A fundamental understanding of the transition from fluid-like to gel-like behavior is critical for a range of applications including personal care, pharmaceuticals, food products, batteries, painting, biomaterials, and concrete. The pipe flow behavior of a Herschel–Bulkley fluid is examined by a combination of rheology, ultrasound imaging velocimetry, and pressure measurements together with modeling. The system is a solution of 0.50 wt. % polyelectrolytes of sulfated polysaccharides in water that solidifies on cooling. Fluids with different ionic strengths were pumped at various rates from a reservoir at 80 °C into a pipe submerged in a bath maintained at 20 °C. The fluid velocity, pressure drop $\Delta P$, and temperature were monitored. The same quantities were extracted by solving continuity, energy, and momentum equations. Moreover, the modeling results demonstrate that the local pressure gradient along the pipe $ d P d x | x$ is related to the local yield stress near the pipe wall $ \tau y wall | x$, which explains the variations of $ d P d x | x$ along the pipe. Experimental results show much lower values for $\Delta P$ compared to those from modeling. This discrepancy is exacerbated at higher ionic strengths and smaller flow rates, where fluid shows a higher degree of solidification. The tabulated experimental $\Delta P$ data against the solidification onset length $ L o n set$ (where the fluid is cool enough to solidify) along with the ultrasound imaging velocimetry associate these discrepancies between experiments and models to a depletion layer of ∼1 $\mu m$, reflecting the lubrication effects caused by the water layer at the wall.

## I. INTRODUCTION

Yield stress fluids are present in a wide range of products such as toothpastes, milk, mayonnaise, paints, cements, and foams as well as in biological systems. The fundamental feature of these materials is that they exhibit a solid-like response when the imposed stress is below a critical value. However, they yield and flow when they are submitted to stresses above this critical yield stress value.^{1}

Gels formed by networks of interconnected particles and macromolecules are an example of such yield stress fluids.^{2,3} Nowadays, there is growing demand for gels made from renewable sources of biodegradable raw materials. In this regard, carrageenans are of particular interest to producing bio-friendly gels. Carrageenans are a family of natural linear sulfated polysaccharides that are extracted from red edible seaweeds.^{4,5} κ-Carrageenan is a type of carrageenan that is composed of alternating α(1–3)-d-galactose-4-sulfated and β(1–4)-3,6-anhydro-d-galactose. It contains one sulfate group per disaccharide unit at carbon 2 of the 1,3 linked galactose unit. The ^{1}C_{4} conformation of the 3,6-anhydro-d-galactose unit allows a helicoidal secondary structure, which is necessary for its gelation.^{6,7} κ-Carrageenan is widely used as a gelling agent, thickening or texture enhancer, as well as a stabilizer in cosmetic, food, and pharmaceutical industries.^{8–14}

The gelation of κ-carrageenan in water is a complicated process. However, it is widely accepted that the gelation takes place through a two-step mechanism. At high temperatures, κ-carrageenan molecules are soluble in water, having a coil configuration. Upon cooling, a configurational transition occurs from a coil to a helical structure.^{15} Finally, the aggregation of these helices leads to gelation.^{16,17} In addition, it is well known that the addition of specific cations such as K^{+}, Rb^{+}, and Cs^{+} strongly favors the aggregation of helices due to the interaction between the sulfate ester groups and cations.^{18,19}

The linear viscoelastic (LVE) measurements have been widely used to probe the sol-gel transition and rheological properties of gels.^{20–25} Despite the significant work on the LVE properties of κ-carrageenan systems, much less is known about the flow properties of these gels.^{26} This is partly due to the fact that in the case of complex heterogeneous fluids, the rheological properties may be affected by flow instabilities and wall slip.^{27–32} Hence, using techniques that are capable of direct measurement of the velocity profile are necessary, including ultrasound imaging velocimetry.^{33–35}

In this work, steady shear tests were used to determine the Herschel–Bulkley^{36,37} flow parameters for a solution of κ-carrageenan (the polyelectrolyte) in water, with different ionic strengths to be able to tune the solidification for the polyelectrolyte. The solidification process was monitored using oscillatory cooling ramps, followed by using elastic modulus $G\u2032$ readings during cooling to obtain a power law equation for yield stress $ \tau y$ vs temperature $T$ while solidification was in progress $ \tau y=a T b$. This equation was included in the flow constitutive equation to be able to simultaneously solve the continuity, energy, and momentum equations for the pressure-driven flow of a hot fluid entering a cool pipe to obtain steady state temperature and velocity profiles across and along the pipe. In addition, an experimental setup was used to measure pressure drop $\Delta P$ along the same geometry used in the modeling (i.e., a circular pipe). A significant discrepancy in $\Delta P$ values was revealed for those from the model vs the experimental ones, with the model results showing much higher $\Delta P$ than those from the experiments for fluids with high ionic strengths and small flow rates (hence, showing a higher degree of solidification passing through the pipe). It is hypothesized that for regions with solidification involved (i.e., cool enough: $T\u2264 T onset$), wall slip or a ∼1 $\mu m$ water film caused the inconsistency. Plotting the experimental $\Delta P$ data against the solidification onset length $ L onset$ (where fluid $T$ at pipe wall reaches $ T onset$), the influence of such wall slip/depletion layer was tabulated. In addition, we used ultrasound imaging velocimetry to investigate the flow behavior of the polyelectrolyte solutions at different cation concentrations, with varying flow rates, and at different axial locations (with respect to the onset of solidification $ L onset$). The ultrasound imaging velocimetry revealed a plug flow for regions with $ x > L onset$, reflecting the lubrication effects caused by the slip/depletion layer at the wall.

## II. MATERIALS AND METHODS

Sulfated polysaccharide (κ-carrageenan) and potassium chloride (KCl) were purchased from Sigma-Aldrich. Solutions of 0.50 wt. % κ-carrageenan in water (the polyelectrolyte solution) with 0.00–0.20 M KCl were prepared using a magnet stirrer at room temperature with adequate mixing time (>4 h). Figure 1s in the supplementary material depicts the hydration process for the polyelectrolyte in water.

Steady shear and oscillatory temperature sweeps were run using a DHR-3 TA Instruments rheometer equipped with a cone plate geometry (diameter 40 mm, cone angle 2°, and truncation 52 *μ*m). A layer of silicone on the outer edge of the sample was used to minimize sample evaporation. The steady shear data for the samples at 80 °C (i.e., in liquid state) and 20 °C (followed by cooling from 80 °C and upon completion of the solidification while conditioned at 20 °C) are shown in Fig. 1. The ionic strength (i.e., 0.00–0.20 M KCl) did not influence the steady shear data at 80 °C. However, the higher the ionic strength, the higher the strength of the gel at 20 °C, consistent with findings published elsewhere.^{16–19}

T (°C) . | Ionic strength (M) . | $ \tau y$ (Pa) . | $K$ ( $Pa\u2009 s n$) . | $n$ . |
---|---|---|---|---|

80 | 0.00–0.20 | 1.45 | 2.50 | 0.95 |

20 | 0.00 | 101.34 | 28.42 | 0.75 |

20 | 0.05 | 395.23 | 191.96 | 0.69 |

20 | 0.10 | 1146.16 | 768.43 | 0.62 |

20 | 0.15 | 2177.70 | 1580.53 | 0.59 |

20 | 0.20 | 2613.23 | 2079.62 | 0.57 |

T (°C) . | Ionic strength (M) . | $ \tau y$ (Pa) . | $K$ ( $Pa\u2009 s n$) . | $n$ . |
---|---|---|---|---|

80 | 0.00–0.20 | 1.45 | 2.50 | 0.95 |

20 | 0.00 | 101.34 | 28.42 | 0.75 |

20 | 0.05 | 395.23 | 191.96 | 0.69 |

20 | 0.10 | 1146.16 | 768.43 | 0.62 |

20 | 0.15 | 2177.70 | 1580.53 | 0.59 |

20 | 0.20 | 2613.23 | 2079.62 | 0.57 |

^{38,39}

Ionic strength (M) . | $ \tau y i$ (Pa) . | $ \tau y sat$ (Pa) . | $a$ ( $Pa/ \xb0 C b$) . | $b$ . | $ T ost$ (°C) . | $ T sat$ (°C) . |
---|---|---|---|---|---|---|

0.0 | 1.45 | 101.34 | 4.78 × 10^{17} | −11.53 | 33.06 | 22.88 |

0.05 | 1.45 | 395.23 | 1.62 × 10^{30} | −19.11 | 37.36 | 27.86 |

0.10 | 1.45 | 1146.16 | 3.45 × 10^{37} | −22.38 | 46.78 | 34.72 |

0.15 | 1.45 | 2177.70 | 1.45 × 10^{39} | −22.71 | 52.15 | 37.79 |

0.20 | 1.45 | 2613.23 | 2.25 × 10^{39} | −22.64 | 53.83 | 38.66 |

Ionic strength (M) . | $ \tau y i$ (Pa) . | $ \tau y sat$ (Pa) . | $a$ ( $Pa/ \xb0 C b$) . | $b$ . | $ T ost$ (°C) . | $ T sat$ (°C) . |
---|---|---|---|---|---|---|

0.0 | 1.45 | 101.34 | 4.78 × 10^{17} | −11.53 | 33.06 | 22.88 |

0.05 | 1.45 | 395.23 | 1.62 × 10^{30} | −19.11 | 37.36 | 27.86 |

0.10 | 1.45 | 1146.16 | 3.45 × 10^{37} | −22.38 | 46.78 | 34.72 |

0.15 | 1.45 | 2177.70 | 1.45 × 10^{39} | −22.71 | 52.15 | 37.79 |

0.20 | 1.45 | 2613.23 | 2.25 × 10^{39} | −22.64 | 53.83 | 38.66 |

Equations (11) and (12) form an initial value problem that can solved using the Euler method. The finite difference method utilized to solve the energy and momentum equation in this paper is described in the supplementary material.

In addition, an experimental setup (described in Fig. 3s of the supplementary material) was used for collecting pressure drop-flow rate data, to be compared with those from the modeling. Moreover, an ultrasound source with a linear probe was used to measure the velocity profile of solutions with different ionic strengths and pumping rates. The setup was calibrated using a Newtonian fluid (corn syrup) with controlled average flow rates. The probe had a 0.35 mm radial resolution without the capability to be submerged under water (in the water bath). Hence, the ultrasound and probe were placed around the pipe with natural convection with the ambient air at $ T \u221e=18\u2009\xb0C$. Similar to the case for forced convection with the bath water, the heat transfer coefficients were determined using the average $T$ for the fluid exiting the pipe.

## III. RESULTS AND DISCUSSION

Example two-dimensional $T$ and $ v x$ profiles are depicted in Figs. 4s and 5s of the supplementary material, respectively. Using Figs. 4s and 5s, example one-dimensional $T$ and $ v x$ profiles are shown in Fig. 3, for the sample with ionic strength 0.20 M at three different flow rates 0.5, 1.0, and 1.5 ml min.

Knowing the numerical values for $ v x$, the velocity gradient along the pipe $ d v x d x$ can be obtained using backward finite differences. Then, using the Euler method for initial value problems, the radial velocity can be obtained (no slip condition at the wall is also assumed to be true for the radial velocity component $ v r$). These calculations are explained in the supplementary material using Eqs. (14s) and (15s).

Using Eq. (13) for a fluid with pronounced solidification passing through the pipe (e.g., with ionic strength 0.20 M at 1.0 ml min), the average $| v r|\u22450.002$ mm/s, a value much smaller than $| v x|\u22452.358$ mm/s. Juxtaposing these values, the assumption to neglect $ v r$ in the equation of motion is justified.

For the heat transfer equation, using the average Peclet number^{40,41} in r-direction $| P e r|= d * | v r | \alpha \u22450.036$, where $d$ is the pipe diameter, and $\alpha $ is the fluid thermal diffusivity, the assumption to neglect the radical heat convection compared with the radial heat conduction is also justified.

Obtaining the two-dimensional $T$ profile for various cases (varying flow rate and ionic strength), the solidification status for the fluid along and across the pipe can be determined using the criteria explained as follows. Solidification not started: $T> T ost$, solidification in progress: $ T sat\u2264T\u2264 T ost$, and solidification completed: $T< T sat$. An example of such calculations is shown in Fig. 4 for the sample with ionic strength 0.20 M and at different flow rates 0.5, 1.0, and 1.5 ml min.

An example of two-dimensional profiles for shear stress and yield stress is shown Fig. 6s of the supplementary material. Using such stress mappings, yield zone graphs can be obtained, such as that of Fig. 5, with yield zone showing $ \tau r x\u2265 \tau y$.

In Fig. 5, the yield zone becoming narrower upon the onset of solidification can be related to the deceleration of the fluid near the wall upon cooling (satisfying the no wall slip condition) and simultaneously the regions closer to the pipe center accelerating (to satisfy the continuity equation). This can be seen in the velocity profile at the axial positions surrounding the solidification onset (Fig. 6).

The pressure gradient along the pipe (Fig. 7) is rather interesting: for a non-solidifying fluid, or without rheopexy, the pressure drop is expected to reduce upon lowering the flow rate. The opposite trend in Fig. 7(a) is related to the solidifying nature of the fluid in this study: with higher residence time passing through the pipe provided at lower flow rates, the pressure needed to make the fluid with a higher degree of solidification yield would be higher. In other words, the shear stress at wall needed for the cooler fluids to yield is higher than those maintaining higher $T$ due to shorter residence time at higher flow rates.

From Fig. 7(b), it is evident that for sufficiently high flow rates (without solidification occurring passing through the pipe), the model and experimental results match quite well. However, as $\Delta P$ increases for flow rates offering long enough residence time passing through the pipe to initiate the solidification, the experimental $\Delta P$ values actually start to drop. This behavior is surprising at first, but could be attributed to the formation of a low viscosity (water-rich) layer at the wall upon the solidification onset. This reasonably occurs at higher flow rates for fluids with larger $ T ost$, with higher ionic strengths. The influence of such water-rich depletion layers on the flow behavior of aqueous yield stress fluids is described by Nazari *et al.*^{28,29}

Hence, using the $ L onset$ values from the model and the experimental pressure drop values, $\Delta P$ vs $ L onset$ data clusters for a given $Q$ were plotted in Fig. 9. The inset in Fig. 9 shows the intercept of such linear regressions is proportional to the flow rate $Q$, in line with Eq. (16). This indicates the validity of our hypothesis about the formation of a water-rich depletion layer upon the solidification onset. Hence, considering 1 mPa s for $ \mu w$, using Eq. (16) and the data in Fig. 9, an estimated value of ∼1 *μ*m can be obtained for the thickness of this water layer $ l w$. This estimated value for the depletion layer thickness is consistent with the literature published on the flow of aqueous xanthan solutions through pores (i.e., reported $ l w\u2009\u223c0.3\u2009\mu m$).^{42} Considering the similarities existing in the structures of *k*-carrageenan and xanthan (as polycations), this consistency in the reported depletion layer thickness $ l w$ is rather interesting. A handful of papers addressed the physics surrounding the formation of a polymer-depleted wall layer upon the migration of the macromolecules away from the high stress locations near the wall in flow configurations with non-homogeneous stress distributions.^{43–45} It has been indicated that “demixing” (i.e., flowing polymeric media becoming non-homogeneous) is dependent on flow rate, polymer concentration, temperature, polymer molecular weight, etc.^{44} In this work, we did not have the means to study the effect of these factors on depletion. Hence, it is suggested to continue this work with varying the aforementioned factors to further study depletion in these solidifying polyelectrolyte systems. Nonetheless, since the polyelectrolyte in our study undergoes a sol-gel transition near the cool wall, it is hypothesized that the formation of a water-rich depletion layer (possibly with no macromolecular presence) is facilitated due to water release from the resulting gel, with no need for major macromolecular diffusion away from the wall (which is the case for polymeric solutions/melts showing depletion).

As described in Sec. II, the ultrasound probe could not be submerged under water. Therefore, the setup was placed around the pipe with natural convection with air at $ T \u221e=18\u2009\xb0C$. The solidification status graphs for three different flow rates and for the fluid with ionic strength 0.20 M are shown in Fig. 8s, while the pipe has convective heat transfer with air. This resulted in solidification onset length $ L onset\u2245$ 54.2, 108.4, and 162.6 mm for flow rate 0.25, 0.50, and 0.75 ml min, respectively. As depicted using the dashed boxes in Fig. 8s, the approximate positions the ultrasound velocimeter was used at was ∼20 mm before and after the solidification onset length $ L onset$ to ensure velocimetry in both yield stress flow and plug flow regions [as described in Fig. 8(a)]. The results are shown in Fig. 10.

From Fig. 10, it is quite evident that for axial locations prior to $ L onset$, the ultrasound imaging velocimetry data (squares) are in line with those from the modeling (solid lines). However, for $ x > L onset$, the ultrasound imaging velocimetry data (circles) do not agree with those from the modeling (dashed lines). Instead, the former data clusters populate around the plug flow equal to the average axial velocity value for each flow rate. This supports the picture depicted in Fig. 8(a) regarding the abrupt transition of the yield stress flow (with no wall slip) to a plug flow $ V p$ at $x= L onset$ due to the formation of a water-rich depletion layer upon the solidification onset. The authors in passing refer to the alternative possibility for the presence of wall slip (see, for example, the fifth line in their conclusions). It is worth emphasizing that due to the rather course resolution of the ultrasound probe used (i.e., ∼0.35 mm radial resolution), we cannot directly comment on the flow pattern near the wall (e.g., wall slippage vs a depletion layer with no slip at wall). However, we plan to address this in the form of future publications by including a Navier-slip condition^{43–47} in our calculations as well as using ultrasound probes with much finer radial resolutions to better compare experimental data with those from the modeling. The recent contributions to understanding the wall slip phenomenon for Herschel–Bulkley fluids^{48–50} along with our future experimental will certainly help provide more physical explanations about the origin and magnitude of the wall slip as well as the nature of the depletion layer (e.g., in terms of polymer concentration, flow pattern near the wall).

## IV. CONCLUSIONS

Comparing data from the model with those from the experiments reveals a significant discrepancy in total pressure drop along the pipe $\Delta P$ values, with the model results showing much higher $\Delta P$ than those from the experiments. The discrepancy is exacerbated at higher ionic strength and smaller flow rates where fluid shows a higher degree of solidification passing through the pipe. This discrepancy for regions with solidification involved (*T < T _{onset}*) is attributed to wall slip or a ∼1

*μ*m water film. The influence of such wall slip/depletion layer was tabulated by plotting the experimental $\Delta P$ data against the solidification onset length L

_{onset}(where fluid

*T*at the pipe wall reaches

*T*).

_{onset}As depicted in Fig. 8(b), the modeling results showed the pressure gradient along the pipe $ d P d x$ is related to *τ _{y}* near the pipe wall

*τ*(illustrated using the dotted line). This explains the rapid pressure drop in the saturated region (

_{y}^{wall}*T < T*) where

_{sat}*τ*has reached the highest value possible

_{y}^{wall}*τ*. In the region with solidification in progress (

_{y}^{sat}*T*

_{sat}*≤ T ≤ T*), $ d P d x$ ∼

_{onset}*τ*increases due to an increase in

_{y}^{wall}*τ*upon cooling. For the region where solidification has not started yet (

_{y}*T > T*), $ d P d x$ ∼

_{onset}*τ*remains constant since

_{y}^{wall}*τ*is nearly constant (i.e., ∼1.45 Pa). This explains the much lower $ d P d x$ in this region compared to those in the solidification in progress and saturated regions. On the contrary, due to a depletion layer of ∼1

_{y}*μ*m for regions with

*x*>

*L*, $ d P d x$ from the experimental data are much lower than those from the model [illustrated using the dashed line in Fig. 8(a)], reflecting the lubrication effects caused by the water layer at the wall.

_{onset}This perspective was substantiated using the ultrasound imaging velocimetry data, showing an abrupt transition from a yield stress flow (with no wall slip) to a plug flow at $x= L onset$ due to the formation of a water-rich depletion layer upon the solidification onset.

## SUPPLEMENTARY MATERIAL

See the supplementary material for additional figures, a list of variables used, and the finite difference solution used.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Behzad Nazari:** Conceptualization (lead); Investigation (lead); Writing – original draft (lead); Writing – review & editing (lead). **Esmaeel Moghimi:** Formal analysis (equal); Writing – review & editing (equal). **Douglas Bousfield:** Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available within the article and its supplementary material.

## REFERENCES

*Food Polysaccharides and Their Applications*

*kofta*

*Ultrasonic Doppler Velocity Profiler for Fluid Flow*