Film Thickness Parameter⚠ unverified

Understanding the Parameters

• Dynamic viscosity $\eta$ — the lubricant's resistance to shear at the contact inlet. More viscous oil is dragged into the gap more effectively and builds a thicker film — the strongest lever on film thickness.

• Entrainment velocity $u$ — the mean speed with which the two surfaces sweep oil into the converging gap, $u = (u_1 + u_2)/2$. Film thickness grows with $u$: a stationary contact has no hydrodynamic film, which is why bearings are vulnerable at start-up and low speed.

• Pressure–viscosity coefficient $\alpha$ — the exponential rise of viscosity with pressure, $\eta(p) = \eta_0 e^{\alpha p}$. It is the reason EHL works: at the GPa pressures of a rolling contact the oil momentarily becomes glass-stiff, resisting being squeezed out and holding the surfaces apart. Mineral oils have $\alpha \approx 15$–$25\ \text{GPa}^{-1}$.

• Effective elastic modulus $E$ — the reduced (combined) modulus of the two bodies. Stiffer materials deform less, giving a smaller, higher-pressure contact; it enters EHL film thickness only weakly.

• Effective radius $R$ — the reduced radius of curvature of the contact, combining both bodies' radii. It sets the geometric scale of the converging wedge.

Derivation (Approaching a Proof)

The rigorous result the inputs are pointing at is the Dowson–Higginson minimum-film-thickness equation for EHL line contact, written in dimensionless groups:

$$\frac{h_{\min}}{R} = 2.65\; U^{0.7}\, G^{0.54}\, W^{-0.13},$$

where the dimensionless speed $U = \eta_0 u/(E' R)$, materials $G = \alpha E'$, and load $W = w'/(E' R)$ groups each combine several of the five inputs above. The physics is a coupled solution of the Reynolds hydrodynamic equation (oil pressure from the converging wedge), the elasticity equations (surfaces deform under that pressure — the "E" in EHL), and the pressure–viscosity law (oil stiffens under pressure). Film thickness therefore depends most strongly on viscosity and speed (exponent $0.7$), moderately on the materials parameter $\alpha E'$, and only weakly (and inversely) on load.

The calculator's $\lambda = \sqrt{\eta\,u\,\alpha\,E\,R}$ keeps all five drivers and their qualitative sense (rising with viscosity, speed, pressure-viscosity, radius) but collapses the exponents to a plain product-under-a-root. It is best read as a rough film-forming potential index for ranking contacts, not as a metric film thickness or a true separation ratio — for those, evaluate the dimensionless groups above.

History and Development

Osborne Reynolds derived the hydrodynamic lubrication equation in 1886. The elastohydrodynamic extension — recognising that contact pressures elastically deform the surfaces and pressurise-stiffen the oil — was solved by A. N. Grubin (1949) and brought to engineering maturity by Dowson and Higginson (Elasto-Hydrodynamic Lubrication, 1966), whose film-thickness formulas remain the standard. The lambda ratio $\lambda = h_{\min}/\sigma$ as a lubrication-regime indicator (boundary, mixed, full-film) was established by Tallian and others in the 1960s and underlies modern bearing life modifiers.

Related Concepts: Elastohydrodynamic Film, Minimum Film Thickness, Viscosity Required, Hydrodynamic Film Pressure, Stribeck Curve, Hertzian Contact Pressure

Notes: Heuristic composite index — see the honesty note. For quantitative work use the Dowson–Higginson film thickness and the $\kappa = \nu/\nu_1$ viscosity ratio (Viscosity Required), and compare $\lambda = h_{\min}/\sigma$ against surface roughness to identify the lubrication regime.