This study theoretically predicted the response of superimposed squeeze and rotational flows (SSRF) of fluids with different viscous behaviors (i.e., Newtonian, shear-thinning, and shear-thickening fluids). The theoretical predictions were verified using the plate-plate geometry for the SSRF measurements with the Newtonian and power-law fluids. In all the cases, the squeeze force increased as the gap decreased, but the response was very different for each rheological behavior. The variation in the squeeze force with the gap was not affected by the superimposed rotational shear stress value, owing to the nondependency of the viscosity on the shear for Newtonian fluids. However, for the power-law fluids, the squeeze force variation with the gap value was based on the value of the superimposed shear stress value. The decrease and increase in the viscosity with the shear stress for the shear-thinning and shear-thickening fluids, respectively, resulted in opposite trends of the squeeze force with the gap value variation. For the shear-thinning fluids, the squeeze force for each gap value decreased with increasing superimposed rotational shear stress. The opposite trend was observed for the shear-thickening fluid. In the absence of wall slip, the theoretical predictions well agreed with the experimental results.

## I. INTRODUCTION

When two coaxial parallel plates with a fluid between them approach each other, a squeeze flow is afforded. Squeeze flows assist in the rheological characterization of matter and can be very useful in material processing. Although the squeeze flows of Newtonian and shear-thinning fluids have been extensively analyzed theoretically [1–3] and experimentally [4–8], those of shear-thickening fluids have not been comprehensively analyzed [9].

Squeeze flows are unavoidably intermixed with shear flows; hence, flow conditions that are more complex than the conventional simple shear flow conditions should be included in the rheological characterization of fluids. For example, a device capable of superimposing oscillatory squeeze and oscillatory rotational flows has been proposed for characterizing the viscoelastic response of fluids [10]. This study focused on the response of superimposed squeeze and rotational flows (SSRF) of fluids with different viscous behaviors (Newtonian, shear-thinning, and shear-thickening fluids). SSRF were realized with a plate-plate geometry because this geometry is representative of different practical situations, such as those appearing in damping, antivibration, or absorbing impact devices. Using the SSRF rheological test, a first-order approximation for the three-dimensional (3D) description of the viscous flow behavior of fluids can be obtained [11]. Since a wall slip influences the recorded squeeze force [12,13], nonslip conditions should be checked. Under steady-state conditions, two independent components of the rate-of-strain tensor $( \gamma \u02d9)$ develop during the SSFR rheological test: the rotation of the upper plate around the common axis induces a shear rate $ \gamma \u02d9 \theta z$, and the squeeze flow of the upper plate toward the down plate induces a radial shear rate $ \gamma \u02d9 r z$.

First, the theoretical SSRF problem was analytically solved for Newtonian and power-law fluids assuming the absence of wall slip. Then, the theoretical predictions for the dependence of the squeeze force on the gap between plates were verified. Honey and the Newtonian plateaus of a shear-thinning ophthalmic ointment and a shear-thickening fumed silica suspension were employed as the Newtonian fluids. A foot cream and an ophthalmic ointment were utilized as the shear-thinning fluids. Finally, a fumed silica suspension was employed as the shear-thickening fluid. This study aimed to present and test an analytical solution for the SSRF problem that can be used for analyzing and predicting the behavior of the three most representative fluid viscous behaviors (i.e., Newtonian, shear-thinning, and shear-thickening) in 3D-flow situations.

## II. EXPERIMENTAL

Rheological experiments were conducted on a controlled stress rheometer MARS III (Haake Thermo Electron Corporation GmbH, Karlsruhe, Germany). The sample was maintained at $20 \xb0 C$ using a Peltier system.

Steady flow curves were obtained using a cone-plate system with a $ 1 \xb0$ cone angle, $20 mm$ diameter, and $52\mu m$ gap. The correlation between the apparent viscosity and shear rate was recorded. The steady flow condition was assumed when the relative variation of the response, that is, shear stress, to each step by step applied shear rate was less than $1%$ for $10 s$. This condition was easily achieved for the considered shear rate intervals. The liquid media used in this study were Newtonian eucalyptus honey (Apisol S.A., Valencia, Spain), two non-Newtonian shear-thinning fluids, that is, a moisturizing foot cream (Babaria, Berioska S.L., Valencia, Spain) and an ophthalmic ointment (Recugel®, Bausch + Lomb Inc., New Jersey, USA), and a shear-thickening suspension. The shear-thickening suspension was prepared by adding polypropylene glycol (PPG400, from Sigma-Aldrich, Spain) with an average molar mass of $400 g / mol$ to hydrophilic fumed silica particles (Aerosil®200, from Evonik, Germany) and then mixing at $800 rpm$ for $10 min$ with an RZR1 stirrer (Heidolph Instruments, Germany) using a PR30 pitched-blade impeller. The amount of the solid phase (A200® particles) in the liquid phase (PPG400) was $20%w/w$, and a constant temperature of $20 \xb0 C$ was maintained during the experiment; thus, the shear-thickening suspension was labelled as 20A200PPG40020. The absence of wall slip in the rotational shear tests was verified by comparing the cone-plate results with the results of a plate-plate system with $20 mm$ (Fig. 1) and two different gaps ( $1$ and $0.5 mm$) [14]. Although the results do not prove the absence of wall slip for the SSFR rheological tests, they indicate the realization of such a condition, at least as a first approximation. Moreover, the nondependence of the results on the cone angle was validated for the cone-plate system with a $ 2 \xb0$ cone angle, $35 mm$ diameter, and $105\mu m$ gap (Fig. 1).

The Newtonian behavior of the employed honey was confirmed (Fig. 2), with a constant viscosity value of $ \eta N$ $=( 22.0 \xb1 0.5) Pa s$ at $20 \xb0 C$. Furthermore, the dependence of the foot cream viscosity (Fig. 2) on the shear rate was obtained under isothermal conditions $( 20 \xb0 C)$, and the results fitted very well to the power law $( R 2 = 0.9985)$ in the full shear rate interval considered herein $( \eta ( \gamma \u02d9 ) = ( 87.5 \xb1 1.4 ) \gamma \u02d9 \u2212 ( 0.62 \xb1 0.01 ))$. The ophthalmic ointment exhibited Newtonian behavior with a constant viscosity of $ \eta N=( 4400 \xb1 100) Pa s$ at shear rates lower than $0.006 s \u2212 1$ and well fitted the power law $( R 2=0.9993)$ in the interval $( 0.006 \u2212 100 s \u2212 1)( \eta ( \gamma \u02d9 ) = ( 43.1 \xb1 0.7 ) \gamma \u02d9 \u2212 ( 0.86 \xb1 0.01 ))$. Moreover, the shear-thinning behavior of the ophthalmic ointment was more abrupt $( n = 0.14)$ than that of the foot cream $( n = 0.38)$. In principle, the consistencies of the two materials cannot be compared because their units are very different ( $ Pa s 0.14$ for the ophthalmic ointment and $ Pa s 0.38$ for the foot cream). Nevertheless, as the numerical value of the consistency index coincided with the viscosity value when the shear rate was $1 s \u2212 1$, the foot cream was inferred to be more consistent than the ophthalmic ointment. Finally, the 20A200PPG40020 suspension displayed Newtonian behavior with a constant viscosity of $ \eta N=( 1.31 \xb1 0.05) Pa s$ at shear rates lower than $20 s \u2212 1$ and exhibited discontinuous shear-thickening behavior (Fig. 2) in a very narrow shear rate interval $( 20 \u2212 40 s \u2212 1)$. Additionally, the results in this shear rate interval were fitted to a power law $( R 2 = 0.9532)$, depicting the dependence $\eta ( \gamma \u02d9)$ $=( 10 \u2212 23 \xb1 10 \u2212 25) \gamma \u02d9 ( 16 \xb1 1 )$ of the viscosity with the shear rate.

The fluids were squeezed out using the axial test included in the rheometer’s Rheowin 4.60 software. A previous study [15] found that better reproducible results were obtained with the rheometer MARS III when the squeeze velocity was controlled than when the squeeze force was monitored. Therefore, this study controlled the squeeze velocity using the constant area method between two parallel coaxial plates (Fig. 3), that is, while the lower plate was maintained stationary, the upper plate was displaced toward the lower plate under constant velocity. Prior to the SSRF test, a rotational shear stress was applied, which was maintained during the SSRF test until a steady state was achieved. This was done to develop the microstructure before the start of the SSRF test and maintain it during the SSRF test. The plate diameter was $2 R 0=20 mm$ and the initial gap was $ h 0=1.5 mm$. Consequently, the geometric parameter $\zeta = h R 0$ related to the variable gap during the squeeze and the plate radius was always sufficiently small; therefore, the terms of $ O( \zeta 2)$ in the movement equation could be neglected.

## III. RESULTS AND DISCUSSION

### A. Newtonian fluids

The results are justified as follows. The increase in the shear rate with decreasing gap did not influence the viscosity value simply because the fluid is Newtonian. Therefore, the superimposed rotational shear does not have any additional influence on the relationship between the squeeze force and velocity compared to squeeze test results in the absence of the superimposed rotational flow (Stefan equation). Figure 4 illustrates that the superimposed rotational shear does not modify the quantitative dependence of $ F z$ with *h*, even when different shear stress values are applied. Linear plots [Fig. 4(a)] and log-log plots [Fig. 4(b)] are also presented to highlight the good fit of the model predictions [Eq. (9)] with the experimental results. Specifically, the slope of the $ logF\u2212 logh$ plot $ ( d log F z d log h = \u2212 3 )$ well agreed with the predictions of Eq. (9).

### B. Power-law fluid

*K*is the consistency and

*n*is the power-law index. Substitution of Eqs. (A22) and (A23) into the motion equation yielded

*r*to be determined. Since $ \gamma \u02d9 r z= \u2202 v r \u2202 z=\u2212 r 2 d 2 v z d z 2$ and $ \gamma \u02d9 \theta z= \u2202 v \theta \u2202 z=r \Omega h h$, the generalized shear rate was

*r*. Equation (7) in terms of nondimensional variables is

*z*can be neglected. Therefore,

*C*and

*D*are constants. As $ v r( r , 0)=\u2212 r 2 d v z d z=0$, $D=0$. Integration of Eq. (23) using the boundary conditions Eq. (A19) yielded

Equation (25) reduces to the Newtonian prediction [Eq. (9)] when $n=1$ and $K= \eta N$, indicating that the superimposed rotation does not influence the squeeze flow behavior of Newtonian fluids.

The experimental results and theoretical predictions [Eq. (25)] for the dependence of the squeeze force on the gap for the SSRF of power-law fluids were compared. The predictions for both shear-thinning and shear-thickening behaviors were analyzed. Figure 5 displays the shear stress intervals where the shear-thinning and shear-thickening behaviors were observed. The shear-thinning behaviors of the foot cream and ophthalmic ointment were observed in the shear stress intervals $( 15 \u2212 500 Pa)$ and $( 20 \u2212 100 Pa)$, respectively, while the shear-thickening behavior of the 20A200PPG40020 suspension was observed in the shear stress interval $( 30 \u2212 5000 Pa)$. Note that the superimposition of the shear stress instead of the shear rate to the squeeze flow was preferred as the employed rheometer in a stress-controlled device, which ensures that the squeezed microstructure was the same during all the SSRF tests.

The rotational term $ ( \Omega h h )$ in Eq. (25) was calculated from the experimental raw data (Table I). As shown in Fig. 6, this term slightly increased as the gap decreased for the shear-thinning fluids, and regardless of the superimposed rotational shear stress value, it was practically independent of the rotational shear stress and gap value for the shear-thickening fluid. These two significantly different variations can be justified as follows. When the rotational superimposed shear stress increased, the microstructure in the shear-thinning fluids weakened. Moreover, for a given gap value, the corresponding viscosity value decreased and the angular velocity increased. When the gap value decreased, the rotational term $ \Omega h h$ increased. The observations for the shear-thickening fluid were very different. The 20A200PPG40020 suspension exhibited discontinuous shear-thickening behavior (Fig. 1). As shown in Fig. 7, for the interval of shear stress values where the 20A200PPG40020 suspension exhibited shear-thickening behavior in accordance with the power-law equation $( 30 \u2212 2500 ) Pa$, the shear rate variation range was extremely short $( 25 \u2212 35 s \u2212 1)$. Thus, despite the high variation in the superimposed shear stress values in the shear-thickening region, the variation in the corresponding rotational term was comparatively low.

σ(Pa)
. | $ \Omega h h( rad / s \u22c5 m)$ . | r^{2}
. |
---|---|---|

Shear-thinning fluid (foot cream) | ||

50 | $( 26.7 \xb1 0.6)+( 315 \xb1 8) exp( \u2212 h / ( ( 2.66 \xb1 0.06 ) \u22c5 10 \u2212 4 ))$ | 0.9962 |

75 | $( 52.3 \xb1 0.6)+( 508 \xb1 8) exp( \u2212 h / ( ( 2.62 \xb1 0.04 ) \u22c5 10 \u2212 4 ))$ | 0.9985 |

100 | $( 138.3 \xb1 0.7)+( 830 \xb1 8) exp( \u2212 h / ( ( 2.74 \xb1 0.03 ) \u22c5 10 \u2212 4 ))$ | 0.9994 |

250 | $( 1322 \xb1 9)+( 6700 \xb1 1400) exp( \u2212 h / ( ( 1.3 \xb1 0.1 ) \u22c5 10 \u2212 4 ))$ | 0.9124 |

500 | $( 7290 \xb1 40)+( 12100 \xb1 700) exp( \u2212 h / ( ( 2.4 \xb1 0.1 ) \u22c5 10 \u2212 4 ))$ | 0.9836 |

Shear-thinning fluid (ophthalmic ointment) | ||

40 | $( 68 \xb1 2)+( 1280 \xb1 40) exp( \u2212 h / ( ( 2.0 \xb1 0.1 ) \u22c5 10 \u2212 4 ))$ | 0.9971 |

50 | $( 97 \xb1 2)+( 1790 \xb1 50) exp( \u2212 h / ( ( 1.8 \xb1 0.1 ) \u22c5 10 \u2212 4 ))$ | 0.9983 |

60 | $( 124 \xb1 3)+( 2180 \xb1 70) exp( \u2212 h / ( ( 1.9 \xb1 0.1 ) \u22c5 10 \u2212 4 ))$ | 0.9982 |

Shear-thickening fluid (20A200PPG40020) | ||

100 | 3600 ± 300 | — |

500 | 3800 ± 200 | — |

1000 | 3900 ± 400 | — |

2500 | 4100 ± 600 | — |

σ(Pa)
. | $ \Omega h h( rad / s \u22c5 m)$ . | r^{2}
. |
---|---|---|

Shear-thinning fluid (foot cream) | ||

50 | $( 26.7 \xb1 0.6)+( 315 \xb1 8) exp( \u2212 h / ( ( 2.66 \xb1 0.06 ) \u22c5 10 \u2212 4 ))$ | 0.9962 |

75 | $( 52.3 \xb1 0.6)+( 508 \xb1 8) exp( \u2212 h / ( ( 2.62 \xb1 0.04 ) \u22c5 10 \u2212 4 ))$ | 0.9985 |

100 | $( 138.3 \xb1 0.7)+( 830 \xb1 8) exp( \u2212 h / ( ( 2.74 \xb1 0.03 ) \u22c5 10 \u2212 4 ))$ | 0.9994 |

250 | $( 1322 \xb1 9)+( 6700 \xb1 1400) exp( \u2212 h / ( ( 1.3 \xb1 0.1 ) \u22c5 10 \u2212 4 ))$ | 0.9124 |

500 | $( 7290 \xb1 40)+( 12100 \xb1 700) exp( \u2212 h / ( ( 2.4 \xb1 0.1 ) \u22c5 10 \u2212 4 ))$ | 0.9836 |

Shear-thinning fluid (ophthalmic ointment) | ||

40 | $( 68 \xb1 2)+( 1280 \xb1 40) exp( \u2212 h / ( ( 2.0 \xb1 0.1 ) \u22c5 10 \u2212 4 ))$ | 0.9971 |

50 | $( 97 \xb1 2)+( 1790 \xb1 50) exp( \u2212 h / ( ( 1.8 \xb1 0.1 ) \u22c5 10 \u2212 4 ))$ | 0.9983 |

60 | $( 124 \xb1 3)+( 2180 \xb1 70) exp( \u2212 h / ( ( 1.9 \xb1 0.1 ) \u22c5 10 \u2212 4 ))$ | 0.9982 |

Shear-thickening fluid (20A200PPG40020) | ||

100 | 3600 ± 300 | — |

500 | 3800 ± 200 | — |

1000 | 3900 ± 400 | — |

2500 | 4100 ± 600 | — |

The experimental results well qualitatively agreed the theoretical predictions obtained using Eq. (25) for the shear-thinning (Figs. 8 and 9) and shear-thickening (Fig. 10) fluids. For the shear-thinning fluids, the viscosity decreased with increasing superimposed rotational shear stress, resulting in a less pronounced increases in the normal force with decreasing gap value. In contrast, for the shear-thickening fluid, the viscosity increased with the superimposed rotational shear stress, resulting in the increase in the normal force based on the squeeze flow corresponding to the gap value. This denotes that the energy dissipation in hydraulic dumpers using shear-thickening fluids will be more effective when the superimposed rotational shear stress is higher. Certainly, the superposition of the rotational shear stress will incur some costs, which could make a device based on this idea unviable. In other words, expending energy (superimposed rotation) to dissipate energy (squeeze flow) sounds absurd. Therefore, superimposed rotation without an additional energetic spending needs to be realized, but this is matter for inventors.

*R*is related to two apparent shear rate values corresponding to different gaps between plates,

## IV. CONCLUSIONS

The theoretical predictions for SSRF were verified using a plate-plate system with Newtonian and power-law fluids. The response of the Newtonian fluids was independent of the superimposed rotational shear stress values. In contrast, for the power-law fluids, the superimposed stress influenced the variation in the squeeze force with the gap value. These very different behaviors stem from the viscosity dependence with shear characterizing each time-independent viscous behavior (Newtonian, shear-thinning, and shear-thickening).

**ACKNOWLEDGMENTS**

The authors gratefully acknowledge the Consejería de Economía, Innovación, Ciencia y Empleo (Junta de Andalucía, Spain) for providing funding under Grant No. PROYEXCEL_00181 for support of this project. The authors thank the Universidad de Málaga for providing funding for the Open Access charge. The authors wish to acknowledge valuable comments by one of the reviewers.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

### APPENDIX: NONDIMENSIONAL GOVERNING EQUATIONS

*r*direction $ ( r ^ = r R 0 )$ and with the distance between disks

*h*in the

*z*direction $ ( z ^ = z h )$. Additionally, the angle is scaled with $ \gamma \u02d9 R o h \u02d9h$, where $ \gamma \u02d9 R o$ $= R o \Omega h h$ is the rotational shear rate at the disk rim and $ \Omega h$ is the angular velocity of the rotor when the gap is

*h*, that is, $ ( \theta ^ = \theta h \u02d9 \gamma \u02d9 R o h )$. Velocities are scaled with the squeeze velocity $ h \u02d9$ in the

*z*direction $ ( v ^ z = v z h \u02d9 )$, with the compression rate $ ( \epsilon \u02d9 = h \u02d9 h )$ times the disk radius $ R 0( h \u02d9 R 0 / h)$ in the

*r*direction $ ( v ^ r = v r h R 0 h \u02d9 )$, and with the shear rate at the rim times the disk radius $( \gamma \u02d9 R o R 0)$ in the $\theta $ direction $ ( v ^ \theta = v \theta R 0 \gamma \u02d9 R o )$. The pressure is scaled with $ \eta c h \u02d9 R 0 2/ h 3$, that is, $ ( p ^ = p h 3 \eta c h \u02d9 R 0 2 )$, where $ \eta c$ is some fluid characteristic viscosity value. Finally, the shear stress $ \sigma r z$ is scaled with $ \eta c h \u02d9 R 0/ h 2$, but shear stresses $ \sigma r \theta $ and $ \sigma z \theta $ are scaled with $ \eta c \gamma \u02d9 R o R 0/h$ and normal stresses are scaled with $ \eta c h \u02d9/h$. Consequently, the nondimensional continuity and movement equations are

*z*plane is different due to the friction between the fluid layers. Moreover, the maximum angular velocity is generally a function of

*h*. In other words, the angular velocity of the rotor varies as the gap decreases due to the eventual variation in the fluid viscosity. Substitution of Eq. (A17) into the continuity Eq. (A2) yields the radial component of the velocity,

*r*and $\theta $ directions, with

*z*being the gradient direction in both cases, that is,

## REFERENCES

*Dynamics of Polymeric Liquids*