**3. Materials and Methods**

#### *3.1. Experimental Setup*

Experimental studies to investigate the structure of the viscoelastic fluid flow were carried out on a test bench (FRC KazSC of RAS, Kazan, Russia) shown in Figure 1. The test sections were made of transparent plexiglass with the diameter D1 = 39 (inner)/D1 = 45 (outer) mm (Figure 1c). The inlet and outlet sections had the length of 43 D<sup>1</sup> sufficient to exclude the inflow and outflow effects. The test section was placed in a rectangular duct (Figure 1d) to compensate for optical distortions. The space between the test section and the duct was filled with a fluid with a refractive index close to that of the test section and the duct. The actual fluid temperature was controlled using a resistance temperature device DTS034-RT100.A3.25/1.5 (OVEN, Moscow, Russia) with the uncertainty of 0.1 ◦C

(Figure 1f,g). The fluid temperature was kept constant using a KENTATSU (KENTATSU DENKI, Guangdong, China) air conditioner installed in the room. (Figure 1f,g). The fluid temperature was kept constant using a KENTATSU (KENTATSU DENKI, Guangdong, China) air conditioner installed in the room.

*Polymers* **2022**, *14*, x FOR PEER REVIEW 5 of 11

( ) 2 *k k k <sup>k</sup> <sup>a</sup> r*

λ= −

λ

ρ

<sup>=</sup>

ρ ρ

ρ

η η

ϕϕ

σ

Then

dient

where dependences

**3. Materials and Methods** 

*3.1. Experimental Setup* 

( ) <sup>0</sup>

,

1

ρ

*w*

1

 γρ

η

λ

γ

ρ

ρ

( ) ( )( )

 γρ

*z k k*

*v dr d d*

*dv*

A more detailed description of the method is provided in [17].

η*k k* =η η<sup>0</sup> , / λ

1 1 11

1 ()

=− +

*m k k k*

( ) ( ) ( )( )

*r Kr*

1 1 ... *<sup>z</sup> <sup>n</sup>*

= = ==

γ

*dr Wi Wi*

Experimental studies to investigate the structure of the viscoelastic fluid flow were

carried out on a test bench (FRC KazSC of RAS, Kazan, Russia) shown in Figure 1. The test sections were made of transparent plexiglass with the diameter D1 = 39 (inner)/D1 = 45 (outer) mm (Figure. 1c). The inlet and outlet sections had the length of 43 D1 sufficient to exclude the inflow and outflow effects. The test section was placed in a rectangular duct (Figure 1d) to compensate for optical distortions. The space between the test section and the duct was filled with a fluid with a refractive index close to that of the test section and the duct. The actual fluid temperature was controlled using a resistance temperature device DTS034-RT100.A3.25/1.5 (OVEN, Moscow, Russia) with the uncertainty of 0.1 °C

λ

*n N kk*

/ /

 ρ λρ

1

=

 ηλ

 λ η

*m m rz k k*

 σ

*<sup>k</sup>* ( ) <sup>1</sup> can be obtained from the definition of the shear rate gra-

γ

> λ

*n*

(15)

 ρ

 λλ

*k ka* <sup>=</sup> . (0 1) ≤ ≤

ρ

1

−

*<sup>k</sup>* (13)

(14)

**Figure 1.** Photo of the test bench. (**a**) tank with polymer solution in distilled water; (**b**) valve; (**c**) Plexiglass pipe; (**d**) rectangular duct; (**e**) laser; (**f**) resistance thermocouple; (**g**) temperature moni-**Figure 1.** Photo of the test bench. (**a**) tank with polymer solution in distilled water; (**b**) valve; (**c**) Plexiglass pipe; (**d**) rectangular duct; (**e**) laser; (**f**) resistance thermocouple; (**g**) temperature monitor.

#### tor. *3.2. Visualization*

*3.2. Visualization*  Velocity profiles were estimated using the Smoke Image Velocimetry (SIV, Kazan, Russia) method developed at our institute. It is an optical measurement method. Articles [18,19] give a detailed description of the SIV method, estimation of its accuracy, and applicability to the measurement of various flow characteristics. The SIV method was chosen due to the peculiarities of the image-processing algorithm. In particular, the SIV method requires neither uniform seeding of the flow with tracers nor smoothing the maximum of the cross-correlation function when refining the displacement of tracers with subpixel accuracy. These aspects simplify the preparation of the working fluid and increase the measurement accuracy. Figure 2 illustrates the tracer's motion at different times. For example, the motion of one particle is highlighted with a red rectangle. The detailed description of the SIV method is presented in [18,19]. The used frequency value was sufficient to fulfill the condition for the permissible displacement of the tracer images with respect to the size of the compared interrogation windows at the highest flow rates and provided the convenience of a virtual decrease in the shooting frequency at lower flow rates. The virtual decrease in the shooting frequency was used to increase the displacement of tracers in the compared pairs of frames to the optimal values from the standpoint of measurement accuracy (about half the longitudinal size of the interrogation window). The size of the compared interrogation windows in all cases was 28 <sup>×</sup> 20 pix<sup>2</sup> and was selected based on the patterns of the distribution of tracers in the flow visualization frames. The scaling factor was 24.4 pix/mm. The spacing between grid nodes was 20 pix (0.82 mm). The shooting time and the number of captured frames were sufficient for time averaging of the data. For most of the investigated modes, 1000 pairs of frames were compared.

**Figure 2.** The tracer's motion at different time: (**a**) <sup>0</sup> *t t* = ; (**b**) <sup>0</sup> *tt t* = +Δ . **Figure 2.** The tracer's motion at different time: (**a**) *t* = *t*0; (**b**) *t* = *t*<sup>0</sup> + ∆*t*.

The laser light sheet in the experiments was created by a continuous diode laser SSP-ST-532-NB-LED (Changchun New Industries Optoelectronics Tech. Co. Ltd, Changchun, China) (Figure 1e). The laser light sheet was no more than 0.6 mm thick. Filming was carried out with a Phantom Miro C110 digital video camera (Vision Research, Inc., Wayne, New Jersey, USA) with a frequency of 60 fps in steady flow regimes. The image processing procedure included elimination of static objects and noise filtering. The procedures for searching and filtering erroneous velocity vectors were applied to the The laser light sheet in the experiments was created by a continuous diode laser SSP-ST-532-NB-LED (Changchun New Industries Optoelectronics Tech. Co. Ltd, Changchun, China) (Figure 1e). The laser light sheet was no more than 0.6 mm thick. Filming was carried out with a Phantom Miro C110 digital video camera (Vision Research, Inc., Wayne, New Jersey, USA) with a frequency of 60 fps in steady flow regimes. The image processing procedure included elimination of static objects and noise filtering. The procedures for searching and filtering erroneous velocity vectors were applied to the primary quantitative data. The average number of erroneous vectors in the entire volume of the obtained data was no more than 0.56%.

Velocity profiles were estimated using the Smoke Image Velocimetry (SIV, Kazan,

Russia) method developed at our institute. It is an optical measurement method. Articles [18,19] give a detailed description of the SIV method, estimation of its accuracy, and applicability to the measurement of various flow characteristics. The SIV method was chosen due to the peculiarities of the image-processing algorithm. In particular, the SIV method requires neither uniform seeding of the flow with tracers nor smoothing the maximum of the cross-correlation function when refining the displacement of tracers with subpixel accuracy. These aspects simplify the preparation of the working fluid and increase the measurement accuracy. Figure 2 illustrates the tracer's motion at different times. For example, the motion of one particle is highlighted with a red rectangle. The detailed description of the SIV method is presented in [18,19]. The used frequency value was sufficient to fulfill the condition for the permissible displacement of the tracer images with respect to the size of the compared interrogation windows at the highest flow rates and provided the convenience of a virtual decrease in the shooting frequency at lower flow rates. The virtual decrease in the shooting frequency was used to increase the displacement of tracers in the compared pairs of frames to the optimal values from the standpoint of measurement accuracy (about half the longitudinal size of the interrogation window). The size of the compared interrogation windows in all cases was 28 × 20 pix2 and was selected based on the patterns of the distribution of tracers in the flow visualization frames. The scaling factor was 24.4 pix / mm. The spacing between grid nodes was 20 pix (0.82 mm). The shooting time and the number of captured frames were sufficient for time averaging of the data. For most of the investigated modes, 1000 pairs of frames

#### primary quantitative data. The average number of erroneous vectors in the entire volume *3.3. Polymer Solution Preparation*

were compared.

of the obtained data was no more than 0.56%. *3.3. Polymer Solution Preparation*  An aqueous solution of polyacrylamide with polyamide particles (average diameter 20 µm) was used as a viscoelastic liquid. The influence of gravity and inertia forces on the position of tracers relative to the carrier medium during the measurements was negligible. The fluid was prepared in the following order. Polyamide particles with a concentration of 0.013% of the total mass were added to warm distilled water and slightly (manually) mixed. Then, the mixing process was carried out using an overhead stirrer Heidolph Hei-TORQUE Value 100 (Heidolph Instruments GmbH & Co. KG, Schwabach, Germany) with a ViscoJet stirrer (Heidolph Instruments GmbH & Co. KG, Schwabach, Germany). After two minutes of mixing, the polyacrylamide powder was gradually added. It took 15 min to mix 700 mL volume. The uncertainty in weighing of distilled water, polyacrylamide powder, and polyamide particles was less than 0.1%. Prior to the experiment, the finished liquid was stored in 5 L canisters in a dark cabinet.

#### *3.4. Rheological Measurements*

The viscosity curve, as well as dynamic moduli of two concentration of polymer solutions (2500 and 7500 ppm weight), were measured using MCR 102 rheometer (Anton Paar, Graz, Austria) equipped with a Peltier (H-PTD200) (Anton Paar, Graz, Austria) temperature control system with an accuracy of 0.01 ◦C; parallel-plate geometry was employed with a plate diameter of 50 mm with 1 mm gap.

#### **4. Results and Discussion** samples of polymer solutions (the temperature value was taken from experimental

**4. Results and Discussion** 

*3.4. Rheological Measurements* 

Figure 3 shows the experimental viscosity curve and dynamic moduli of tested samples of polymer solutions (the temperature value was taken from experimental studies of flow visualization). Intermediate calculations showed that four modes are sufficient to approximate the dynamic modulus curves and the viscosity curve, which is consistent with the literature [11] (Figure 3). The literature data [20] were used to find the linear spectrum of the relaxation time for the four-modal rheological equations of state of a viscoelastic medium. The search for a set of nonlinear parameters of rheological models (3) and (4) is based on the flow curve approximation; e.g., for the Giesekus model, a detailed algorithm was presented in our earlier published work [21]. For the reader's convenience, Table 1 presents the parameters of the Giesekus and eXt. Pom-Pom models that we found, which characterize the rheological behavior of the tested samples (2500 and 7500 ppm). From the figure, we can see that the eXt. Pom-Pom model, in comparison with the Giesekus model, best approximates the experimental viscosity curves for both concentrations, especially in the shear rate range 0.1 < . *γ* < 60. A slight deviation in the storage modulus approximation in the interval *ω* < 0.02 was caused by the use of four modes; however, as is shown below, this deviation does not significantly affect the final result. studies of flow visualization). Intermediate calculations showed that four modes are sufficient to approximate the dynamic modulus curves and the viscosity curve, which is consistent with the literature [11] (Figure 3). The literature data [20] were used to find the linear spectrum of the relaxation time for the four-modal rheological equations of state of a viscoelastic medium. The search for a set of nonlinear parameters of rheological models (3) and (4) is based on the flow curve approximation; e.g., for the Giesekus model, a detailed algorithm was presented in our earlier published work [21]. For the reader's convenience, Table 1 presents the parameters of the Giesekus and eXt. Pom-Pom models that we found, which characterize the rheological behavior of the tested samples (2500 and 7500 ppm). From the figure, we can see that the eXt. Pom-Pom model, in comparison with the Giesekus model, best approximates the experimental viscosity curves for both concentrations, especially in the shear rate range 0.1 60 < < γ . A slight deviation in the storage modulus approximation in the interval ω < 0.02 was caused by the use of four modes; however, as is shown below, this deviation does not significantly affect the final result.

experiment, the finished liquid was stored in 5 L canisters in a dark cabinet.

employed with a plate diameter of 50 mm with 1 mm gap.

*Polymers* **2022**, *14*, x FOR PEER REVIEW 7 of 11

An aqueous solution of polyacrylamide with polyamide particles (average diameter 20 μm) was used as a viscoelastic liquid. The influence of gravity and inertia forces on the position of tracers relative to the carrier medium during the measurements was negligible. The fluid was prepared in the following order. Polyamide particles with a concentration of 0.013% of the total mass were added to warm distilled water and slightly (manually) mixed. Then, the mixing process was carried out using an overhead stirrer Heidolph Hei-TORQUE Value 100 (Heidolph Instruments GmbH & Co. KG, Schwabach, Germany) with a ViscoJet stirrer (Heidolph Instruments GmbH & Co. KG, Schwabach, Germany). After two minutes of mixing, the polyacrylamide powder was gradually added. It took 15 min to mix 700 mL volume. The uncertainty in weighing of distilled water, polyacrylamide powder, and polyamide particles was less than 0.1%. Prior to the

The viscosity curve, as well as dynamic moduli of two concentration of polymer solutions (2500 and 7500 ppm weight), were measured using MCR 102 rheometer (Anton Paar, Graz, Austria) equipped with a Peltier (H-PTD200) (Anton Paar, Graz, Austria) temperature control system with an accuracy of 0.01 °C; parallel-plate geometry was

Figure 3 shows the experimental viscosity curve and dynamic moduli of tested

**Figure 3.** Viscosity curve (**a**) and dynamic moduli (**b**); experiment and fitting with four-mode Giesekus and eXt. Pom-Pom models (2500 ppm at *T* = 22.8 ◦C and 7500 ppm at *T* = 20.0 ◦C).


**Table 1.** Parameters of Giesekus and eXt. Pom-Pom models (2500 ppm at *T* = 22.8 ◦C, 7500 ppm at *T* = 20.0 ◦C).

= 20.0 °C).

**Concentration** 

λ*i* **[s]** 

**[ppm]** 

2500

7500

section according to the formula

Since the specific values of the average bulk velocity (*Va*) used in flow visualization were calculated during post-processing by integrating the velocity profile over the pipe section according to the formula then analytical solutions were obtained using the initial experimental data. For 2500 ppm, we visualized four variants of the flow with the values *Va* = 1.866;

*Polymers* **2022**, *14*, x FOR PEER REVIEW 8 of 11

η

0.1406 0.1253

0.1106 0.4699

*i* **[Pa∙s]** 

**Figure 3.** Viscosity curve (**a**) and dynamic moduli (**b**); experiment and fitting with four-mode

**Table 1.** Parameters of Giesekus and eXt. Pom-Pom models (2500 ppm at *T* = 22.8 °C, 7500 ppm at *T* 

 **Giesekus eXt. Pom-Pom** 

α

0.991 0.71 0.495 2 0.4 0.5

7.1911 4.064 0.495 1 0.1 0.7

62.6357 17.1691 0.495 1 0.1 0.6

1.0032 3.8742 0.495 2 0.4 0.1

9.0587 31.7676 0.495 1 0.1 0.6

98.5914 191.802 0.495 1 0.1 0.4

2

*V V rdr R a z* <sup>=</sup> (16)

*<sup>i</sup> <sup>i</sup> q <sup>k</sup>*

0.495 1 0.1 0.9

0.495 1 0.1 0.6

ε

μ

166 10<sup>−</sup> = ×

α*i*

*<sup>N</sup>* **[Pa∙s]** 

0.0411

0.0962

were calculated during post-processing by integrating the velocity profile over the pipe

0

2 *R*

Since the specific values of the average bulk velocity (*Va* ) used in flow visualization

Giesekus and eXt. Pom-Pom models (2500 ppm at *T* = 22.8 °C and 7500 ppm at *T* = 20.0 °C).

η

$$V\_d = 2\int\_0^R V\_\mathcal{Z} r dr / R^2 \tag{16}$$

then analytical solutions were obtained using the initial experimental data. *Va* = 0.993; 1.561 [mm/s] were investigated, which corresponded to the following *Wi* =

For 2500 ppm, we visualized four variants of the flow with the values *Va* = 1.866; 4.579; 7.511; 9.588 [mm/s], which corresponded to the following Weissenberg numbers: *Wi* = 4.80; *Wi* = 11.76; *Wi* = 19.29; *Wi* = 24.63. For 7500 ppm, two variants with the values *Va* = 0.993; 1.561 [mm/s] were investigated, which corresponded to the following *Wi* = 4.29; 6.74. 4.29; 6.74. The visualization method was preliminarily tested on the flow of 40% glycerol and 60% propylene glycol (Figure 4). Good agreement of the experimental data with the

The visualization method was preliminarily tested on the flow of 40% glycerol and 60% propylene glycol (Figure 4). Good agreement of the experimental data with the parabolic velocity profile characterizing the laminar flow of Newtonian fluid in a round pipe was obtained. The relative uncertainty did not exceed 0.6%. parabolic velocity profile characterizing the laminar flow of Newtonian fluid in a round pipe was obtained. The relative uncertainty did not exceed 0.6%.

**Figure 4.** Dimensionless axial velocity profiles in the pipe for various *v*a [mm/s] (Newtonian flow, the mixture of 40% glycerol and 60% propylene glycol with dynamic viscosity <sup>3</sup> **Figure 4.** Dimensionless axial velocity profiles in the pipe for various *v*<sup>a</sup> [mm/s] (Newtonian flow, the mixture of 40% glycerol and 60% propylene glycol with dynamic viscosity *<sup>µ</sup>* = 166 <sup>×</sup> <sup>10</sup>−<sup>3</sup> Pa·s; *T* = 22.6 ◦C).

Pa∙s; *T* = 22.6 °C). Figures 5 and 6 show the profiles of dimensionless axial velocity in the longitudinal section of the channel, which characterize the laminar flow of tested polymer solutions in the circular pipe. The illustration of the obtained data shows that the experimental data Figures 5 and 6 show the profiles of dimensionless axial velocity in the longitudinal section of the channel, which characterize the laminar flow of tested polymer solutions in the circular pipe. The illustration of the obtained data shows that the experimental data are best predicted using the eXt. Pom-Pom model, while the relative uncertainty consequently does not exceed 0.9 and 0.96% for 2500 and 7500 ppm. For the Giesekus model, there is a tendency to overestimate the axial velocity values on the channel axis with a decrease in the Weissenberg number. For 2500 ppm, the relative uncertainty on the channel axis does not exceed 1.2% at *Wi* = 24.63 and it is 4.66% at *Wi* = 4.80. For 7500 ppm the discrepancy is more pronounced, so the relative uncertainty equals 8.77% at *Wi* = 4.29 and 7.49% at *Wi* = 6.74. Additional calculation was performed for a particular case, i.e., *Wi*→0 (tending to Newtonian flow), to check the correctness of the developed method. At *Wi* = 0.01, the calculated velocity profiles predicted by both the Giesekus and eXt. Pom-Pom models coincided with the parabolic profile. The latter indicates that even for the simplest case of viscoelastic fluid flow in a round pipe, the Giesekus model overestimates the value of axial velocity in the central region of the channel in the range of Weissenberg numbers 0.1 < *Wi* < 25. The obtained results agree with the results of other authors who investigated more complex geometries [22].

authors who investigated more complex geometries [22].

are best predicted using the eXt. Pom-Pom model, while the relative uncertainty consequently does not exceed 0.9 and 0.96% for 2500 and 7500 ppm. For the Giesekus model, there is a tendency to overestimate the axial velocity values on the channel axis with a decrease in the Weissenberg number. For 2500 ppm, the relative uncertainty on the channel axis does not exceed 1.2% at *Wi* = 24.63 and it is 4.66% at *Wi* = 4.80. For 7500 ppm the discrepancy is more pronounced, so the relative uncertainty equals 8.77% at *Wi* = 4.29 and 7.49% at *Wi* = 6.74. Additional calculation was performed for a particular case, i.e., *Wi*→0 (tending to Newtonian flow), to check the correctness of the developed method. At *Wi* = 0.01, the calculated velocity profiles predicted by both the Giesekus and eXt. Pom-Pom models coincided with the parabolic profile. The latter indicates that even for the simplest case of viscoelastic fluid flow in a round pipe, the Giesekus model overestimates the value of axial velocity in the central region of the channel in the range of Weissenberg numbers 0.1 < *Wi* < 25. The obtained results agree with the results of other

**Figure 5.** Dimensionless axial velocity profiles in the pipe for various *Wi* ( <sup>1</sup> 50.087 s λ*a* <sup>−</sup> = ): (**a**) *Wi* = 4.80; (**b**) *Wi* = 11.76; (**c**) *Wi* = 19.29; (**d**) *Wi* = 24.63. **Figure 5.** Dimensionless axial velocity profiles in the pipe for various *Wi* (*λ<sup>a</sup>* = 50.087 s −1 ): (**a**) *Wi* = 4.80; (**b**) *Wi* = 11.76; (**c**) *Wi* = 19.29; (**d**) *Wi* = 24.63.

**Figure 6.** Dimensionless axial velocity profiles in the pipe for various *Wi* ( <sup>1</sup> 84.25 s λ*a* <sup>−</sup> = ): (**a**) *Wi* = 4.29; (**b**) *Wi* = 6.74. **Figure 6.** Dimensionless axial velocity profiles in the pipe for various *Wi* (*λ<sup>a</sup>* = 84.25 s −1 ): (**a**) *Wi* = 4.29; (**b**) *Wi* = 6.74.

All authors have read and agreed to the published version of the manuscript.

with the financial support of the state assignment for the FRC KazSC of RAS.

Our proposed parametric method for multimode Giesekus fluid flows remains sta-

The best agreement with the experimental data was obtained using the eXt. Pom-Pom model for example of pipe flows of aqueous polyacrylamide solutions with 2500 and 7500 ppm concentrations. The SIV method used to visualize turbulent flows demonstrated its efficiency for visualization of viscoelastic fluid flows and can be successfully

**Author Contributions:** Supervision E.V.; visualization N.D.; data curation E.K.; methodology A.K.

**Funding:** The experimental investigation was carried out with the financial support of the Russian Science Foundation (Project no. 19-11-00220). The adaptation of the SIV method was carried out

**Data Availability Statement:** The data presented in this study are available on request from the

applied to the analysis of more complex flows.

**Institutional Review Board Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflicts of interest.

1. Cruz, D.O.A.; Pinho, F.T. Skewed Poiseuille-Couette Flows of sPTT Fluids in Concentric Annuli and Channels. *J. Non-Newt.* 

2. Cruz, D.O.A.; Pinho, F.T.; Oliveira, P.J. Analytical solutions for fully developed laminar flow of some viscoelastic liquids with a Newtonian solvent contribution, *J. Non-Newtonian Fluid Mech.* **2005**, *132*, 28–35. https://doi.org/10.1016/j.jnnfm.2005.08.013. 3. Oliveira, P.J. An exact solution for tube and slit flow of a FENE-P fluid. *Acta Mechanica*. **2002**, *158*, 157–167.

4. Schleiniger, G.; Weinacht, R.J. Steady Poiseuille flows for a Giesekus fluid. *J. Non-Newt. Fluid Mech*. **1991**, *40*, 79–102.

5. Oliveira, P.J.; Coelho, P.M.; Pinho, F.T. The Graetz problem with viscous dissipation for FENE-P fluids. *J. Non-Newt. Fluid Mech*.

6. Coelho, P.M.; Pinho, F.T.; Oliveira, P.J. Thermal entry flow for a viscoelastic fluid: The Graetz problem for the PTT model", *Int.* 

7. Coelho, P.M.; Pinho, F.T.; Oliveira, P.J. Fully-developed forced convection of the Phan-Thien-Tanner fluid in ducts with a constant wall temperature. *Int. J. Heat and Mass Transfer.* **2002**. 45, 1413–1423. https://doi.org/10.1016/s0017-9310(01)00236-8.

*J. Heat and Mass Transfer*. **2003**, *46*, 3865–3880. https://doi.org/10.1016/S0017-9310(03)00179-0.

**Informed Consent Statement:** Not applicable.

corresponding author.

*Fluid Mech*, **2004**, *121*, 1–14. https://doi.org/10.1016/j.jnnfm.2004.03.007.

https://doi.org/10.1007/BF01176906.

https://doi.org/10.1016/0377-0257(91)87027-U.

**2004**, *121*, 69–72. https://doi.org/10.1016/j.jnnfm.2004.04.005.

**References** 

**5. Conclusions** 

#### **5. Conclusions**

Our proposed parametric method for multimode Giesekus fluid flows remains stable in a wide range of Weissenberg numbers, but the velocity profiles predicted by this model are slightly higher than the experimental data in the central region of the channel. The best agreement with the experimental data was obtained using the eXt. Pom-Pom model for example of pipe flows of aqueous polyacrylamide solutions with 2500 and 7500 ppm concentrations. The SIV method used to visualize turbulent flows demonstrated its efficiency for visualization of viscoelastic fluid flows and can be successfully applied to the analysis of more complex flows.

**Author Contributions:** Supervision E.V.; visualization N.D.; data curation E.K.; methodology A.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** The experimental investigation was carried out with the financial support of the Russian Science Foundation (Project no. 19-11-00220). The adaptation of the SIV method was carried out with the financial support of the state assignment for the FRC KazSC of RAS.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

