Lambda Ratio Calculator

Estimate lubrication regime from tribological test conditions

Calculates the lambda ratio (λ = h_min / σ) from test configuration, load, entrainment speed, lubricant properties, and surface roughness using EHL theory (Hamrock-Dowson / Dowson-Higginson).

1Select test method

Rotating

Reciprocating

Equations and standards

Hamrock–Dowson (1981)Dowson–Higginson (1966)Hertz

\bar{U} = \frac{\eta_0\,u_e}{E'R}, \qquad \bar{G} = \alpha E', \qquad \bar{W} = \frac{W}{E'R^{2}}

The dimensionless speed, material, and load groups. Line contact uses W' = load / (E' * R) instead.

H_{\min} = 3.63\,\bar{U}^{0.68}\,\bar{G}^{0.49}\,\bar{W}^{-0.073}\left(1 - e^{-0.68k}\right)

Point contact. For a circular contact k = 1; the dimensional film follows from h_min = H_min · R.

H_{\min} = 2.65\,\bar{U}^{0.7}\,\bar{G}^{0.54}\,\bar{W}'^{-0.13}

Line contact, used for cylinder-on-flat style geometries.

\frac{1}{E'} = \frac{1-\nu_1^{2}}{E_1} + \frac{1-\nu_2^{2}}{E_2}

Reduced modulus. Here nu is Poisson's ratio, not viscosity.

a = \left(\frac{3WR}{4E'}\right)^{1/3}, \qquad p_{\max} = \frac{3W}{2\pi a^{2}}

Hertzian contact radius and peak pressure for point contact. Line contact uses b = sqrt(4w'R / (pi E')) and p_max = 2w' / (pi b).

\lambda = \frac{h_{\min}}{\sigma}, \qquad \sigma = \sqrt{\sigma_1^{2} + \sigma_2^{2}}

Classified here as boundary below 1, mixed from 1 to 3, and full film at 3 and above.

Symbols and units

$\eta_0$	Dynamic viscosity at ambient pressure and operating temperature	Pa·s
$u_e$	Entrainment speed (u1 + u2)/2 — not the sliding speed	m/s
$\alpha$	Pressure–viscosity coefficient	GPa⁻¹
$E'$	Reduced elastic modulus	Pa
$R$	Reduced radius of curvature	m
$W$	Normal load (per unit length, w', for line contact)	N
$h_{\min}$	Minimum film thickness	nm
$\sigma$	Composite RMS surface roughness	nm
$\lambda$	Film thickness ratio	—

Valid range and limitations

·Film thickness is governed by entrainment speed. Feeding a sliding speed from a pure-sliding test overestimates the film.
·Assumes isothermal, Newtonian behaviour — no shear heating or shear thinning at high slide-to-roll ratios.
·The pressure–viscosity coefficient depends strongly on base oil and temperature; a borrowed literature value can shift the film by tens of percent.
·The regime boundaries at lambda = 1 and 3 are conventional rules of thumb; texture and additive films move them.

Reproduce in Excel

Reduced modulus E'

=1/((1-$B$1^2)/$A$1+(1-$B$2^2)/$A$2)

A1 = E1 in Pa, B1 = Poisson 1, A2 = E2 in Pa, B2 = Poisson 2.

Minimum film thickness, point contact

=3.63*(($A$5*$A$6)/($A$3*$A$4))^0.68*($A$7*$A$3)^0.49*($A$8/($A$3*$A$4^2))^-0.073*(1-EXP(-0.68))*$A$4

A3 = E' in Pa, A4 = R in m, A5 = eta0 in Pa·s, A6 = u_e in m/s, A7 = alpha in 1/Pa, A8 = load in N. Result in metres.

Lambda ratio

=A10/SQRT(A11^2+A12^2)

A10 = h_min, A11 = sigma1, A12 = sigma2 — all in the same unit.

Assumptions & Limitations

This calculator assumes Newtonian behavior and isothermal conditions. Real lubricants exhibit shear-thinning and temperature/pressure-dependent viscosity, which can significantly alter the actual lambda ratio.
Film thickness is governed by the entrainment (mean) speed u_e = (u₁ + u₂) / 2. Sliding speed affects film thickness only indirectly, via inlet shear heating.
Lambda ratio predictions carry inherent uncertainty due to variability in effective viscosity and deformed surface roughness. Use results as a comparative guide, not an absolute prediction.

What is the Lambda Ratio (λ)?

The lambda ratio (λ) is a dimensionless number that characterises the lubrication condition in a tribological contact. It is defined as the ratio of the minimum oil film thickness h_min to the composite surface roughness σ* of the contacting bodies: λ = h_min / σ*. When λ < 1, the asperities of the surfaces are in direct contact (boundary lubrication). For 1 ≤ λ < 3, a mixed regime occurs where partial asperity contact coexists with hydrodynamic pressure. At λ ≥ 3, the surfaces are fully separated by the oil film (full-film or hydrodynamic lubrication), resulting in minimal wear.

Calculation Method

This calculator applies Hamrock–Dowson equations for elastohydrodynamic lubrication (EHL) to estimate h_min for point and line contact geometries. The reduced elastic modulus E' and Hertzian contact parameters (contact radius a, maximum pressure p₀) are computed first. Viscosity at operating temperature is interpolated from tabulated data using the Williams–Landel–Ferry (WLF) model. The resulting λ value classifies the regime and guides lubricant selection or film thickness optimisation.

Related Calculators

Viscosity Index & Temperature Calculator

ASTM D2270 VI and the ASTM D341 (Walther) viscosity–temperature curve

Specific Wear Rate Calculator

Compute wear volume and specific wear rate k from surface data, load, and sliding distance

Oil Blend Calculator

Compute the kinematic viscosity of a two-oil blend using the Refutas equation

Assumptions & Limitations

This calculator assumes Newtonian behavior and isothermal conditions. Real lubricants exhibit shear-thinning and temperature/pressure-dependent viscosity, which can significantly alter the actual lambda ratio.

Film thickness is governed by the entrainment (mean) speed u_e = (u₁ + u₂) / 2. Sliding speed affects film thickness only indirectly, via inlet shear heating.

Lambda ratio predictions carry inherent uncertainty due to variability in effective viscosity and deformed surface roughness. Use results as a comparative guide, not an absolute prediction.

What is the Lambda Ratio (λ)?

Calculation Method