Viscoelastic flows with a type of small controlled deviation from viscometricity are investigated. The calculations provide a basis for exploring the use of simple constitutive equations in some steady flows. The two‐dimensional flow of a low‐order Rivlin‐Ericksen fluid through a channel of slowly varying width is worked out. Inversion formulas enabling the deduction of material functions from experimental observations are also given, which may provide a new means of measuring one normal stress difference.