Elastohydrodynamic Film⚠ unverified

Understanding the Parameters

• Entrainment velocity $U$ and base viscosity $\mu_0$ — appear in the speed parameter $U^* = \mu_0 U/(E^* R)$ raised to the powerful 0.7. This is the classical hydrodynamic wedge action: faster entrainment and more viscous oil drag more lubricant into the contact. Speed is the dominant lever — which is why EHL films collapse at low speed (start/stop, reversal), the moment of highest wear risk.

• Pressure–viscosity coefficient $\alpha$ — in the materials parameter $G^* = \alpha E^*$ raised to 0.54. This is the term that makes EHL work: at the GPa contact pressure the oil's viscosity rises by many orders of magnitude ($\mu = \mu_0 e^{\alpha p}$), momentarily behaving like a solid and resisting being squeezed out. Without pressure–viscosity, the film would vanish.

• Load per unit length $w$ — in the load parameter $W^* = w/(E^* R)$ raised to only $-0.13$. Load's weak, inverse influence is EHL's most surprising feature: pressing harder barely thins the film (the surfaces just flatten, spreading the load), so rolling contacts maintain lubrication across a wide load range.

• Effective radius $R$ and modulus $E^*$ — the Hertzian contact geometry (as in Hertzian Contact Pressure); they set the contact size and the elastic flattening.

Derivation (Approaching a Proof)

EHL couples three physics that classical hydrodynamics keeps separate:

1. Reynolds hydrodynamics — the entrained oil generates pressure in the converging inlet (Hydrodynamic Film Pressure).

2. Elastic (Hertzian) deformation — that pressure is so high it flattens the surfaces, enlarging the contact and creating a nearly parallel film with a characteristic constriction at the outlet (Hertzian Contact Pressure).

3. Pressure–viscosity — the oil's viscosity rises exponentially with pressure, $\mu(p) = \mu_0 e^{\alpha p}$, locking the film in.

Solving the coupled Reynolds + elasticity + pressure–viscosity equations is intractable by hand, so Dowson and Higginson (1966) computed it numerically and curve-fit the minimum film thickness to three dimensionless groups — speed $U^*$, materials $G^*$, and load $W^*$:

$$\frac{h_{min}}{R} = 2.65\,(U^*)^{0.7}(G^*)^{0.54}(W^*)^{-0.13}.$$

Substituting $U^* = \mu_0 U/(E^* R)$, $G^* = \alpha E^*$, $W^* = w/(E^* R)$ gives the registry form. The exponents are the fitted results, and their message is physical: film thickness is governed by inlet conditions (speed, viscosity, pressure–viscosity), which is why it barely depends on the load carried in the high-pressure Hertzian zone. The computed film ($h \sim 0.1$–$1\ \mu\text{m}$) compared with surface roughness gives the Lambda Ratio that decides bearing/gear surface-fatigue life.

Dimensional check. Each of $U^*$, $G^*$, $W^*$ is dimensionless, so $h/R$ is dimensionless and $[h] = [R] = \text{m}$. ✓

History and Development

Grubin (1949) first recognised the elastic-deformation coupling; Dowson and Higginson (Elasto-Hydrodynamic Lubrication, 1966) produced the definitive numerical solutions and film-thickness formulas, with Hamrock–Dowson later extending them to elliptical (point) contacts. EHL explained how ball bearings and gears — nominal point/line contacts at GPa pressure — can run for billions of cycles on a sub-micron oil film. It is the theoretical basis of the viscosity ratio (Viscosity Required) and the lubrication life factors in bearing rating.

Related Concepts: Hertzian Contact Pressure, Lambda Ratio, Viscosity Required, Minimum Film Thickness, Hydrodynamic Film Pressure, Gear Contact Stress

Notes: Dowson–Higginson line-contact fit; the h_min input is an unused placeholder (see note). Film depends strongly on speed/viscosity/pressure-viscosity, weakly on load. Compare with roughness via the Lambda Ratio.