# Critical Shear Rate of Polymer-Enhanced Hydraulic Fluids

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Rheological Measurements

#### 2.2. Molecular Dynamics Simulations

## 3. Results and Discussion

#### 3.1. Fluid Identification and Formulation

#### 3.2. Rheological Behavior

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

EMD | equilibrium molecular dynamics |

GK | Green–Kubo |

MD | molecular dynamics |

NEMD | non-equilibrium molecular dynamics |

NPT | isothermal–isobaric ensemble |

NVE | canonical ensemble |

NVT | microcanonical ensemble |

PAO | polyalphaolefin |

PAO2 | polyalphaolefin of 2 mm${}^{2}$/s at 100 ${}^{\circ}$C |

PAO4 | polyalphaolefin of 4 mm${}^{2}$/s at 100 ${}^{\circ}$C |

PAO8 | polyalphaolefin of 8 mm${}^{2}$/s at 100 ${}^{\circ}$C |

PIB | polyisobutylene |

PIB1300 | polyisobutylene of 1300 g/mol |

PIB6000 | polyisobutylene of 6000 g/mol |

TraPPE | transferable potentials for phase equilibria |

UA | united atom |

VI | viscosity index |

VM | viscosity modifier |

Nomenclature | |

${c}_{p}$ | concentration of polymer |

${c}_{s}$ | concentration of of solvent or base oil |

${k}_{B}$ | Boltzmann constant |

M | molecular weight |

n | power-law exponent |

${P}_{\alpha \beta}$ | pressure or stress tensor |

${P}_{xz}$ | shear stress |

${R}_{g}$ | universal gas constant |

T | temperature |

${u}_{x}$ | velocity of atoms in the x-direction |

V | volume |

$Wi$ | Weissenberg number |

$\eta $ | shear viscosity of fluid |

${\eta}_{0}$ | Newtonian viscosity of fluid |

${\eta}_{0p}$ | Newtonian viscosity of polymer |

${\eta}_{0s}$ | Newtonian viscosity of solvent or base oil |

${\eta}_{\infty}$ | viscosity at infinite shear rate of fluid |

$\dot{\gamma}$ | shear rate |

${\dot{\gamma}}_{cr}$ | critical shear rate |

$\lambda $ | time constant or rotational relaxation time |

$\rho $ | mass density |

${\rho}_{p}$ | density of polymer |

${\rho}_{s}$ | density of solvent or base oil |

$\partial {u}_{x}/\partial z$ | flow gradient or momentum flux |

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**Figure 1.**Predicted critical shear rate and Newtonian viscosity at 80 ${}^{\circ}$C of polymer and base oil blends as functions of polymer molecular weight for three different polymer concentrations in (

**a**) PAO2, (

**b**) PAO4, and (

**c**) PAO8. The ideal fluid is highlighted by the dashed lines and stars in (

**b**).

**Figure 2.**Molecular structures of (

**a**) decene trimer, (

**b**) decene tetramer, (

**c**) PIB1300, and (

**d**) PIB6000 in united atom representation. The yellow, cyan, red, pink, blue, and green spheres represent CH${}_{3}$, CH${}_{2}$, CH, C, sp${}^{2}$ CH${}_{2}$, and sp${}^{2}$ C, respectively.

**Figure 3.**Average Newtonian viscosity of the three fluids at (

**a**) 50 ${}^{\circ}$C and (

**b**) 80 ${}^{\circ}$C and standard deviation obtained from 20 NVE trajectories in equilibrium molecular dynamics (EMD) simulations.

**Figure 4.**Viscosity as a function of shear rate from experiments at two different shear rate ranges (hollow symbols) and simulations (solid symbols) for the fluids studied at (

**a**) 50 ${}^{\circ}$C and (

**b**) 80 ${}^{\circ}$C. The dotted lines are the Carreau model fit to the viscosity profile of each fluid. The shaded region indicates the range of shear rates in the key lubricating gaps within the pump.

**Figure 5.**Normalized viscosity as a function of change in average polymer length in the direction of shear rate for the fluids studies here at (

**a**) 50 ${}^{\circ}$C and (

**b**) 80 ${}^{\circ}$C. Each point corresponds to a different shear rate.

Fluid ID | Fluid 1 | Fluid 2 | Fluid 3 |
---|---|---|---|

PAO4 | 88 wt.% | 84 wt.% | - |

PAO8 | - | - | 88 wt.% |

PIB1300 | - | - | 12 wt.% |

PIB6000 | 12 wt.% | 16 wt.% | - |

**Table 2.**Simulation plan for the three fluids with the number of molecules of each type for each model.

Fluid ID | Fluid 1 | Fluid 2 | Fluid 3 |
---|---|---|---|

Decene trimer, C${}_{30}$H${}_{62}$ | 261 | 244 | - |

Decene tetramer, C${}_{40}$H${}_{82}$ | 196 | 168 | 386 |

Polymer, PIB1300 (C${}_{96}$H${}_{192}$) | - | - | 22 |

Polymer, PIB6000 (C${}_{428}$H${}_{856}$) | 5 | 6 | - |

**Table 3.**Comparison of simulated Newtonian viscosity of the three fluids with experimentally Newtonian viscosities. The error associated with all experimental data is ±0.2 mPa·s.

Temperature | Fluid ID | Viscosity (mPa·s) | |
---|---|---|---|

Experiment | Simulation | ||

50 ${}^{\circ}$C | Fluid 1 | 40.2 | 39.7 ± 4.0 |

Fluid 2 | 62.3 | 58.9 ± 3.2 | |

Fluid 3 | 35.45 | 38.6 ± 3.4 | |

80 ${}^{\circ}$C | Fluid 1 | 15.9 | 15.7 ± 1.1 |

Fluid 2 | 22.3 | 23.1 ± 1.0 | |

Fluid 3 | 13.5 | 14.3 ± 1.0 |

**Table 4.**Critical shear rate of the fluids obtained from the Carreau fit to experiment and simulation viscosity data as compared to the theoretical prediction.

Temperature | Fluid ID | Critical Shear Rate (1/s) | |
---|---|---|---|

Experimental/Simulation | Prediction | ||

50 ${}^{\circ}$C | Fluid 1 | $5.27\times {10}^{6}$ | $1.19\times {10}^{6}$ |

Fluid 2 | $2.78\times {10}^{5}$ | $9.24\times {10}^{5}$ | |

Fluid 3 | $2.71\times {10}^{7}$ | $1.69\times {10}^{7}$ | |

80 ${}^{\circ}$C | Fluid 1 | $2.41\times {10}^{6}$ | $3.38\times {10}^{6}$ |

Fluid 2 | $1.29\times {10}^{6}$ | $2.91\times {10}^{6}$ | |

Fluid 3 | $1.01\times {10}^{8}$ | $4.54\times {10}^{7}$ |

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**MDPI and ACS Style**

Panwar, P.; Michael, P.; Devlin, M.; Martini, A.
Critical Shear Rate of Polymer-Enhanced Hydraulic Fluids. *Lubricants* **2020**, *8*, 102.
https://doi.org/10.3390/lubricants8120102

**AMA Style**

Panwar P, Michael P, Devlin M, Martini A.
Critical Shear Rate of Polymer-Enhanced Hydraulic Fluids. *Lubricants*. 2020; 8(12):102.
https://doi.org/10.3390/lubricants8120102

**Chicago/Turabian Style**

Panwar, Pawan, Paul Michael, Mark Devlin, and Ashlie Martini.
2020. "Critical Shear Rate of Polymer-Enhanced Hydraulic Fluids" *Lubricants* 8, no. 12: 102.
https://doi.org/10.3390/lubricants8120102