentration solutions of XG ($c < 0.03\,\%$) appear to have $\mathcal {R}(U_0)^+$ values that overlap with water at high $Re$. However, for $Re<20$ and high XG concentrations, the values of $\mathcal {R}(U_0)^+$ appear to be independent of the Reynolds number and relatively constant. The higher concentration XG solutions of 0.04 % and 0.05 % appear to have subtly larger values of $\mathcal {R}(U_0)^+$ than water and the other XG solutions; however, the difference is not substantial. Generally, $\mathcal {R}(U_0)^+$ for XG appears to be independent of concentration and similar to water at higher $Re$ values. Figure 11(b) presents the average shape factor $SF = \langle U_0 \rangle /\langle U \rangle$ for water and XG at different $Re$ and $c$ within the PCT. Relative to water, XG flows at all $c$ and $Re$ have lower $SF$ values. As the $c$ of XG increases, $SF$ decreases. The reducing trend in $SF$ aptly summarizes how shear-thinning makes the profile more blunt as the concentration of XG increases.

Figure 11. (a) Standard deviation in the centreline velocity divided by the mean centreline velocity, and (b) the shape factor with respect to different $Re$ for XG solutions and water.

Vorticity contours for the XG flows with $c = 0.02\,\%$ and 0.05 % are shown in figure 12. Similar to the flows of water in the PCT, $\omega _{\theta }^+$ attains a maximum value near the wall and at $x^+ = 0.5$. Both the XG flows with $c = 0.02\,\%$ and $0.05\,\%$ have a noticeably attenuated $\omega _{\theta }^+$ in regions farther from the tube centreline. In other words, the thickness (along $r^+$) of the region near the pipe centreline with $\omega _{\theta }^+ = 0$ is larger for XG relative to water. The thickness also grows with increasing $c$. The attenuated $\omega _{\theta }^+$ is attributed to the more uniform profiles of $u_x^+$ caused by shear-thinning.

Figure 12. Vorticity normalized by the average wall shear rate $\dot {\gamma }_w$ for different $c$ and $Re$ of XG. The solid black line is the sinusoidal wall profile.

4.3. Polyacrylamide solutions

Relative to water and XG, different patterns in the velocity are encountered for the flow of PAM within the PCT. Figure 13 demonstrates contours of $\|\boldsymbol {u}\|^+$ for PAM at different $c$ and $Re$. In this figure, $c$ increases from bottom to top, and $Re$ increases from left to right. Despite the low-$Re$ flows showing some visual resemblance to the results for water, flows at high $c$ and large $Re$ are asymmetric about $x^+ = 0.5$. For these cases, the large velocity contours take on a triangular or half-chevron appearance leaning towards the upstream direction. Therefore, within the contraction regions (i.e. from $x^+ = 0$ to $0.5$), the maximum velocity is not necessarily situated at the centreline of the PCT. Within the tail of the chevron, streamlines appear to be tilted farther towards the centreline and non-conforming to the sinusoidal profile of the walls. Despite the PAM solutions having seemingly comparable steady shear rheology as the XG solutions (figure 4), the flow of PAM within the PCT produces an entirely different velocity distribution. It is clear that another rheological property, not present in XG, is causing the chevron pattern in the flows of PAM through the PCT.

Figure 13. Velocity magnitude normalized by the average centreline velocity $\langle U_0 \rangle$ for different $c$ and $Re$ of PAM. Solid black lines overlaid on filled contours are streamlines. The solid black line at the limit of the filled contour is the sinusoidal wall profile.

Streamwise velocity profiles $u_x^+$ along different $x^+$ values are shown in figure 14(a) for PAM with $c = 0.03\,\%$. Two different $Re$ are compared to contrast the change in the velocity from when the contours transition to the half-chevron seen in figure 13. For the low-Reynolds-number case of $Re = 3.02$ (red symbols), profiles of $u_x^+$ are similar to those for water or XG. The shape of the $u_x^+$ profiles appear to be subtly more blunted than the parabolic Poiseuille profile and the variations in $U_0$ are slightly larger than the values encountered for water at $Re = 15.7$ seen in figure 7(a). Similar to XG, the more blunted velocity profile for PAM at low $Re$ can likely be explained by shear-thinning. Recall that PAM with $c = 0.03\,\%$ has a lower power-law index than XG with $c = 0.05\,\%$, as seen in Appendix A. Therefore, we would expect $u_x^+$ profiles not to be parabolic, but also not as blunted as the higher concentration XG solutions. Although the low-$Re$ flow reflects some similarities to previous findings for XG, the higher $Re$ flow of PAM (blue symbols) exhibits entirely unique distributions in $u_x^+$. At $x^+ = 0$ the $u_x^+$ profile has two local maxima – one at the centreline, the other at $r^+ = 0.9$. Within the contraction, where $x^+ = 0.25$, the maximum value of $u_x^+$ is no longer situated at the centreline, but at $r^+ = 0.6$. Prior works have observed large velocity overshoot near the wall in gradual planar contraction flows of PAM solutions (Poole et al. Reference Poole, Escudier and Oliveira2005) and numerical investigations that used various viscoelastic constitutive models (Afonso & Pinho Reference Afonso and Pinho2006; Alves & Poole Reference Alves and Poole2007; Poole et al. Reference Poole, Escudier, Afonso and Pinho2007; Poole & Alves Reference Poole and Alves2009). Poole et al. (Reference Poole, Escudier and Oliveira2005) referred to these velocity overshoots as ‘cat's ears’ given their appearance.

Figure 14. (a) Velocity profiles along different $x$ locations for PAM with $c = 0.03\,\%$ at $Re =3.02$ and $Re = 60.5$. Red symbols show $Re = 3.02$ and blue symbols show $Re = 60.5$. Down sampled error bars are shown for the flow of PAM with $c = 0.03\,\%$ and $Re = 3.02$ at $x^+ = 0.5$. (b) Overlaid streamlines of PAM at different $Re$. The black line in panel (b) indicates the wall profile $R_w$.

Coupled with the near-wall velocity overshoots are highly curved streamlines, as shown in figure 14(b). At sufficiently large $Re$, PAM with $c = 0.03\,\%$ has streamlines that are directed away from the PCT core and more towards the tube wall for $x^+ = 0$ to $0.5$. The works by Cable & Boger (Reference Cable and Boger1978a,Reference Cable and Bogerb, Reference Cable and Boger1979) referred to the state of these curved streamlines as ‘divergent flow’. In general, solutions of PAM with $c \geq 0.02\,\%$ and sufficiently large $Re$ are subjected to near-wall velocity overshoots and divergent flow within the contracting portions of the PCT (i.e. $0< x^+<0.5$), as seen in figure 13.

The pattern of $u_x^+$ for PAM is clearly dependent on $Re$ and $c$, as observed in figure 13. Compared to water, it can be observed that variations in $U_0$ are larger for PAM at large $c$ and $Re$ based on figure 14(a). Figure 15(a) demonstrates $\mathcal {R}(U_0)^+$ as a function of $Re$ for different $c$ of PAM. When the concentration of PAM is low ($c = 0.01\,\%$), values of $\mathcal {R}(U_0)^+$ are similar to water. This is expected; contours of velocity for PAM with $c = 0.01\,\%$ do not exhibit a prevalent asymmetric half-chevron pattern in figure 13. At large concentrations, PAM enhances the variations in $U_0$. For more moderate PAM concentrations of $c = 0.02\,\%$ and $0.03\,\%$, $\mathcal {R}(U_0)^+$ increases up until an $Re$ of approximately 35, before decreasing with further growth in $Re$. At large concentrations of $c = 0.04\,\%$ and $0.05\,\%$, values of $\mathcal {R}(U_0)^+$ are similar.

Figure 15. Standard deviation in the centreline velocity divided by the mean centreline velocity with respect to (a) different $Re$ and (b) different Deborah number $De$ for various PAM solutions.

Rheological measurements of PAM solutions demonstrated that the solutions are viscoelastic – see figure 5 and Appendix B. Therefore, values of $\mathcal {R}(U_0)^+$ are also contrasted with the elastic properties of the flow. A Deborah number within the PCT was defined as $De = t_e/t_f$, where $t_f = \lambda /\langle U_0 \rangle$ is the time scale of the flow along the PCT centreline. Figure 15(b) shows values of $\mathcal {R}(U_0)^+$ with respect to $De$ for different concentrations of PAM. For all flows of PAM with $c = 0.01\,\%$, the values of $De$ are less than 0.1 and the corresponding values of $\mathcal {R}(U_0)^+$ are less than 0.2. Larger concentration PAM solutions with $De>0.1$ have large values of $\mathcal {R}(U_0)^+$ that are greater than 0.2 and tend to increase with growing $De$ – that is, up until the point where $Re$ has attained 35, with reference to figure 15(a). Based on the trend in $\mathcal {R}(U_0)^+$ versus $Re$ for water, a decreasing $\mathcal {R}(U_0)^+$ is likely attributed to inertial effects – perhaps producing more stagnant flow or small, unseen recirculations in the expansion regions with adverse pressure gradients and where $R_w = R_o$ (Deiber & Schowalter Reference Deiber and Schowalter1981). However, elasticity acts to augment $\mathcal {R}(U_0)^+$. The increasing–decreasing trend in $\mathcal {R}(U_0)^+$ is most likely a result of the competing effects of elasticity and inertia. When $c$ is sufficiently large, elasticity dominates and the trend in $\mathcal {R}(U_0)^+$ as function of $De$ shows better overlap for different $c$. Cases with $De>0.1$ also tend to have a pronounced half-chevron velocity pattern in figure 13.

Lastly, contours of vorticity $\omega _{\theta }^+$ are shown for the flows of PAM in the PCT in figure 16. At low $c$ and $Re$, $\omega _{\theta }^+$ is everywhere positive, similar to those for water and XG. However, PAM with sufficiently large $c$ and $Re$ exhibits negative values of $\omega _{\theta }^+$ within the PCT contractions. In certain cases, e.g. $c = 0.03\,\%$ and $Re = 60.5$, there is a strong contrast between the tilted negative contour of $\omega _{\theta }^+$ and the surrounding positive $\omega _{\theta }^+$ values.

Figure 16. Vorticity normalized by $\dot {\gamma }_w$ for different $c$ and $Re$ of PAM. The solid black line is the sinusoidal wall profile.

4.4. Surfactant solutions

Velocity contours are shown for the TTAC solutions in figure 17. At the lowest concentration of $c = 0.01\,\%$ and $0.02\,\%$, the contours are similar to those for water. Half-chevron patterns that are similar to the those of PAM appear for all $c$ greater than $0.03\,\%$ and $Re$ that exceed 115. The chevrons result in curved streamlines that are asymmetric with respect to $x^+ = 0.5$. Despite a water-like shear rheogram, shown in figure 4, TTAC demonstrates a complex, non-Newtonian response within the PCT that is similar to flexible polymers and unlike rigid polymers. Therefore, the current measurements show that the rheological trait responsible for the asymmetric chevron pattern in PAM is clearly also inherent in TTAC.

Figure 17. Velocity magnitude normalized by the average centreline velocity $\langle U_0 \rangle$ for different $c$ and $Re$ of TTAC. Solid black lines overlaid on filled contours are streamlines. The solid black line at the limit of the filled contour is the sinusoidal wall profile.

Profiles of $u_x^+$ with respect to $r^+$ and at different $x^+$ are shown in figure 18(a) for TTAC with $c = 0.05\,\%$ and $Re = 119$ and compared with PAM at $c = 0.02\,\%$ and $Re = 83.1$ – the closest possible $Re$. The near-wall velocity overshoots encountered for the PAM flows are also present in the TTAC solution. Within the contraction, from $x^+ = 0$ to $0.25$, distributions of $u_x^+$ can be described by a higher-order polynomial with two local peaks along $r^+$. Streamlines are also compared for TTAC and PAM in figure 18(b). Both solutions demonstrate divergent flow patterns within the PCT contraction (Cable & Boger Reference Cable and Boger1978a,Reference Cable and Bogerb, Reference Cable and Boger1979). Streamlines for PAM are projected farther towards the wall relative to TTAC. This is despite PAM having slightly lower near-wall velocity overshoots. In general, we see good qualitative agreement between the velocity field of PAM and TTAC – cat's ears and divergent flow.

Figure 18. (a) Velocity profiles along different $x$ locations for TTAC with $c = 0.05\,\%$ at $Re = 119$, shown by the red symbols, and PAM with $c = 0.02\,\%$ at $Re = 83.7$, shown with blue symbols. Down sampled error bars are shown for the flow of PAM with $c = 0.02\,\%$ and $Re = 83.7$ at $x^+ = 0.5$. (b) Overlaid streamlines of TTAC and PAM.

Distributions of $\mathcal {R}(U_0)^+$ as a function of $Re$ are shown in figure 19 for TTAC solutions of different $c$. Solutions that do not exhibit asymmetric velocity patterns, namely TTAC with $c=0.01\,\%$ and $0.02\,\%$, have values of $\mathcal {R}(U_0)^+$ that overlap with those of water and demonstrate the same decreasing trend in $\mathcal {R}(U_0)^+$ with increasing $Re$. For more concentrated solutions of TTAC, such as $c = 0.03\,\%$ and $0.04\,\%$, values of $\mathcal {R}(U_0)^+$ overlap with measurements for water at low $Re$. As $Re$ is increased further, values of $\mathcal {R}(U_0)^+$ abruptly increase. This is different than the monotonic increase in $\mathcal {R}(U_0)^+$ with growing $Re$ observed for PAM in figure 15(a). The $Re$ at which $\mathcal {R}(U_0)^+$ abruptly increases appears to be sensitive to small discrepancies in $Re$. It appears as though transition to large $\mathcal {R}(U_0)^+$ occurs earlier for the $c = 0.04\,\%$ TTAC solution compared to the $c = 0.03\,\%$ solution. However, the $Re$ at which $\mathcal {R}(U_0)^+$ increases for $c = 0.03\,\%$ is subtly larger than the $Re$ of the $c = 0.04\,\%$ solution at a comparable flow rate. Evidently, the resolution of $Re$ is too sparse to capture the sudden augmentation in $\mathcal {R}(U_0)^+$. Ultimately, the trend in the velocity pattern, namely $\mathcal {R}(U_0)^+$ as a function of $Re$, is different for TTAC compared to PAM. Beyond a critical $Re$, the asymmetric velocity patterns that are formed by the TTAC solution exhibit qualitatively the same pattern as PAM, with values of $\mathcal {R}(U_0)^+$ that are also larger than the Newtonian and XG flows.

Figure 19. Standard deviation in the centreline velocity divided by the mean centreline velocity as a function of $Re$ for the various TTAC solutions.

Figure 20 presents contours of $\omega _{\theta }^+$ for the TTAC solutions at different $c$ and $Re$. Similar to the PAM solutions, the flows of TTAC with sufficiently large $c$ and $Re$ demonstrate negative values of $\omega _{\theta }^+$ within the contractions of the PCT ($0< x^+<0.5$). As expected, the conditions where half-chevrons appear in velocity contours also reflect negative values in $\omega _{\theta }^+$. The $c = 0.01\,\%$ and $0.02\,\%$ TTAC solutions have $\omega _{\theta }^+$ distributions that are seemingly identical to those for water, seen in figure 8. At large $c$ and $Re$, negative contours of $\omega _{\theta }^+$ begin to appear. It is notable that the TTAC solutions exhibit water-like rheology, yet they respond in a manner similar to PAM within the PCT at larger $c$ and $Re$ conditions.

Figure 20. Vorticity normalized by $\dot {\gamma }_w$ for different $c$ and $Re$ of TTAC. The solid black line is the sinusoidal wall profile.

4.5. Non-Newtonian torque

The non-Newtonian torque was established based on the deficit between the advection of vorticity ($VA$) and the viscous solvent diffusion ($VSD$) – see (2.8). The normalized azimuthal component of the non-Newtonian torque is $T_{\theta }^+ = T_{\theta }/\dot {\gamma }_w^2$. The distributions of $T_{\theta }^+$ are presented in figures 21, 22 and 23 for water, XG, PAM and TTAC. In these figures, the open contours show $T_{\theta }^+$ and are overlaid on filled contours of $\omega _{\theta }^+$. Contour levels for $T_{\theta }^+$ are from $-$0.4 to $+$0.4 in steps of 0.2. Contours greater than or equal to zero are solid lines and negative contours are dotted lines. Values of $T_{\theta }^+$ were not computed or shown near the wall (within 42 % of $R_w$) due to difficulties in computing spatial gradients within this region, as discussed in § 2.5.

Figure 21. Contours of vorticity and the non-Newtonian torque for the flows of (a) water with $Re = 106$ and (b) XG at $c = 0.02\,\%$, $Re = 71.7$. Positive and zero contours are solid lines, while dashed lines are negative contours.

Figure 22. Contours of vorticity and the non-Newtonian torque for the flow of PAM solutions with (a) $c = 0.03\,\%$, $Re = 60.5$ and (b) $c = 0.05\,\%$, $Re = 35.5$. Positive and zero contours are solid lines, while dashed lines are negative contours.

Figure 23. Contours of vorticity and the non-Newtonian torque for the flow of TTAC solutions with (a) $c = 0.05\,\%$, $Re = 119$, and (b) $c = 0.05\,\%$, $Re = 254$. Positive and zero contours are solid lines, while dashed lines are negative contours.

Based on (2.8), $T_{\theta }^+$ should be equal to zero in the flow of water. In other words, the dynamics of vorticity should be entirely described by vorticity advection ($VA$) and diffusion ($VSD$). Figure 21(a) presents contours of $\omega _{\theta }^+$ and $T_{\theta }^+$ for water at $Re = 170$. Contours of $T_{\theta }^+$ are relatively low in magnitude and noisy. Although XG is a non-Newtonian flow, with evidently large amounts of shear-thinning and linear viscoelasticity (see Appendix B), it too does not have contours of $T_{\theta }^+$ with large magnitude, as seen in figure 21(b). Generally, the plug-like flow of XG within the PCT has a larger region where $\omega _{\theta }^+$ and $T_{\theta }^+$ are equal to 0.

Contours of $\omega _{\theta }^+$ and $T_{\theta }^+$ for PAM with $c=0.03\,\%$ and $Re = 60.5$ are shown in figure 22(a). A zone of large $T_{\theta }^+$ values is interspersed between regions of negative and positive $\omega _{\theta }^+$ within the PCT contraction. This $T_{\theta }^+$ zone appears in areas where values of $\omega _{\theta }^+$ significantly vary in space. The opposite can be observed within the PCT expansion ($0.5< x^+<1$); the vorticity reduces with increasing $x^+$, and hence $T_{\theta }^+$ is at its most negative. Similar observations can be made for the flow of PAM with $c = 0.05\,\%$ and $Re = 35.5$ in figure 22(b). Large values of $T_{\theta }^+$ are interspersed between positive and negative contours of $\omega _{\theta }^+$. In both cases, the Newtonian diffusion term ($VSD$) cannot account for the large spatial variations in $\omega _{\theta }^+$, implying that the non-Newtonian torque is needed to balance the vorticity equation.

Figure 23(a) demonstrates contours of $\omega _{\theta }^+$ and $T_{\theta }^+$ for TTAC with $c = 0.05\,\%$ at $Re = 119$. Similar to the flows of PAM, TTAC exhibits large values of $T_{\theta }^+$ intermittent between the regions of positive and negative $\omega _{\theta }^+$. The largest positive value of $T_{\theta }^+$ occurs within the contraction, where $\omega _{\theta }^+$ changes abruptly from negative to positive with increasing $x^+$. The same can be observed for larger $Re$ flows, such as TTAC with $c = 0.05\,\%$ and $Re=254$, seen in figure 23(b). As $Re$ increases – comparing figure 23(a,b) – the large positive contour of $T_{\theta }^+ = 0.4$ within the PCT contraction moves closer towards the centreline. Overall, the large values of $T_{\theta }^+$ are coupled with the strong spatial variations in $\omega _{\theta }^+$. Distributions in $T_{\theta }^+$ are relatively consistent among solutions of flexible polymers and surfactants as the two solutions apply the same mechanism via non-Newtonian torque for disrupting $\omega _{\theta }^+$. This mechanism is potentially associated with a common rheological feature that produces the non-Newtonian torque.