This paper proposes a new approach to quantifying the rheological effects that come into play in traction fluids under different regimes of operation in elastohydrodynamic (EHD) contacts. We take account of shear thinning and viscoelasticity and of the joint temperature and pressure dependence of these effects and show how the resulting model conforms with measured traction coefficients for a typical traction fluid. Understanding the bulk rheology of traction fluids in detail is seen as an aid to designing more effective fluids and predicting performance in real engineering systems. Moreover, the form of the constitutive model used is consistent with molecular dynamics simulation studies, as described in the companion paper. Thus, a fairly complete quantitative picture of traction fluid behaviour from the molecular level through to performance in a macroscopic contact is emerging.