A Newtonian fluid with small variations in the viscosity in the primary flow direction of steady and oscillatory Couette flow is studied. These variations in viscosity create a coupling of the components of the momentum equations between the flow-direction component and the gradient-direction component. The coupling leads to secondary flows even in planar Couette flow where a rectilinear flow may be expected for a purely viscous fluid under creeping flow conditions. A perturbation solution has been applied for small-amplitude oscillations in the viscosity in both steady and oscillatory Couette flow. Because many theological measurements are made assuming rectilinear flow, these results may have important consequences and may allow error caused by heterogeneity to be estimated. Finally, the relation between the momentum and the assumption of a symmetric stress tensor is discussed by introducing an alternative constitutive equation that is linear in the velocity gradient tensor and objective, but gives an asymmetric stress tensor. By adjusting the degree of asymmetry for the stress tensor, the secondary flows can be altered or eliminated.