# Extensional viscosity

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Polymer solution parameter

This article is about extensional viscosity. For other uses, see [Rheology](/source/Rheology).

**Extensional viscosity** (also known as **elongational viscosity**) is a [viscosity](/source/Viscosity) coefficient when the applied [stress](/source/Stress_(physics)) is [extensional stress](/source/Extensional_stress).[1] It is often used for characterizing polymer solutions. Extensional viscosity can be measured using rheometers that apply *extensional stress*. [Acoustic rheometer](/source/Acoustic_rheometer) is one example of such devices.

Extensional viscosity is defined as the ratio of the normal stress difference to the rate of strain. For uniaxial extension along direction z {\displaystyle z} :[2]

- η e = σ z z − 1 2 σ x x − 1 2 σ y y ε ˙ {\displaystyle \eta _{e}={\frac {\sigma _{zz}-{\frac {1}{2}}\sigma _{xx}-{\frac {1}{2}}\sigma _{yy}}{\dot {\varepsilon }}}\,\!}

where

- η e {\displaystyle \eta _{e}\,\!} is the **extensional viscosity** or **elongational viscosity**

- σ n n {\displaystyle \sigma _{nn}\,\!} is the **normal [stress](/source/Stress_(physics))** along direction n.

- ε ˙ {\displaystyle {\dot {\varepsilon }}\,\!} is the **rate of strain**: ε ˙ = ∂ v z ∂ z {\displaystyle {\dot {\varepsilon }}={\frac {\partial v_{z}}{\partial z}}\,\!}

The ratio between the extensional viscosity η e {\displaystyle \eta _{e}} and the [dynamic viscosity](/source/Viscosity) η {\displaystyle \eta } is known as [Trouton's Ratio](/source/Trouton's_ratio_(rheology)), T r = η e / η {\displaystyle \mathrm {Tr} =\eta _{e}/\eta } . For a [Newtonian Fluid](/source/Newtonian_Fluid), the Trouton ratio equals three.

## See also

- [Rheology](/source/Rheology)

## References

1. **[^](#cite_ref-1)** Barnes, H.A. "A handbook of elementary rheology", Institute of non-Newtonian Fluid Mechanics, 2000

1. **[^](#cite_ref-2)** Guyon, E., Hulin, JP. and Petit, L., Physical Hydrodynamics, Oxford University Press (2015), p113

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