For the steady flow of viscoelastic liquids a perturbation theory is developed for a weak secondary flow superimposed on an arbitrary primary flow. The dynamic and kinematic equations are used in a simplified form appropriate to slow flow, but the constitutive equations are not so approximated, being allowed to retain general nonlinear viscoelastic effects. The equations are used to study flow driven by a disk rotating atop a cylindrical tank. The governing equations are solved using Fourier‐Bessel series. Comparison with experimental data is given in part II.