Start-up and transient shear-stress flows are modelled here using the Unified Model for linear viscoelasticity of polymers. The generated constitutive models rewrite Boltzmann superposition principle and use a time-rate separability basis for functions instead of the normally used time–strain separability. To describe stress and viscosity at different induced shear rates, we use a caretCaret” to indicate the effective old viscosity \(\hat{\eta }\) at t = t0 developed at a shear rate \(\left( {\hat{\eta }} \right)\) or \(\left( {\hat{\eta }} \right) = {\upeta }\left( {t_{0} ,{ }\dot{\gamma }_{0} } \right)\) , which reaches \(\mathop {\lim }\limits_{t \to \infty } \left[ {\eta \left( {t,\dot{\gamma }} \right)} \right] = \eta \left( {\dot{\gamma }} \right)\) in the steady-state situation. Depending on the shear history, effective viscosity \(\hat{\eta }\) at time t = t0 may vary considerably from its steady-state limitSteady-state limit” \(\eta \left( {\dot{\gamma }_{0} } \right)\) . In other words, low-quality measurements might yield true effective viscosity \(\hat{\eta }\) , which can differ from true steady-state viscosity η.