# Absolute Rheological Measurements of Model Suspensions: Influence and Correction of Wall Slip Prevention Measures

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

#### Wall Slip Correction

## 3. Geometries

^{2}. The structure depth was varied between $0.1$ and $1.0\mathrm{mm}$, and the column distance between $0.5$ and $1.2\mathrm{mm}$, resulting in seven different columnar systems. Another measuring geometry consists of $0.5\mathrm{mm}$ thick, radially arranged, longwise bars, in the following only referred to as bars. The structure depth was varied between $0.3$ and $1.0\mathrm{mm}$, and the number of bars was incremented from $16$ to $20$. Because of manufacturing restrictions, the plate diameter of all seven bar systems was $30\mathrm{mm}$. The geometric dimensions for all systems are listed in the following tables (Table 1, Table 2 and Table 3). All plates were manufactured using a turning lathe and by precisely milling the structures into the plates.

## 4. Methods

#### 4.1. Zero Gap Determination

#### 4.2. Measurement Preparation

#### 4.3. Measurement Profile

## 5. Results and Discussion

#### 5.1. Newtonian Fluids

#### 5.1.1. Influence of Measuring Gap and Material

#### 5.1.2. Influence of Shear Rate

#### 5.1.3. Determination of Gap Extension $\mathsf{\delta}$

#### 5.1.4. Correction of Viscosity

#### 5.1.5. Influence of Different Structural Features

#### 5.1.6. Comparison to Previous Studies

- The relative influence of different modified geometries is viscosity-independent for the investigated range;
- A reduction of the measuring gap decreases the normalised viscosity due to the increasing influence of $\delta $;
- The values for $\delta $ increase with increasing shear rate and gap, resulting from increasing angular velocity;
- Since the dependency of $\delta $ from the measuring gap and from the material is small, the development of a gap- and material-independent correction function $\delta =f\left(\dot{\gamma}\right)$ is reasonable;
- An increasing structure depth as well as a decreasing number of structure elements lead to increasing values of $\delta $.

#### 5.2. Shear Thinning Fluid

#### 5.2.1. Comparison to Newtonian Fluids

#### 5.2.2. Influence of Measuring Gap and Material

#### 5.2.3. Influence of Shear Rate

#### 5.2.4. Determination of Gap Extension $\mathsf{\delta}$

#### 5.2.5. Correction of Viscosity

#### 5.2.6. Influence of Different Structural Features

#### 5.2.7. Comparison to Previous Studies

- The relative influence of structured geometries is far smaller than that observed for Newtonian fluids and is also independent of the strength of the shear thinning effect in the range of the investigated fluids;
- In contrast to Newtonian fluids, no exact relationship between measuring gap and normalised viscosity was observed, which is the result of two counteracting effects—gap height and shear thinning—combined with the overall small influence of structured geometries;
- The values for $\delta $ decrease with increasing shear rate, resulting from complex shear fields combined with shear thinning properties of xanthan gum solutions;
- The development of a gap-independent correction function $\delta =f\left(\dot{\gamma}\right)$ is possible, since observed deviations due to different gaps heights are small;
- Because of the relatively small changes of $\delta $, it was not possible to reveal if $\delta $ is dependent on changes in structure depth in the case of columnar structures and bars;
- Concerning pyramids, an increase of structure depth, leading to a decrease of number of elements, leads to an increase of $\delta $;
- An increase of bars leads to an increase of $\delta $.

#### 5.3. Suspensions

#### 5.3.1. Comparison to Newtonian Fluids

#### 5.3.2. Correction of Viscosity

#### 5.3.3. Comparison to Previous Studies

- The normalised viscosities for the investigated concentrations, $5\mathrm{wt}.\text{}\%$, $25\mathrm{wt}.\text{}\%$, and $35\mathrm{wt}.\text{}\%$, do not deviate from the normalised viscosities obtained for silicon oils;
- The normalised viscosity of suspensions shows the same gap dependency as silicon oils;
- For the investigated suspensions, no significant deviation in the effective gap extension $\delta $ from silicon oils could be observed;
- Since the suspensions behaved only in weakly shear thinning manner, the assumption can be made that the relative influence of structured geometries mainly depends on the flow properties of the investigated materials;
- The viscosity can be corrected using a gap-independent correction function;
- Since the flow behaviour of the suspensions was only weakly shear thinning, the material-independent correction function derived for Newtonian oils could be applied to correct the suspension viscosity values, showing that slight deviations from Newtonian flow behaviour do not pose a problem.

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Schematic illustration of the velocity profile in a parallel-plate system (gap height $H$, plate radius $R$); (

**a**) showing wall slip (slip velocity ${u}_{S}$) with the particle-depleted area $\epsilon $; (

**b**) with grooved surface to prevent wall slip.

**Figure 2.**Comparison between apparent shear rate ${\dot{\mathsf{\gamma}}}_{\mathrm{a}}$ and true shear rate $\dot{\mathsf{\gamma}}$ in the velocity profile between two parallel plates showing wall slip.

**Figure 3.**CAD construction of the modified parallel-plate systems with pyramidal structure (

**a**), columnar structure (

**b**), and longwise bars (

**c**).

**Figure 4.**Comparison between apparent shear rate ${\dot{\mathsf{\gamma}}}_{\mathrm{a}}$ and true shear rate $\dot{\mathsf{\gamma}}$ in the velocity profile between two parallel plates showing wall slip, measured with columns (system 6), bars (system 12), and pyramids (system 2).

**Figure 5.**Apparent viscosity of silicon oil AK 5000 as a function of shear rate using bars (system 13) at a gap height of $H=1.2\text{}\mathrm{mm}$ and reference viscosity determined with smooth plates.

**Figure 6.**Effective gap extension $\delta $ dependent on shear rate for AK 5000, measured with different pyramidal structures (systems 1–3).

**Figure 7.**Uncorrected, corrected, and reference viscosity from silicon oil AK 5000, measured with pyramids (system 2).

**Figure 8.**Correction function for the silicon oils AK 5000 and AK 12,500, measured with pyramids (systems 2) and columns (system 6).

**Figure 9.**Uncorrected, corrected, and mean correction functions of AK 5000 viscosities, measured with pyramids (system 2).

**Figure 10.**Effective gap extension $\delta $ (gap- and material-averaged) dependent on structure depth, measured with pyramidal structure (system 1–4), columnar structure (systems 5, 8, 10, 11) and bars (system 12–15) for $\dot{\gamma}=5{\text{}\mathrm{s}}^{-1}$.

**Figure 11.**Effective gap extension $\mathsf{\delta}$ (gap- and material-averaged) dependent on column distance (

**a**) (systems 6–9, $L=0.5\mathrm{mm}$) and number of bars (

**b**) (systems 15–18, $L=1.0\mathrm{mm}$) for $\dot{\gamma}=5{\text{}\mathrm{s}}^{-1}$.

**Figure 12.**Comparison between normalised viscosities of xanthan gum solution and silicon oils dependent on gap height, measured with pyramids (system 2, $L=0.5\mathrm{mm}$), columns (system 6, $L=0.5\mathrm{mm}$), and bars (system 12, $L=0.3\mathrm{mm})$ at $\dot{\gamma}=5.05\text{}1/\mathrm{s}$.

**Figure 13.**Comparison between apparent shear rate ${\dot{\mathsf{\gamma}}}_{\mathrm{a}}$ and true shear rate $\dot{\mathsf{\gamma}}$ in the velocity profile between two parallel plates showing wall slip, measured with pyramids (system 2) at $\dot{\gamma}=5.05\text{}1/\mathrm{s}$.

**Figure 14.**Comparison between viscosity values of $1\mathrm{wt}.\text{}\%$ xanthan gum solution obtained for columns (system 6) and reference system at $H=1\mathrm{mm}$.

**Figure 15.**Effective gap extension $\delta $ (gap- and material-averaged) dependent on the shear rate of $1\text{}\mathrm{wt}.\text{}\%$ xanthan gum solution obtained for columns (system 6, $H=1\text{}\mathrm{mm}$).

**Figure 16.**Correction of viscosity values obtained for columns (system 6) at $H=1.4\text{}\mathrm{mm}$ and $H=1.0\text{}\mathrm{mm}$.

**Figure 17.**Influence of structure depth $L$ on $\delta $ measured at $\dot{\mathsf{\gamma}}=5.05\text{}1/\mathrm{s}$.

**Figure 18.**Influence of number of bars (systems 15, 17, and 18) on $\delta $, measured at $\dot{\gamma}=5.05\text{}1/\mathrm{s}$.

**Figure 19.**Normalised viscosities of silicon oils and $25\text{}\mathrm{wt}.\text{}\%$ suspension dependent on the measuring gap, measured with columns (system 2), bars (system 6), and pyramids (system 9) at $\dot{\gamma}=5.05{\text{}\mathrm{s}}^{-1}$.

**Figure 20.**Normalised viscosities of suspensions for different concentrations, measured with pyramidal structure (system 1) at $\dot{\gamma}=5.05{\text{}\mathrm{s}}^{-1}$.

**Figure 21.**Flow curves of silicon oil AK 5000, $1\text{}\mathrm{wt}.\text{}\%$ xanthan gum solution, and $25\text{}\mathrm{wt}.\text{}\%$ suspension, measured with reference system.

**Figure 22.**Uncorrected, corrected, and reference viscosities of $25\text{}\mathrm{wt}.\text{}\%$ suspension, measured with columns (system 5).

Measuring System No. | $\mathbf{Structure}\text{}\mathbf{Depth}\text{}\left(\mathbf{Pyramid}\text{}\mathbf{Height}\right)\text{}\mathit{L}/\mathbf{mm}$ | $\mathbf{Side}\text{}\mathbf{Length}\text{}\mathbf{of}\text{}\mathbf{Base}\text{}\mathbf{Area}\text{}\mathit{b}/\mathbf{mm}$ |
---|---|---|

$1$ | 0.3 | 0.6 |

$2$ | 0.5 | 1.0 |

$3$ | 0.7 | 1.4 |

$4$ | 1.0 | 2.0 |

Measuring System No. | $\mathbf{Structure}\text{}\mathbf{Depth}\text{}\left(\mathbf{Column}\text{}\mathbf{Height}\right)\text{}\mathit{L}/\mathbf{mm}$ | $\mathbf{Distance}\text{}\mathbf{Between}\text{}\mathbf{Columns}\text{}\mathit{a}/\mathbf{mm}$ |
---|---|---|

5 | 0.1 | 1.0 |

6 | 0.5 | 0.5 |

7 | 0.5 | 0.8 |

8 | 0.5 | 1.0 |

9 | 0.5 | 1.2 |

10 | 0.7 | 1.0 |

11 | 1.0 | 1.0 |

Measuring System No. | $\mathbf{Structure}\text{}\mathbf{Depth}\text{}\left(\mathbf{Bar}\text{}\mathbf{Height}\right)\text{}\mathit{L}/\mathbf{mm}$ | $\mathbf{Number}\text{}\mathbf{of}\text{}\mathbf{Bars}\text{}\mathit{N}/-$ |
---|---|---|

12 | 0.3 | 20 |

13 | 0.5 | 20 |

14 | 0.7 | 20 |

15 | 1.0 | 20 |

16 | 1.0 | 19 |

17 | 1.0 | 18 |

18 | 1.0 | 16 |

**Table 4.**Values for $\delta $ for different measuring systems at $\dot{\gamma}=5.05{\text{}\mathrm{s}}^{-1}$.

Measuring System | $\mathbf{Effective}\text{}\mathbf{Gap}\text{}\mathbf{Extension}\text{}\mathit{\delta}/\mathbf{mm}$ | ||
---|---|---|---|

Silicon Oil AK 5000 | Silicon Oil AK 12,500 | $25\mathbf{wt}.\mathit{\%}\text{}\mathbf{Suspension}$ (AK 5000) | |

1 (pyramids) | $0.23\pm 0.006$ | $0.25\pm 0.004$ | $0.22\pm 0.011$ |

4 (pyramids) | $0.91\pm 0.040$ | $0.83\pm 0.032$ | $0.86\pm 0.071$ |

5 (pyramids) | $0.44\pm 0.023$ | $0.49\pm 0.016$ | $0.48\pm 0.029$ |

8 (columns) | $0.20\pm 0.003$ | $0.22\pm 0.009$ | $0.19\pm 0.014$ |

12 (bars) | $0.35\pm 0.009$ | $0.37\pm 0.004$ | $0.35\pm 0.009$ |

17 (bars) | $0.73\pm 0.068$ | $0.71\pm 0.072$ | $0.70\pm 0.089$ |

**Table 5.**Values for $\mathsf{\delta}$ for suspensions of different concentrations, measured with pyramidal structure (system 1) at $\dot{\gamma}=5.05{\text{}\mathrm{s}}^{-1}$.

AK 5000 | AK 12,500 | $5\mathbf{wt}.\mathbf{\%}$ | $25\mathbf{wt}.\%$ | $35\mathbf{wt}.\%$ |
---|---|---|---|---|

$0.23\pm 0.006$ | $0.25\pm 0.004$ | $0.23\pm 0.007$ | $0.22\pm 0.011$ | $0.22\pm 0.015$ |

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**MDPI and ACS Style**

Pawelczyk, S.; Kniepkamp, M.; Jesinghausen, S.; Schmid, H.-J. Absolute Rheological Measurements of Model Suspensions: Influence and Correction of Wall Slip Prevention Measures. *Materials* **2020**, *13*, 467.
https://doi.org/10.3390/ma13020467

**AMA Style**

Pawelczyk S, Kniepkamp M, Jesinghausen S, Schmid H-J. Absolute Rheological Measurements of Model Suspensions: Influence and Correction of Wall Slip Prevention Measures. *Materials*. 2020; 13(2):467.
https://doi.org/10.3390/ma13020467

**Chicago/Turabian Style**

Pawelczyk, Sebastian, Marieluise Kniepkamp, Steffen Jesinghausen, and Hans-Joachim Schmid. 2020. "Absolute Rheological Measurements of Model Suspensions: Influence and Correction of Wall Slip Prevention Measures" *Materials* 13, no. 2: 467.
https://doi.org/10.3390/ma13020467