Viscosity Calculator: Barus and Roelands Equations

Use at your own risk. Verify inputs, assumptions and results independently before applying them.

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In general, lubricants used in industry are subject to high pressures, up to several GPa. At high pressures, lubricant viscosity increases significantly, and this increase should be taken into account when calculating central film thickness or friction in the contact. This calculator estimates viscosity for a given pressure using two common equations: Barus and Roelands. More background is available in the TriboNet article on lubricant viscosity.

Barus equation

This equation is very simple and therefore widely used. It links the viscosity of the oil to the pressure as follows:

μ = μ₀e^αp

where μ₀ is the viscosity at ambient pressure and α is the pressure-viscosity coefficient. For lubricating oils, α is typically about 0.1 × 10^-8 Pa^-1 to 4 × 10^-8 Pa^-1.

The Barus equation is known to become inaccurate at high pressure, above about 1 GPa, where it can predict a larger viscosity increase rate than is observed in practice.

Roelands equation

A more accurate relation, which can also include temperature dependence, is given by Roelands:

μ = μ₀e^{(ln(μ₀) + 9.67)((1 + p / p_r)^z((T₀ – 138) / (T – 138))^S₀ – 1)}

Here:

p_r = 1.962 × 10⁸ Pa
T₀ is the temperature at which μ₀ was measured, in kelvins
S₀ is the viscosity-temperature index, commonly 1.0 to 1.5
z is the pressure-viscosity index, commonly 0 to 0.8
μ and μ₀ are given in Pa·s in this equation

The pressure-viscosity coefficient α used in the Barus equation is related to the pressure-viscosity index z in the Roelands equation as follows:

z = p_rα / (ln(μ₀) + 9.67)

Worked example

In the default Barus viscosity mode, enter ambient viscosity 0.1 cP, pressure–viscosity coefficient 10 GPa⁻¹ and pressure 0.5 GPa. The calculated viscosity is 0.1 exp(10 × 0.5) = 14.84 cP after rounding. In ratio mode, 60 mPa·s, 20 GPa⁻¹ and 100 MPa give a Barus ratio exp(2) = 7.39; the ratio has no units. These are model illustrations. Select calibrated coefficients and review the high-pressure limitations described below before using either relation.