Physics:Second-order fluid

A second-order fluid is a fluid where the stress tensor is the sum of all tensors that can be formed from the velocity field with up to two derivatives, much as a Newtonian fluid is formed from derivatives up to first order. This model may be obtained from a retarded motion expansion truncated at the second-order. For an isotropic, incompressible second-order fluid, the total stress tensor is given by

[math]\displaystyle{ \sigma_{ij} = -p \delta_{ij} + \eta_0 A_{ij(1)} + \alpha_1 A_{ik(1)}A_{kj(1)} + \alpha_2 A_{ij(2)}, }[/math]

where

[math]\displaystyle{ -p \delta_{ij} }[/math] is the indeterminate spherical stress due to the constraint of incompressibility,

[math]\displaystyle{ A_{ij(n)} }[/math] is the [math]\displaystyle{ n }[/math]-th Rivlin–Ericksen tensor,

[math]\displaystyle{ \eta_0 }[/math] is the zero-shear viscosity,

[math]\displaystyle{ \alpha_1 }[/math] and [math]\displaystyle{ \alpha_2 }[/math] are constants related to the zero shear normal stress coefficients.

References

• Bird, RB., Armstrong, RC., Hassager, O., Dynamics of Polymeric Liquids: Second Edition, Volume 1: Fluid Mechanics. John Wiley and Sons 1987 ISBN:047180245X(v.1)
• Bird R.B, Stewart W.E, Light Foot E.N.: Transport phenomena, John Wiley and Sons, Inc. New York, U.S.A., 1960