# A Computational Fluid Dynamics Study on Shearing Mechanisms in Thermal Elastohydrodynamic Line Contacts

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Governing Equations

#### 2.2. Cavitation Modeling

#### 2.3. Lubricant Properties

#### 2.3.1. Density Equations

#### 2.3.2. Viscosity Equations for Newtonian Fluid Behavior

#### 2.3.3. Rheological Models

#### 2.4. Surface Temperature

#### 2.5. The Film Thickness Equation

#### 2.6. Mesh Generation and Numerical Method

## 3. Results and Discussion

#### 3.1. Mesh Verification Test

#### 3.2. Isothermal Conditions, Newtonain Fluid Behavior, Low Pressure

#### 3.3. Thermal Conditions, Non-Newtonain Fluid Behavior, High Fluid Pressure

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$\alpha $ | Pressure–viscosity coefficient | ${\mathrm{Pa}}^{-1}$ |

${\alpha}_{k}$ | Volume fraction of phase k | - |

$\beta $ | Temperature–viscosity coefficient | ${\mathrm{K}}^{-1}$ |

${\beta}_{K}$ | Temperature coefficient of ${K}_{0}$ | ${\mathrm{K}}^{-1}$ |

$\mathsf{\Gamma}$ | Diffusion coefficient | ${\mathrm{m}}^{2}/\mathrm{s}$ |

$\dot{\gamma}$ | Shear strain rate | ${\mathrm{s}}^{-1}$ |

$\Delta {\mathrm{x}}_{\mathrm{min}}$ | Min. cell size in X-direction | $\mathrm{m}$ |

$\epsilon $ | Thermal expansion coefficient | ${\mathrm{K}}^{-1}$ |

${\eta}_{Carreau}$ | Dynamic viscosity according to Carreau rheological model | $\mathrm{Pa}\mathrm{s}$ |

${\eta}_{H}$ | Dynamic viscosity according to Houpert | $\mathrm{Pa}\mathrm{s}$ |

${\eta}_{R}$ | Dynamic viscosity according to Roelands | $\mathrm{Pa}\mathrm{s}$ |

${\eta}_{Ree-Eyring}$ | Dynamic viscosity according to Ree–Eyring rheological model | $\mathrm{Pa}\mathrm{s}$ |

$\mathsf{\Lambda}$ | Limiting stress pressure coefficient | - |

${\lambda}_{R}$ | Relaxation time at ambient pressure and reference temperature ${T}_{R}$ | - |

$\mu $ | Limiting low-shear viscosity | $\mathrm{Pa}\mathrm{s}$ |

${\mu}_{0}$ | Dynamic viscosity at ambient pressure and reference temperature ${T}_{R}$ | $\mathrm{Pa}\mathrm{s}$ |

${\mu}_{k}$ | Dynamic viscosity of phase k | $\mathrm{Pa}\mathrm{s}$ |

${\mu}_{R}$ | Low shear viscosity at ambient pressure and reference temperature ${T}_{R}$ | $\mathrm{Pa}\mathrm{s}$ |

${\mu}_{v}$ | Dynamic viscosity of vapor phase | $\mathrm{Pa}\mathrm{s}$ |

${\mu}_{\infty}$ | Viscosity extrapolated to infinite temperature | $\mathrm{Pa}\mathrm{s}$ |

$\mathsf{\nu}$ | Velocity | $\mathrm{m}/\mathrm{s}$ |

$\upsilon $ | Poisson ratio | - |

$\rho $ | Density | $\mathrm{kg}/{\mathrm{m}}^{3}$ |

${\rho}_{k}$ | Density of phase k | $\mathrm{kg}/{\mathrm{m}}^{3}$ |

${\rho}_{0}$ | Lubricant density at ambient pressure and reference temperature ${T}_{R}$ | $\mathrm{kg}/{\mathrm{m}}^{3}$ |

$\stackrel{\xb7}{\tau}$ | Stress tensor | $\mathrm{Pa}$ |

${\tau}_{L}$ | Limiting shear stress | $\mathrm{Pa}$ |

${\tau}_{0}$ | Eyring stress | $\mathrm{Pa}$ |

$\phi $ | Dimensionless viscosity scaling parameter | - |

${\phi}_{\infty}$ | Viscosity scaling parameter for unbounded viscosity | - |

${a}_{V}$ | Thermal expansivity defined for volume linear with $T$ | ${\mathrm{K}}^{-1}$ |

$b$ | Hertzian half width | $\mathrm{m}$ |

${B}_{F}$ | Fragility parameter in the new viscosity equation | - |

$C$ | Specific heat capacity | $\mathrm{J}/\left(\mathrm{kgK}\right)$ |

$E$ | Modulus of elasticity | $\mathrm{Pa}$ |

${E}^{\prime}$ | Reduced elastic modulus | $\mathrm{Pa}$ |

${E}_{k}$ | Energy per unit mass of phase k | $\mathrm{J}/\mathrm{kg}$ |

$g$ | Thermodynamic interaction parameter | - |

$h$ | Film thickness | $\mathrm{m}$ |

${h}_{0}$ | Minimum gap between the solid surfaces in undeformed state | $\mathrm{m}$ |

$\mathrm{I}$ | Unit tensor | - |

$k$ | Thermal conductivity | $\mathrm{W}/\left(\mathrm{mK}\right)$ |

${k}_{eff}$ | Effective thermal conductivity | $\mathrm{W}/\left(\mathrm{mK}\right)$ |

${K}_{0}$ | Isothermal bulk modulus at $p=0$ | $\mathrm{Pa}$ |

${K}_{0}^{\prime}$ | Pressure rate of change of isothermal bulk modulus at $p=0$ | - |

${K}_{00}$ | ${K}_{0}$ at zero absolute temperature | $\mathrm{Pa}$ |

$L$ | Characteristic length scale | $\mathrm{m}$ |

$n$ | Power law exponent | - |

$p$ | Pressure | $\mathrm{Pa}$ |

${p}_{vap}$ | Vapor pressure | $\mathrm{Pa}$ |

$Pe$ | Peclet number | - |

${q}_{f}$ | Heat flux from fluid to solid wall | $\mathrm{W}/{\mathrm{m}}^{2}$ |

${R}^{\prime}$ | Reduced radius of curvature | $\mathrm{m}$ |

$t$ | Time | $\mathrm{s}$ |

$T$ | Temperature | $\mathrm{K}$ |

${T}_{CJ}$ | Surface temperature according to Carlaw-Jaeger | $\mathrm{K}$ |

${T}_{R}$ | Reference temperature | $\mathrm{K}$ |

${u}_{ent}$ | Entrainment speed | $\mathrm{m}/\mathrm{s}$ |

${u}_{s}$ | Surface velocity | $\mathrm{m}/\mathrm{s}$ |

$V$ | Fluid volume | ${\mathrm{m}}^{3}$ |

${V}_{0}$ | Fluid volume at ambient pressure | ${\mathrm{m}}^{3}$ |

${V}_{R}$ | Fluid volume at ambient pressure and reference temperature ${T}_{R}$ | ${\mathrm{m}}^{3}$ |

$w$ | Applied load | $\mathrm{N}/\mathrm{m}$ |

$x$ | Coordinate | $\mathrm{m}$ |

$\widehat{x}$ | Relative coordinate | $\mathrm{m}$ |

$z$ | Coordinate | $\mathrm{m}$ |

$Z$ | Roelands pressure–viscosity index | - |

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**Figure 3.**Comparison of the isothermal results for slide-to-roll ratio (SRR) = 0 against the results presented by Srirattayawong [16]. (

**a**) Pressure distribution; (

**b**) film thickness distribution.

**Figure 4.**Contour plots for isothermal conditions and SRR = 0. (

**a**) Static pressure; (

**b**) velocity streamlines; (

**c**) density; (

**d**) viscosity; (

**e**) strain rate; (

**f**) shear stress; (

**g**) oil volume fraction; (

**h**) vapor volume fraction.

**Figure 5.**Comparison of model combinations of Tait and Carreau and of Houpert and Ree–Eyring for thermal conditions and SRR = 1. (

**a**) Pressure distribution; (

**b**) film thickness distribution.

**Figure 6.**Comparison of contour plots for model combinations of Tait and Carreau and of Houpert and Ree–Eyring for thermal conditions and SRR = 1. (

**a**) Static pressure; (

**b**) velocity streamlines; (

**c**) strain rate; (

**d**) viscosity; (

**e**) shear stress; (

**f**) temperature.

**Figure 7.**Comparison of the wall temperature for thermal conditions and SRR = 1. (

**a**) model combination of Tait and Carreau; (

**b**) model combination of Houpert and Ree–Eyring.

**Figure 8.**Contour plot of shear stress to limiting shear stress ratio (model combination of Tait and Carreau).

**Figure 9.**The change of viscosity with pressure for zero strain rate and arbitrarily chosen constant values of temperature.

**Figure 10.**The change of viscosity with pressure at reference temperature (${\mathrm{T}}_{\mathrm{R}}=313.15\mathrm{K}$) and arbitrarily chosen constant values of strain rate.

Min. Cell Size of in X-dir. | Total no. of Cells | Max. Pressure in Pa | CPU Time in s |
---|---|---|---|

$\Delta {\mathrm{x}}_{\mathrm{min}}=5.00\times {10}^{-7}$ m | 13,480 | $1.941\times {10}^{7}$ | 304 |

$\Delta {\mathrm{x}}_{\mathrm{min}}=2.50\times {10}^{-7}$ m | 25,480 | $1.943\times {10}^{7}$ | 370 |

$\Delta {\mathrm{x}}_{\mathrm{min}}=1.25\times {10}^{-7}$ m | 49,480 | $1.943\times {10}^{7}$ | 496 |

Parameter | Value | Unit | |
---|---|---|---|

Operating conditions | |||

External load, $w$ | $50$ | $\mathrm{kN}/\mathrm{m}$ | |

Entrainment speed, ${u}_{ent}$ | $2.5$ | $\mathrm{m}/\mathrm{s}$ | |

Reference temperature, ${T}_{R}$ | $313.15$ | $\mathrm{K}$ | |

Reduced radius of curvature, ${R}^{\prime}$ | $0.01$ | $\mathrm{m}$ | |

Properties of solids | |||

steel [16] | ceramics [16] | ||

Modulus of elasticity, $E$ | $210$ | $450$ | $\mathrm{GPa}$ |

Poisson ratio, $\upsilon $ | $0.3$ | $0.15$ | - |

Density, $\rho $ | $7850$ | $3800$ | $\mathrm{kg}/{\mathrm{m}}^{3}$ |

Specific heat capacity, $C$ | $460$ | $1050$ | $\mathrm{J}/\left(\mathrm{kgK}\right)$ |

Thermal conductivity, $k$ | $47$ | $29$ | $\mathrm{W}/\left(\mathrm{mK}\right)$ |

Properties of liquids | |||

oil [16] | Squalane | ||

Dynamic viscosity at ambient pressure and ${T}_{R}$, ${\mu}_{0}$ | $0.01$ | $0.0156$ [21] | $\mathrm{P}\mathrm{a}\mathrm{s}$ |

Density, $\rho $ | $850$ | $794.6$ [26] | $\mathrm{kg}/{\mathrm{m}}^{3}$ |

Dynamic viscosity of vapor phase, ${\mu}_{v}$ | $8.97\times {10}^{-6}$ | $8.97\times {10}^{-6}$^{1} | $\mathrm{Pa}\mathrm{s}$ |

Density of vapor phase, ${\rho}_{v}$ | $0.0288$ | $0.0288$^{1} | $\mathrm{kg}/{\mathrm{m}}^{3}$ |

Specific heat capacity, $C$ | - | $2104$ [26] | $\mathrm{J}/\left(\mathrm{kgK}\right)$ |

Thermal conductivity, $k$ | - | $0.21$ [27] ^{2} | $\mathrm{W}/\left(\mathrm{mK}\right)$ |

Thermal expansivity, $\epsilon $ | - | $8.36\times {10}^{-4}$ [21] | ${\mathrm{K}}^{-1}$ |

Houpert and Ree–Eyring input parameters | |||

oil [16] | Squalane | ||

Temperature–viscosity coefficient, $\beta $ | - | $0.038$ [28] | ${\mathrm{K}}^{-1}$ |

Eyring stress, ${\tau}_{0}$ | - | $3.0\times {10}^{6}$ [29] ^{3} | $\mathrm{Pa}$ |

Roelands pressure–viscosity index, $Z$ | 0.689 | $0.6442$^{4} | - |

Tait and Carreau input parameters | |||

Squalane [21] | |||

Pressure rate of change of isothermal bulk modulus at $p=0$, ${K}_{0}^{\prime}$ | $11.74$ | - | |

Thermal expansivity defined for volume linear with $T$, ${a}_{V}$ | $8.36\times {10}^{-4}$ | ${\mathrm{K}}^{-1}$ | |

${K}_{0}$ at zero absolute temperature, ${K}_{00}$ | $8.658\times {10}^{9}$ | $\mathrm{Pa}$ | |

Temperature coefficient of ${K}_{0}$, ${\beta}_{K}$ | $6.332\times {10}^{-3}$ | ${\mathrm{K}}^{-1}$ | |

Thermodynamic interaction parameter, $g$ | $3.921$ | - | |

Viscosity scaling parameter for unbounded viscosity, ${\phi}_{\infty}$ | $0.1743$ | - | |

Fragility parameter in the new viscosity equation, ${B}_{F}$ | $24.50$ | - | |

Viscosity extrapolated to infinite temperature, ${\mu}_{\infty}$ | $0.9506\times {10}^{-4}$ | $\mathrm{Pa}\mathrm{s}$ | |

Relaxation time at ${T}_{R}$ and ambient pressure, ${\lambda}_{R}$ | $2.26\times {10}^{-9}$ | $\mathrm{s}$ | |

Power law exponent, $n$ | $0.463$ | - | |

Limiting stress pressure coefficient, $\mathsf{\Lambda}$ | $0.075$ | - |

^{1}Vapor phase parameters for Squalane are assumed to be the same as for oil.

^{2}Thermal conductivity is taken from the referenced paper by taking an average value from the thermal conductivity vs. pressure graph.

^{3}Eyring stress is derived from the shear stress vs. strain rate graph from the referenced paper.

^{4}Roelands pressure–viscosity index is calculated by using Equation (24) and pressure–viscosity coefficient of $\alpha =18.1{\mathrm{GPa}}^{-1}$ [30].

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Tošić, M.; Larsson, R.; Jovanović, J.; Lohner, T.; Björling, M.; Stahl, K.
A Computational Fluid Dynamics Study on Shearing Mechanisms in Thermal Elastohydrodynamic Line Contacts. *Lubricants* **2019**, *7*, 69.
https://doi.org/10.3390/lubricants7080069

**AMA Style**

Tošić M, Larsson R, Jovanović J, Lohner T, Björling M, Stahl K.
A Computational Fluid Dynamics Study on Shearing Mechanisms in Thermal Elastohydrodynamic Line Contacts. *Lubricants*. 2019; 7(8):69.
https://doi.org/10.3390/lubricants7080069

**Chicago/Turabian Style**

Tošić, Marko, Roland Larsson, Janko Jovanović, Thomas Lohner, Marcus Björling, and Karsten Stahl.
2019. "A Computational Fluid Dynamics Study on Shearing Mechanisms in Thermal Elastohydrodynamic Line Contacts" *Lubricants* 7, no. 8: 69.
https://doi.org/10.3390/lubricants7080069