# Thermal Effects in Slender EHL Contacts

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Experimental Investigation

- pure rolling $SRR=0$;
- negative sliding $SRR=-1.5$ (counter steel body moves faster than glass disk);
- positive sliding $SRR=+1.5$ (glass disk moves faster than counter steel body).

#### 2.2. Numerical Investigation

#### 2.2.1. Governing Equations

#### 2.2.2. Computational Domain and Meshing

## 3. Results and Discussion

#### 3.1. Film Thickness

#### 3.2. Temperature, Viscosity and Velocity

#### 3.3. Side Flow

## 4. Conclusions

- The film thickness is lower for slender EHL contacts than for circular ones, considering the same Hertzian pressure.
- The different thermal effusivities of glass and steel result in a viscosity wedge, which is particularly pronounced at high positive sliding with the glass disk moving faster.
- A strong viscosity wedge diverts the oil flow to the contact sides and limits the amount of oil maintaining the oil film in the central region.
- At high positive sliding, a continuous decrease in film thickness is observed in the gap length direction of the slender EHL contact.
- At high positive sliding, the influence of entrainment speed on minimum film thickness is almost negligible, especially for slender contacts.
- To support EHL film formation in slender EHL contacts at higher sliding, the solid body made of the material with higher thermal effusivity has to move faster.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Latin symbols | ||

${A}_{c1,\dots ,5}$ | Coefficients of the lubricant heat capacity model | $-$ |

${A}_{\eta},{B}_{\eta},{C}_{\eta}$ | Coefficients of the lubricant Vogel temperature model | $-$ |

$a$ | Hertzian half-width, circular contact | $\mathrm{m}$ |

${a}_{\mathrm{x}}$ | Semi-major Hertzian contact length, slender contact | $\mathrm{m}$ |

${a}_{\mathrm{y}}$ | Semi-minor Hertzian contact length, slender contact | $\mathrm{m}$ |

$\overrightarrow{\mathrm{C}}$ | Compliance matrix | $\mathrm{Pa}$ |

${c}_{p}$ | Specific heat capacity | $\mathrm{J}/\left(\mathrm{kg}\mathrm{K}\right)$ |

${D}_{{\rho}_{0}},{D}_{{\rho}_{1}},{D}_{{\rho}_{2}},{D}_{{\rho}_{3}}$ | Coefficients of the lubricant Bode density model | $-$ |

$e$ | Thermal effusivity | $\mathrm{J}/\left(\mathrm{K}\sqrt{\mathrm{s}}{\mathrm{m}}^{2}\right)$ |

${d}_{\lambda 1},{d}_{\lambda 2}$ | Pressure coefficients of the lubricant thermal conductivity model | $1/\mathrm{Pa}$ |

$E$ | Young’s Modulus | $\mathrm{Pa}$ |

${F}_{N}$ | Normal force | $\mathrm{N}$ |

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

${h}_{c}$ | Central film thickness | $\mathrm{m}$ |

${h}_{m}$ | Minimum film thickness | $\mathrm{m}$ |

$H$ | Dimensionless film thickness | $-$ |

${H}_{0}$ | Dimensionless constant parameter of the film thickness equation | $-$ |

$k$ | Ellipticity ratio | $-$ |

${n}_{e}^{{\mathsf{\Omega}}_{P}}$ | Number of mesh elements in domain ${\mathsf{\Omega}}_{P}$ | $-$ |

${n}_{e}^{f}$ | Number of mesh layers in gap height direction in domain $f$ | $-$ |

${n}_{e}^{{\mathsf{\Omega}}_{T}}$ | Number of mesh elements in domain ${\mathsf{\Omega}}_{T}$ | $-$ |

${n}_{e}^{{\mathsf{\Omega}}_{\delta}}$ | Number of mesh elements in domain ${\mathsf{\Omega}}_{\delta}$ | $-$ |

${n}_{e}^{{\mathsf{\Omega}}_{T,1}}$ | Number of mesh elements in domain ${\mathsf{\Omega}}_{T,1}$ | $-$ |

${n}_{e}^{{\mathsf{\Omega}}_{T,2}}$ | Number of mesh elements in domain ${\mathsf{\Omega}}_{T,2}$ | $-$ |

${n}_{e}$ | Total number of mesh elements | $-$ |

$p$ | Hydrodynamic pressure | $\mathrm{Pa}$ |

$P$ | Dimensionless hydrodynamic pressure | $-$ |

${p}_{H}$ | Hertzian pressure | $\mathrm{Pa}$ |

$\dot{{Q}_{\mathrm{y}p}}$ | Mass flow rate crossing the frontier $\mathcal{F}$ in y-direction | $\mathrm{kg}/\left({\mathrm{m}}^{2}\mathrm{s}\right)$ |

${p}_{{\eta}_{0}}$ | Coefficient of the Roelands’ equation | $\mathrm{Pa}$ |

$P{e}^{e}$ | Peclet number for the energy equation | $-$ |

${R}_{\mathrm{x}},{R}_{\mathrm{y}}$ | Radii of curvature in x- and y-direction | $\mathrm{m}$ |

$SRR$ | Slide-to-roll ratio | $-$ |

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

${T}_{max}$ | Maximum temperature | $\mathrm{K}$ |

${T}_{M}$ | Bulk temperature | $\mathrm{K}$ |

$\overrightarrow{U}$ | Displacement vector | $\mathrm{m}$ |

${v}_{g,\mathrm{x}}$ | Sliding speed in x-direction. | $\mathrm{m}/\mathrm{s}$ |

${v}_{m,\mathrm{x}}$ | Entrainment speed in x-direction | $\mathrm{m}/\mathrm{s}$ |

${v}_{\Sigma ,\mathrm{x}}$ | Sum speed in x-direction | $\mathrm{m}/\mathrm{s}$ |

${v}_{1}$ | Speed of the counter steel body | $\mathrm{m}/\mathrm{s}$ |

${v}_{2}$ | Speed of the glass disk | $\mathrm{m}/\mathrm{s}$ |

$\mathrm{x},\mathrm{y},\mathrm{z}$ | Coordinates | $\mathrm{m}$ |

$\mathrm{X},\mathrm{Y},\mathrm{Z}$ | Dimensionless coordinates | $-$ |

Greek symbols | ||

${\alpha}_{p}$ | Pressure viscosity coefficient | $1/\mathrm{Pa}$ |

${\alpha}_{s}$ | Coefficient of the lubricant Bode density model | $1/\mathrm{K}$ |

$\delta $ | Deformation of the equivalent body | $\mathrm{m}$ |

$\overline{\delta}$ | Dimensionless deformation of the equivalent body | $-$ |

$\overrightarrow{\epsilon}$ | Strain tensor | $-$ |

$\dot{{\gamma}_{x}}$ | Shear rate in x-direction | $1/s$ |

$\vartheta $ | Temperature | $\mathbb{C}$ |

${\vartheta}_{oil}$ | Oil temperature | $\mathbb{C}$ |

$\eta $ | Dynamic viscosity | $\mathrm{Pa}\xb7\mathrm{s}$ |

$\overline{\eta}$ | Dimensionless dynamic viscosity | $-$ |

${\eta}_{0}$ | Dynamic viscosity of the lubricant at ${T}_{M}$ and atmospheric pressure | $\mathrm{Pa}\xb7\mathrm{s}$ |

$\lambda $ | Thermal conductivity | $\mathrm{W}/\left(\mathrm{m}\mathrm{K}\right)$ |

${\lambda}_{0}$ | Thermal conductivity of the lubricant at ${T}_{M}$ and atmospheric pressure | $\mathrm{W}/\left(\mathrm{m}\mathrm{K}\right)$ |

$v$ | Poisson’s ratio | $-$ |

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

$\overline{\rho}$ | Dimensionless density | $-$ |

${\rho}_{0}$ | Fluid density at ${T}_{M}$ and atmospheric pressure | $\mathrm{kg}/{\mathrm{m}}^{3}$ |

${\rho}_{s}$ | Coefficient of the lubricant Bode density model | $\mathrm{kg}/{\mathrm{m}}^{3}$ |

$\overrightarrow{\sigma}$ | Stress tensor of the equivalent body | $\mathrm{Pa}$ |

$\dot{\Sigma {Q}_{\mathrm{y}p}}$ | Total mass flow rate crossing the frontier $\mathcal{F}$ in y-direction | $\mathrm{kg}/\left({\mathrm{m}}^{2}\mathrm{s}\right)$ |

$\tau $ | Shear stress | $\mathrm{Pa}$ |

${\tau}_{c}$ | Eyring shear stress | $\mathrm{Pa}$ |

${\overline{\tau}}_{\mathrm{z},\mathrm{x}}^{0},{\overline{\tau}}_{\mathrm{z},\mathrm{y}}^{0}$ | Dimensionless shear stress components in the x- and y-directions over the domain ${\Omega}_{P}$ | $-$ |

Special symbols | ||

$\mathcal{F}$ | Frontier used for the side flow analysis | $-$ |

Indices | ||

$1$ | Lower solid body (counter steel body) | $-$ |

$2$ | Upper solid body (glass disk) | $-$ |

$f$ | Fluid | $-$ |

## References

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**Figure 1.**Mechanical layout of the used optical EHL tribometer based on [16].

**Figure 2.**FEM model of the TEHL point contact problem. (

**a**) Geometry of the 3D computation domains; (

**b**) Discretized domains; (

**c**) Discretized oil film domain.

**Figure 4.**Solution of a slender EHL contact test case with fine mesh. (

**a**) Pressure distribution; (

**b**) Film thickness distribution; (

**c**) Temperature distribution.

**Figure 5.**Experimental and numerical results for the slender EHL contact with different slide-to-roll ratios $SRR$ and entrainment speeds ${v}_{m,\mathrm{x}}$ (subfigures (

**a**–

**i**)): Measured and calculated interferograms (

**top**), derived film thickness profiles ${h}_{exp}$ and ${h}_{num}$, and calculated presure profiles ${p}_{num}$ in gap length direction (

**bottom**).

**Figure 6.**Experimental and numerical results for the circular EHL contact with different slide-to-roll ratios $SRR$ and entrainment speeds ${v}_{m,\mathrm{x}}$ (subfigures (

**a**–

**i**)): Measured and calculated interferograms (

**top**), derived film thickness profiles ${h}_{exp}$ and ${h}_{num}$, and calculated pressure profiles ${p}_{num}$ in gap length direction (

**bottom**).

**Figure 7.**Experimental and numerical results for ${h}_{m}$ and ${h}_{c}$ over entrainment speed ${v}_{m,\mathrm{x}}$ for $SRR=0$ (

**left**), $SRR=-1.5$ (

**middle**) and $SRR=+1.5$ (

**right**). (

**a**) Slender EHL contact; (

**b**) Circular EHL contact; (

**c**) Ratio of ${h}_{m}$ to ${h}_{c}$ for slender and circular EHL contact.

**Figure 8.**Numerical results for temperature (

**a**), viscosity (

**b**) and velocity (

**c**) distributions in the slender EHL contact at ${v}_{m,\mathrm{x}}=1.8\mathrm{m}/\mathrm{s}$ and different $SRR$.

**Figure 9.**Schematic representation of the frontier $\mathcal{F}$ used for the side flow analysis (

**top**). Variation of side flow ratio for different ${v}_{m,\mathrm{x}}$ and $SRR$ in the slender (

**bottom left**) and circular EHL contact (

**bottom right**).

Slender Contact | Circular Contact | ||
---|---|---|---|

${F}_{N}$ | $15.5$ | $54$ | $\mathrm{N}$ |

${p}_{H}$ | $0.63$ | $0.63$ | $\mathrm{GPa}$ |

${v}_{m,\mathrm{x}}$ | $\left\{0.6,1.2,1.8\right\}$ | $\left\{0.6,1.2,1.8\right\}$ | $\mathrm{m}/\mathrm{s}$ |

$SRR$ | $\left\{0,-1.5,+1.5\right\}$ | $\left\{0,-1.5,+1.5\right\}$ | $-$ |

${\vartheta}_{oil}$ | $40$ | $40$ | $\mathbb{C}$ |

Steel | Glass | ||
---|---|---|---|

$E$ | $210$ | $81$ | $\mathrm{GPa}$ |

$\upsilon $ | $0.3$ | $0.208$ | $-$ |

$\rho $ | $7850$ | $2500$ | $\mathrm{kg}/{\mathrm{m}}^{3}$ |

${c}_{p}$ | $470$ | $858$ | $\mathrm{J}/\left(\mathrm{kgK}\right)$ |

$\lambda $ | $21$ [20] | 1.1 | $\mathrm{W}/\left(\mathrm{mK}\right)$ |

$e=\sqrt{\lambda \rho {c}_{p}}$ | $8802$ | $1536$ | $\mathrm{J}/\left(\mathrm{K}\sqrt{\mathrm{s}}{\mathrm{m}}^{2}\right)$ |

**Table 3.**Properties of MIN100 [19].

${A}_{\eta}$ | $0.047$ | $\mathrm{mPa}\xb7\mathrm{s}$ |

${B}_{\eta}$ | $1006$ | $\mathbb{C}$ |

${C}_{\eta}$ | $0$ | $\mathbb{C}$ |

${p}_{{\eta}_{0}}$ | $0$ | $\mathrm{Pa}$ |

${E}_{{\alpha}_{p}1}$ | $0.181$ | ${\mathrm{m}}^{2}/\mathrm{N}$ |

${E}_{{\alpha}_{p}2}$ | $-0.0059$ | $1/\mathrm{N}$ |

$\rho \left(15\mathbb{C}\right)$ | $885$ | $\mathrm{kg}/{\mathrm{m}}^{3}$ |

${\rho}_{s}$ | $1042$ | $\mathrm{kg}/{\mathrm{m}}^{3}$ |

${\alpha}_{s}$ | $0.00053$ | $1/\mathrm{K}$ |

${D}_{{\rho}_{0}}$ | $0.0786$ | $-$ |

${D}_{{\rho}_{1}}$ | $315.8$ | $\mathrm{N}/{\mathrm{mm}}^{2}$ |

${D}_{{\rho}_{2}}$ | $0$ | $\mathrm{N}/\mathrm{mm}/\mathrm{K}$ |

${D}_{{\rho}_{3}}$ | $0.00035$ | $\mathrm{N}/\mathrm{mm}/{\mathrm{K}}^{2}$ |

${\eta}_{0}$ | $11.0$ | $\mathrm{mPa}\xb7\mathrm{s}$ |

${\alpha}_{p}\left(90\mathbb{C}\right)$ | $0.021$ | ${\mathrm{mm}}^{2}/\mathrm{N}$ |

Mesh Case | ${\mathit{n}}_{\mathit{e}}^{{\mathit{\Omega}}_{\mathit{P}}}$ | ${\mathit{n}}_{\mathit{e}}^{\mathit{f}}$ | ${\mathit{n}}_{\mathit{e}}^{{\mathit{\Omega}}_{\mathit{T}}}$ | ${\mathit{n}}_{\mathit{e}}^{{\mathit{\Omega}}_{\mathit{\delta}}}$ | ${\mathit{n}}_{\mathit{e}}^{{\mathit{\Omega}}_{\mathit{T},1}}$ | ${\mathit{n}}_{\mathit{e}}^{{\mathit{\Omega}}_{\mathit{T},2}}$ | ${\mathit{n}}_{\mathit{e}}{}^{1}$ |
---|---|---|---|---|---|---|---|

Extra coarse | 1301 | 3 | 3903 | 671 | 5891 | 5826 | 16,291 |

Coarse | 1521 | 3 | 4563 | 2489 | 7143 | 7028 | 21,233 |

Normal | 1866 | 4 | 7464 | 7289 | 8734 | 8792 | 32,279 |

Fine | 2658 | 5 | 13,290 | 13,385 | 12,345 | 12,336 | 51,356 |

Extra fine | 2658 | 10 | 26,580 | 40,958 | 12,345 | 12,336 | 92,219 |

^{1}Total number of mesh elements ${n}_{e}$ is calculated as: ${n}_{e}={n}_{e}^{{\mathsf{\Omega}}_{T}}+{n}_{e}^{{\mathsf{\Omega}}_{\delta}}+{n}_{e}^{{\mathsf{\Omega}}_{T,1}}+{n}_{e}^{{\mathsf{\Omega}}_{T,2}}$.

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**MDPI and ACS Style**

Tošić, M.; Larsson, R.; Lohner, T.
Thermal Effects in Slender EHL Contacts. *Lubricants* **2022**, *10*, 89.
https://doi.org/10.3390/lubricants10050089

**AMA Style**

Tošić M, Larsson R, Lohner T.
Thermal Effects in Slender EHL Contacts. *Lubricants*. 2022; 10(5):89.
https://doi.org/10.3390/lubricants10050089

**Chicago/Turabian Style**

Tošić, Marko, Roland Larsson, and Thomas Lohner.
2022. "Thermal Effects in Slender EHL Contacts" *Lubricants* 10, no. 5: 89.
https://doi.org/10.3390/lubricants10050089