Viscosity Temperature⚠ unverified
Mechanical / Lubrication · Compute the viscosity at temperature using a simplified ASTM relation
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| mu40 | μ40 | Pa.s | 1.0 | Dynamic viscosity at 40 degC |
| T | T | degC | 1.0 | Operating temperature |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | μ | Pa.s | Dynamic viscosity at temperature ``T``, in pascal-seconds (Pa.s) |
The science & history
Understanding the Parameters
-
Reference viscosity $\mu_{40}$ — the viscosity at the 40 °C standard reference (the temperature at which ISO VG grades are defined). It anchors the curve.
-
Temperature $T$ — the operating temperature. Viscosity drops exponentially above the reference and rises below it: at the constant $0.03$/°C, viscosity halves roughly every 23 °C rise ($\ln 2/0.03 \approx 23$). This steep sensitivity is why bearings that run hot lose their oil film and why cold starts are hard on lubrication.
-
The decay constant $0.03$/°C — sets how fast viscosity changes. Real oils differ: a high viscosity index (VI) oil (multigrade, with VI improvers) has a smaller effective decay constant — it holds viscosity better across temperature, which is the whole point of multigrade engine oils.
Derivation (Approaching a Proof)
Viscosity of a liquid arises from molecules needing to overcome an energy barrier to slide past one another; the fraction with enough thermal energy follows a Boltzmann/Arrhenius law, giving the classic exponential temperature dependence:
$$\mu(T) = \mu_0\, e^{E_a/(k_B T)},$$
where $E_a$ is an activation energy for viscous flow. Over a modest temperature span this can be linearised about a reference temperature $T_{\text{ref}} = 40$ °C. Expanding the exponent to first order gives a simple exponential in $(T - T_{\text{ref}})$:
$$\mu \approx \mu_{40}\, e^{-\beta(T - 40)},$$
with $\beta \approx 0.03$/°C a fitted decay rate for typical mineral oils. The negative sign is physical: higher temperature → more molecules clear the flow barrier → lower viscosity. (This is the same Arrhenius origin as the viscosity term in EHL and the reason grease and oil life fall with temperature — though note the sign discipline: viscosity genuinely decreases with $T$, unlike the erroneous increasing form flagged in Grease Life Factor.) For accuracy across a wide range the Walther equation, which uses a double-logarithmic straight line, replaces this single exponential.
Dimensional check. The exponent $0.03(T-40)$ is dimensionless (per-°C times °C), so $[\mu] = [\mu_{40}] = \text{Pa}\cdot\text{s}$. ✓
History and Development
The exponential/Arrhenius viscosity–temperature relation dates to the late 19th century; the engineering standard is the Walther equation (1931), adopted as ASTM D341, which plots viscosity on double-log-vs-log-T axes as a straight line and defines the viscosity index (VI). Managing viscosity across temperature — via base-oil choice and VI improvers — is central to lubricant formulation and to selecting an oil that maintains an adequate film (Viscosity Required, Lambda Ratio) at operating temperature.
Related Concepts: Viscosity Required, Sommerfeld Number, Elastohydrodynamic Film, Stribeck Curve, Grease Life Factor
Notes: Simplified local exponential (viscosity halves ~every 23 °C here); use Walther/ASTM D341 or Vogel for wide ranges. Reference is 40 °C (ISO VG). High-VI oils change viscosity less with temperature.