Lambda Ratio Calculator
Estimate lubrication regime from tribological test conditions
Calculates the lambda ratio (λ = hmin / σ) from test configuration, load, entrainment speed, lubricant properties, and surface roughness using EHL theory (Hamrock-Dowson / Dowson-Higginson).
Select test method
Rotating
Reciprocating
Enter test conditions
Geometry
Load
Motion
Mean of the two surface velocities, u_e = (u₁ + u₂) / 2. For pin-on-flat with one moving surface, entrainment speed ≈ (moving surface speed) / 2. For two surfaces moving together, use the surface speed directly.
Lubricant
Materials
Surface roughness
Surface roughness — Body 1 (Ball)
Surface roughness — Body 2 (Disk)
Results
Results will appear automatically as you enter values.
Equations and standards
The dimensionless speed, material, and load groups. Line contact uses W' = load / (E' * R) instead.
Point contact. For a circular contact k = 1; the dimensional film follows from h_min = H_min · R.
Line contact, used for cylinder-on-flat style geometries.
Reduced modulus. Here nu is Poisson's ratio, not viscosity.
Hertzian contact radius and peak pressure for point contact. Line contact uses b = sqrt(4w'R / (pi E')) and p_max = 2w' / (pi b).
Classified here as boundary below 1, mixed from 1 to 3, and full film at 3 and above.
Symbols and units
| Dynamic viscosity at ambient pressure and operating temperature | Pa·s | |
| Entrainment speed (u1 + u2)/2 — not the sliding speed | m/s | |
| Pressure–viscosity coefficient | GPa⁻¹ | |
| Reduced elastic modulus | Pa | |
| Reduced radius of curvature | m | |
| Normal load (per unit length, w', for line contact) | N | |
| Minimum film thickness | nm | |
| Composite RMS surface roughness | nm | |
| 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$4A3 = 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.
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