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Film Thickness Formula for Thermal EHL Line Contact Considering a New Reynolds–Carreau Equation


This article presents thermal EHL calculations for line contacts using a new analytical form of the Reynolds equation for lubricants whose rheological behaviour follows a modified Carreau model proposed by Bair. The isothermal calculation process was presented in: de la Guerra (Tribol Int 82:133–141, 2015). A new parametric formula is hereby developed using the aforementioned Reynolds–Carreau equation and adding the thermal effects to the solving process. The accuracy of this formula is discussed by comparing the estimates with the experimental and numerical results available. This analytical formula provides a fast and easy calculation methodology with good accuracy within a reasonably wide range of operating conditions.

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a :

Hertzian contact half-width (m)

c l :

Specific heat of the lubricant (J/kg K)

c b :

Specific heat of the bodies (J/kg K)

d 1,2 :

Thermally affected depth of the bodies (m)


Young’s reduced modulus (Pa)

G :

Shear modulus (Pa)

\(\bar{G}\) :

Dimensionless material parameter

h :

Film thickness profile (m)

h N :

Newtonian central film thickness (m)

h 0 :

Central film thickness (m)

k l :

Thermal conductivity of the lubricant (W/m K)

k b :

Thermal conductivity of the bodies (W/m K)

L :

Contact length in flow direction (m)

L T :

Thermal loading factor

n :

Carreau exponent

p :

Pressure (Pa)

p m :

Average Hertz pressure (Pa)

p 0 :

Maximum Hertz pressure (Pa)

Q :

Flow rate per unit length (m2/s)

R :

Reduced contact radius (m)

\(\bar{R}\) :

Anuradha and Kumar factor for shear-thinning under pure rolling conditions

\(\bar{S}\) :

Anuradha and Kumar factor for shear-thinning under rolling and sliding conditions

T :

Temperature of the lubricant (K)

T 0 :

Reference temperature of the lubricant (K)

T 1 :

Temperature of the upper body (K)

T 2 :

Temperature of the lower body (K)

T b :

Lubricant bath temperature (K)

u :

Velocity of the lubricant (m/s)

\(\bar{U}\) :

Dimensionless velocity parameter

u 1,2 :

Velocity of the surfaces (m/s)

u m :

Average velocity of the contacting surfaces (m/s)

Δu :

Sliding velocity of the contacting surfaces (m/s)

W :

Normal load per unit length (N/m)

\(\bar{W}\) :

Dimensionless load parameter

x :

Coordinate in flow direction (m)

z :

Coordinate across the film thickness (m)

α :

Viscosity–pressure coefficient (Pa−1)

β :

Viscosity–temperature coefficient (K−1)

\(\dot{\gamma }\) :

Shear rate (s−1)

ε :

Thermal expansion coefficient (K−1)

η :

Viscosity (Pa s)

κ :

Shear-thinning parameter

μ :

Low-shear viscosity (Pa s)

μ 0 :

Low-shear viscosity at ambient pressure and reference temperature (Pa s)

ρ l :

Density of the lubricant (kg/m3)

ρ b :

Density of the bodies (kg/m3)

Σ :

Δu/u m , slide-to-roll ratio

τ :

Shear stress (Pa)

τ m :

Mid-plane shear stress (Pa)

φ NN :

Shear-thinning factor under pure rolling conditions

φ SRR :

Shear-thinning factor under rolling and sliding conditions

φ T :

Thermal film thickness factor


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This work was carried out as a part of the Research Project DPI2013-48348-C2-2-R, financed by the Spanish Ministry of Economy and Competitiveness. We would also like to thank the Lubricants Laboratory of Repsol.

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Correspondence to Eduardo de la Guerra Ochoa.

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de la Guerra Ochoa, E., Echávarri Otero, J., Sánchez López, A. et al. Film Thickness Formula for Thermal EHL Line Contact Considering a New Reynolds–Carreau Equation. Tribol Lett 66, 31 (2018).

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  • Reynolds–Carreau equation
  • Shear-thinning
  • Thermal effects
  • Film thickness