Lubricant viscosity vs pressure
Calculates the rise of dynamic viscosity under high pressure using the Barus equation -- relevant to elastohydrodynamic (EHL) contacts, not ordinary, near-atmospheric hydrodynamic plain bearings.
Formula
eta_p = eta_0 * exp(alpha * p)
In heavily loaded point or line contacts (e.g. rolling contacts, gear flanks or highly loaded sliding points), local pressures of several hundred MPa up to a few GPa occur. At such pressures, lubricant viscosity rises exponentially; the Barus equation is the simplest approximation for this, using the fluid-specific pressure-viscosity coefficient alpha.
What this model does not show
The Barus equation is a simple approximation, not a full EHL model: it does not capture film-shape change (elastic deformation of the contacting bodies), contact heating, or shear-thinning at high shear rates. The coefficient alpha itself is temperature- and fluid-specific and must be known for the actual lubricant at operating temperature.
Worked example
A mineral oil has eta_0 = 30 mPa*s at operating temperature and a pressure-viscosity coefficient alpha = 20 GPa^-1. In the EHL contact, p = 500 MPa = 0.5 GPa. So alpha*p = 20 GPa^-1 x 0.5 GPa = 10 (dimensionless). Ratio eta_p/eta_0 = exp(10) ~ 22,026. So eta_p ~ 30 mPa*s x 22,026 ~ 660,800 mPa*s ~ 661 Pa*s -- an enormous rise typical of an EHL contact, compared with the atmospheric-pressure value.
Common mistakes
- Combining alpha and p in incompatible units (e.g. alpha in Pa^-1 but p in MPa) -- the product alpha*p must be dimensionless.
- Applying this equation to an ordinary hydrodynamic plain bearing, where oil pressure is only a few MPa and the Barus correction is negligible.
- Reusing an alpha value measured at one temperature unchanged at a different operating temperature.
Frequently asked questions
When does the Barus equation actually matter?
At very high local pressures, as occur in elastohydrodynamic (EHL) contacts -- e.g. rolling bearings, gear flanks or cam followers -- not in ordinary hydrodynamic plain bearings, where oil pressure usually reaches only a few MPa.
Where do I get the pressure-viscosity coefficient alpha?
Alpha is a measured, fluid- and temperature-specific property, usually from manufacturer datasheets or tribology literature for the actual lubricant at operating temperature, typically 10 to 30 GPa^-1 for mineral oils.
Why does viscosity rise exponentially, not linearly, with pressure?
At high pressure, lubricant molecules are compressed much closer together, raising intermolecular friction resistance much more than proportionally. The Barus equation empirically captures this observed exponential relationship.
Is the Barus equation a complete EHL model?
No. Full EHL models (e.g. Dowson-Higginson) additionally account for elastic deformation of the contacting bodies and film-shape change; the Barus equation only supplies the viscosity rise as an input to such models.
Does alpha change with temperature?
Yes, alpha generally falls as temperature rises. An alpha value measured at 40 C must not be reused unchanged at a 100 C operating temperature.
Why is pressure entered in GPa rather than MPa in the formula?
Alpha is usually given in GPa^-1. For the product alpha*p to stay dimensionless, p must also be in GPa -- the calculator accepts input in MPa and internally divides by 1000 to convert.