Johnson--Segalman--Oldroyd model

The Johnson--Segalman--Oldroyd model is a linear relaxation-retardation equation made nonlinear through a Gordon--Schowalter objective derivative. Its slip parameter can produce a rate-dependent steady shear viscosity.

For a tensor $\mathbf{\sigma }$, the Gordon--Schowalter derivative used here iswhich interpolates among objective convected derivatives as the dimensionless slip parameter $a$ changes.

The retardation time controls the delayed strain-rate term in an Oldroyd constitutive equation. Together with the relaxation time it determines the solvent-like high-frequency or high-rate response.

In small-amplitude oscillatory shear,

For Gordon--Schowalter parameter $0<a<2$, the steady shear viscosity isIt is shear thinning when ${\lambda }_1>{\lambda }_2$.