# On Waviness and Two-Sided Surface Features in Thermal Elastohydrodynamically Lubricated Line Contacts

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Elastohydrodynamic Model

#### 2.2. Thermal Model

#### 2.3. Lubricant Properties

#### 2.4. Numerical Approach

## 3. Results and Discussion

#### 3.1. Sinusoidal Roughness

#### 3.2. Two-Sided Surface Features

- an asperity on the slower surface overtaken by an asperity on the faster surface ($A+A$),
- a dent on the slower surface overtaken by a dent on the faster surface ($D+D$),
- an asperity on the slower surface overtaken by a dent on the faster surface ($A+D$),
- a dent on the slower surface overtaken by an asperity on the faster surface ($D+A$).

## 4. Conclusions

- (i)
- In case of sinusoidal roughness, differences in outlet film thickness between the thermal non-Newtonian and isothermal Newtonian approaches were for all studied wavelengths observed. These differences were explained by a reduced viscosity in the contact inlet due to shear thinning and shear heating, and further resulted in the isothermal Newtonian approach to overestimate the minimum film thickness by up to $20\phantom{\rule{0.166667em}{0ex}}\%$.
- (ii)
- Differences in the high pressure region were only found to be significant in case of short wavelength roughness. These differences were explained by a reduced viscosity in the contact inlet due to shear thinning and shear heating, further disturbing the formation of the complementary function. This resulted in central film thickness estimations to differ between $20\phantom{\rule{0.166667em}{0ex}}\%$ and $-14\phantom{\rule{0.166667em}{0ex}}\%$ by using the thermal non-Newtonian approach compared to the isothermal Newtonian approach.
- (iii)
- Following the relatively small influence on pressure by the complementary function, only small differences in maximum pressure between the thermal non-Newtonian and isothermal Newtonian approaches were noticed.
- (iv)
- In case of two-sided surface features overtaking within the contact, it was found that an interference between the film thickness perturbations did not occur due to the complementary function being generated in the inlet of the contact and travelling with the speed of the lubricant. On the other hand, due to the particular integral being directly connected to the roughness feature, an interference between the pressure and temperature variations were obvious.
- (v)
- In case of the overtaking event occurring in the inlet of the contact, one single complementary function was formed with larger amplitude than the other cases. Moreover, if overtaking instead took place in the outlet of the contact, a significantly reduced film thickness was noted.
- (vi)
- For the studied cases, it can be concluded that a thermal non-Newtonian approach is required for quantitatively accurate outlet film thickness predictions and may also be necessary for accurate predictions of film thickness within the contact if the complementary function is affected during formation. Otherwise, in the case of long wavelength roughness or low sliding conditions, an isothermal approach may be sufficient due to the inlet dominated film formation.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

A | Conductivity scaling parameter $\left(-\right)$ |

${A}_{\mathcal{R},i}$ | Amplitude of surface roughness $\left(\mathrm{m}\right)$ |

${\overline{A}}_{\mathcal{R},i}$ | Dimensionless amplitude of surface roughness, $\left(-\right)$ |

a | Hertzian contact radius $\left(\mathrm{m}\right)$ |

${a}_{v}$ | Thermal expansivity $\left({\mathrm{K}}^{-1}\right)$ |

${B}_{F}$ | Fragility parameter in viscosity equation $\left(-\right)$ |

${C}_{0}$ | Parameter for calculation of heat capacity $\left(\mathrm{J}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}{\mathrm{K}}^{-1}\right)$ |

${C}_{k}$ | Parameter in conductivity function $\left(\mathrm{W}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}\right)$ |

${C}_{lub}$ | Volumetric heat capacity of lubricant $\left(\mathrm{J}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}{\mathrm{K}}^{-1}\right)$ |

${c}_{lub}$ | Specific heat capacity of lubricant $\left(\mathrm{J}\phantom{\rule{0.166667em}{0ex}}{\mathrm{kg}}^{-1}{\mathrm{K}}^{-1}\right)$ |

${c}_{s1},\phantom{\rule{0.166667em}{0ex}}{c}_{s2}$ | Specific heat capacity of upper- and lower solid $\left(\mathrm{J}\phantom{\rule{0.166667em}{0ex}}{\mathrm{kg}}^{-1}{\mathrm{K}}^{-1}\right)$ |

${E}_{1},\phantom{\rule{0.166667em}{0ex}}{E}_{2}$ | Young’s modulus of upper- and lower body, respectively $\left(\mathrm{Pa}\right)$ |

F | Applied load per unit width $\left(\mathrm{N}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-1}\right)$ |

${G}_{r}$ | Shear thinning parameter $\left(\mathrm{Pa}\right)$ |

g | Thermodynamic interaction parameter $\left(-\right)$ |

h | Lubricant film thickness $\left(\mathrm{m}\right)$ |

${h}_{0}$ | Rigid body displacement $\left(\mathrm{m}\right)$ |

H | Dimensionless lubricant film thickness $\left(-\right)$ |

${K}_{00}$ | Isothermal bulk modulus at $p=0$ and zero absolute temperature $\left(\mathrm{Pa}\right)$ |

${K}_{0}^{\prime}$ | Pressure rate of change of ${K}_{0}$ $\left(-\right)$ |

${k}_{lub}$ | Thermal conductivity of lubricant $\left(\mathrm{W}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}\right)$ |

${k}_{s1},\phantom{\rule{0.166667em}{0ex}}{k}_{s2}$ | Thermal conductivity of upper- and lower solid $\left(\mathrm{W}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}\right)$ |

m | Parameter in heat capacity function $\left(\mathrm{J}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}{\mathrm{K}}^{-1}\right)$ |

n | Shear thinning parameter $\left(-\right)$ |

${p}_{h}$ | Maximum Hertzian pressure (1D) $\left(\mathrm{Pa}\right)$ |

p | Hydrodynamic pressure in lubricant film $\left(\mathrm{Pa}\phantom{\rule{0.166667em}{0ex}}\mathrm{m}\right)$ |

P | Dimensionless hydrodynamic pressure $\left(-\right)$ |

q | Coefficient in conductivity scaling parameter $\left(-\right)$ |

R | Equivalent radius of curvature $\left(\mathrm{m}\right)$ |

${R}_{1},\phantom{\rule{0.166667em}{0ex}}{R}_{2}$ | Radius of upper- and lower body, respectively $\left(\mathrm{m}\right)$ |

s | Exponent in conductivity model $\left(-\right)$ |

${S}_{upper},\phantom{\rule{0.166667em}{0ex}}{S}_{lower}$ | Visualised upper- and lower surface, respectively $\left(-\right)$ |

$SRR$ | Slide-to-roll ratio $\left(-\right)$ |

$T,\phantom{\rule{0.166667em}{0ex}}{T}_{r}$ | Temperature and reference temperature, respectively $\left(\mathrm{K}\right)$ |

$\overline{T}$ | Dimensionless temperature $\left(-\right)$ |

t | Time $\left(\mathrm{s}\right)$ |

$u,\phantom{\rule{0.166667em}{0ex}}w$ | Deformation in x and z, respectively $\left(-\right)$ |

${u}_{e}$ | Mean entrainment velocity of lubricant $\left(\mathrm{m}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}\right)$ |

${u}_{1},\phantom{\rule{0.166667em}{0ex}}{u}_{2}$ | Speed of upper- and lower surface, respectively $\left(\mathrm{m}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}\right)$ |

${u}_{f}$ | Lubricant velocity in $xz$-plane $\left(\mathrm{m}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}\right)$ |

V | Volume of lubricant $\left(-\right)$ |

${V}_{0},\phantom{\rule{0.166667em}{0ex}}{V}_{r}$ | Volume of lubricant at ${P}_{0}$ and ${T}_{r}$, respectively $\left({\mathrm{m}}^{3}\right)$ |

$x,\phantom{\rule{0.166667em}{0ex}}y,\phantom{\rule{0.166667em}{0ex}}z$ | Spatial coordinates $\left(\mathrm{m}\right)$ |

${x}_{\mathcal{R},1},\phantom{\rule{0.166667em}{0ex}}{x}_{0}$ | Current- and initial position of roughness, respectively $\left(\mathrm{m}\right)$ |

$X,\phantom{\rule{0.166667em}{0ex}}Y,\phantom{\rule{0.166667em}{0ex}}Z$ | Dimensionless spatial coordinates $\left(-\right)$ |

${X}_{\mathcal{R}},\phantom{\rule{0.166667em}{0ex}}{X}_{0}$ | Current- and initial dimensionless position of asperity, respectively $\left(\mathrm{m}\right)$ |

${\beta}_{K}$ | Temperature coefficient $\left({\mathrm{K}}^{-1}\right)$ |

$\overline{\eta}$ | Dimensionless shear dependent lubricant viscosity $\left(-\right)$ |

$\eta $ | Shear dependent lubricant viscosity $\left(\mathrm{Pa}\phantom{\rule{0.166667em}{0ex}}\mathrm{s}\right)$ |

$\dot{\gamma},\phantom{\rule{0.166667em}{0ex}}{\dot{\gamma}}_{xz}$ | Lubricant resultant shear rate and shear rate in $xz$-plane, respectively $\left({\mathrm{s}}^{-1}\right)$ |

${\lambda}_{r}$ | Relaxation time at ${T}_{r}$ and ambient pressure $\left(\mathrm{s}\right)$ |

${\lambda}_{\mathcal{R},i}$ | Wavelength of asperity $\left(\mathrm{m}\right)$ |

${\Lambda}_{\mathcal{R},i}$ | Dimensionless wavelength of asperity $\left(-\right)$ |

$\mu ,\phantom{\rule{0.166667em}{0ex}}{\mu}_{r}$ | Lubricant Newtonian- and reference viscosity, respectively $\left(\mathrm{Pa}\phantom{\rule{0.166667em}{0ex}}\mathrm{s}\right)$ |

${\mu}_{\infty}$ | Extrapolated viscosity to infinite temperature $\left(\mathrm{Pa}\phantom{\rule{0.166667em}{0ex}}\mathrm{s}\right)$ |

${\nu}_{1},\phantom{\rule{0.166667em}{0ex}}{\nu}_{2}$ | Possion’s ratio of upper- and lower body, respectively $\left(-\right)$ |

$\phi ,\phantom{\rule{0.166667em}{0ex}}{\phi}_{\infty}$ | Scaling parameters for viscosity $\left(-\right)$ |

${\mathcal{R}}_{i}$ | Geometry of surface roughness $\left(\mathrm{m}\right)$ |

$\overline{{\mathcal{R}}_{i}}$ | Dimensionless geometry of surface roughness $\left(-\right)$ |

$\overline{\rho}$ | Dimensionless lubricant density $\left(-\right)$ |

${\rho}_{s1}\phantom{\rule{0.166667em}{0ex}}{\rho}_{s2}$ | Density of upper- and lower body $\left(\mathrm{kg}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}\right)$ |

$\rho ,\phantom{\rule{0.166667em}{0ex}}{\rho}_{r}$ | Lubricant density and reference density, respectively $\left(\mathrm{kg}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}\right)$ |

$\sigma $ | Stress tensor $\left(-\right)$ |

$\tau $ | Shear stress $\left(\mathrm{Pa}\right)$ |

$\Theta $ | Dimensionless time $\left(-\right)$ |

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**Figure 1.**Illustration of the two cylinders in contact being reduced to one equivalent curvature in contact with a flat plane.

**Figure 2.**The three different roughness profiles studied with a fixed dimensionless amplitude of ${\overline{A}}_{\mathcal{R},1}=0.015$ and dimensionless wavelength values of ${\Lambda}_{\mathcal{R},1}=1/2$ (

**top**), ${\Lambda}_{\mathcal{R},1}=1/4$ (

**middle**) and ${\Lambda}_{\mathcal{R},1}=1/6$ (

**bottom**).

**Figure 3.**Film thickness and pressure for the thermal non-Newtonian (T+NN) and isothermal Newtonian (IT+N) approaches (

**left column**) and temperature throughout the film thickness (

**right column**) for the case ${\Lambda}_{\mathcal{R},1}=1/2$ and ${\overline{A}}_{\mathcal{R},1}=0.015$, showing the steady state solution, that is, $\Theta =0$ (

**top**), and the transient solution frozen in time at $\Theta =2.25$ (

**middle**) and $\Theta =2.5$ (

**bottom**).

**Figure 4.**Film thickness and pressure for the thermal non-Newtonian (T+NN) and isothermal Newtonian (IT+N) approaches (

**left column**) and temperature throughout the film thickness (

**right column**) for the case ${\Lambda}_{\mathcal{R},1}=1/4$ and ${\overline{A}}_{\mathcal{R},1}=0.015$, showing the steady state solution, that is, $\Theta =0$ (

**top**), and the transient solution frozen in time at $\Theta =2.25$ (

**middle**) and $\Theta =2.5$ (

**bottom**).

**Figure 5.**Film thickness and pressure for the thermal non-Newtonian (T+NN) and isothermal Newtonian (IT+N) approaches

**(left column**) and temperature throughout the film thickness (

**right column**) for the case ${\Lambda}_{\mathcal{R},1}=1/6$ and ${\overline{A}}_{\mathcal{R},1}=0.015$, showing the steady state solution, that is, $\Theta =0$ (

**top**), and the transient solution frozen in time at $\Theta =2.25$ (

**middle**) and $\Theta =2.5$ (

**bottom**).

**Figure 6.**Viscosity variations throughout the film thickness at $\Theta =2.5$ for the case ${\Lambda}_{\mathcal{R},1}=1/2$ (

**top left**), ${\Lambda}_{\mathcal{R},1}=1/4$ (

**top right**) and ${\Lambda}_{\mathcal{R},1}=1/6$ (

**bottom**).

**Figure 7.**Minimum and central film thickness values over time (

**left column**) and relative difference between the thermal non-Newtonian and isothermal Newtonian (IT+N) solutions (

**right column**) for the case ${\Lambda}_{\mathcal{R},1}=1/2$ (

**top**), ${\Lambda}_{\mathcal{R},1}=1/4$ (

**middle**) and ${\Lambda}_{\mathcal{R},1}=1/6$ (

**bottom**).

**Figure 8.**Maximum pressure over time for the three different cases showing the thermal non-Newtonian solution in comparison to the corresponding isothermal Newtonian solution (IT+N).

**Figure 9.**Asperity geometry (

**left**) and dent geometry (

**right**) using Equation (9) with the values ${\overline{A}}_{\mathcal{R},i}=\pm 0.03$, ${\Lambda}_{\mathcal{R},1}={\Lambda}_{\mathcal{R},2}=0.5$ and ${X}_{0}=0$.

**Figure 10.**Pressure and film thickness (

**left column**), and temperature throughout the film (

**right column**) for the case $A+A$ showing the transient solution frozen in time for the positions ${X}_{\mathcal{R},1}=-2$, ${X}_{\mathcal{R},1}=-1$, ${X}_{\mathcal{R},1}=-0.5$, ${X}_{\mathcal{R},1}=-0$, ${X}_{\mathcal{R},1}=0.5$ and ${X}_{\mathcal{R},1}=1$ from top to bottom.

**Figure 11.**Pressure and film thickness (

**left column**), and temperature throughout the film (

**right column**) for the case $A+D$ showing the transient solution frozen in time for the positions ${X}_{\mathcal{R},1}=-2$, ${X}_{\mathcal{R},1}=-1$, ${X}_{\mathcal{R},1}=-0.5$, ${X}_{\mathcal{R},1}=-0$, ${X}_{\mathcal{R},1}=0.5$ and ${X}_{\mathcal{R},1}=1$ from top to bottom.

**Figure 12.**Film thickness in space over time for the case $A+A$ (

**top left**), $D+D$ (

**top right**), $A+D$ (

**bottom left**) and $D+A$ (

**bottom right**).

**Figure 13.**Central and minimum film thickness over time for the four studied overtaking scenarios, following the position of the surface feature located on the upper surface, that is, ${X}_{\mathcal{R},1}$.

**Figure 14.**Pressure in space over time for the case $A+A$ (

**top left**), $D+D$ (

**top right**), $A+D$ (

**bottom left**) and $D+A$ (

**bottom right**).

**Figure 15.**Pressure and film thickness (

**left column**), and temperature throughout the film (

**right column**) for the case $A+A$ with overtaking at the outlet, that is, at $X=1$, showing the transient solution frozen in time for the positions ${X}_{\mathcal{R},1}=-2$, ${X}_{\mathcal{R},1}=-1$, ${X}_{\mathcal{R},1}=-0.5$, ${X}_{\mathcal{R},1}=-0$, ${X}_{\mathcal{R},1}=0.5$ and ${X}_{\mathcal{R},1}=1$ from top to bottom.

**Table 1.**Input data related to material properties and operating conditions used in the simulations.

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

${E}_{1},\phantom{\rule{0.166667em}{0ex}}{E}_{2}$ | 206 | $\mathrm{GPa}$ | F | $0.600$ | $\mathrm{MN}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-1}$ |

${\rho}_{1},\phantom{\rule{0.166667em}{0ex}}{\rho}_{2}$ | 7850 | $\mathrm{kg}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}$ | ${p}_{h}$ | $1.201$ | $\mathrm{GPa}$ |

${\nu}_{1},\phantom{\rule{0.166667em}{0ex}}{\nu}_{2}$ | $0.3$ | − | R | 15 | $\mathrm{mm}$ |

${k}_{1},\phantom{\rule{0.166667em}{0ex}}{k}_{2}$ | 45 | $\mathrm{W}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$ | ${T}_{r}$ | 40 | ${}^{\circ}\mathrm{C}$ |

${c}_{1},\phantom{\rule{0.166667em}{0ex}}{c}_{2}$ | 450 | $\mathrm{J}\phantom{\rule{0.166667em}{0ex}}{\mathrm{kg}}^{-1}{\mathrm{K}}^{-1}$ | ${u}_{e}$ | 1 | $\mathrm{m}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$ |

a | $0.318$ | $\mathrm{mm}$ | $SRR$ | $-0.5$ | − |

© 2020 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**

Hultqvist, T.; Vrček, A.; Marklund, P.; Larsson, R.
On Waviness and Two-Sided Surface Features in Thermal Elastohydrodynamically Lubricated Line Contacts. *Lubricants* **2020**, *8*, 64.
https://doi.org/10.3390/lubricants8060064

**AMA Style**

Hultqvist T, Vrček A, Marklund P, Larsson R.
On Waviness and Two-Sided Surface Features in Thermal Elastohydrodynamically Lubricated Line Contacts. *Lubricants*. 2020; 8(6):64.
https://doi.org/10.3390/lubricants8060064

**Chicago/Turabian Style**

Hultqvist, Tobias, Aleks Vrček, Pär Marklund, and Roland Larsson.
2020. "On Waviness and Two-Sided Surface Features in Thermal Elastohydrodynamically Lubricated Line Contacts" *Lubricants* 8, no. 6: 64.
https://doi.org/10.3390/lubricants8060064