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This paper analyzes the flow of non-Newtonian fluids between infinitely long coaxial cylinders, when the inner cylinder rotates with given angular velocity @W, and a given axial flow rate, Q, is superimposed on this rotational motion. Such helical flow is of significance in the modelling of the action of a recirculating coaxial cylinder rheometer, where the axial flow is superimposed on the standard cup-and-bob rheometer to allow accurate rheological measurements involving fluids that are settling in nature. Such fluids are encountered in many mineral, food, and chemical processes of industrial significance. The analysis presented here applies the perturbation approach to analyze the above flows in two situations of physical interest: 1.the case of low axial flow rates; and 2.the case where the radial separation of the cylinders is small. In each case, the appropriate perturbation parameter is identified, and appropriate expressions for the velocity field are obtained for non-Newtonian fluids of interest. More significantly, this analysis allows the construction of approximate forms of the Reiner-Rivlin relation for this flow, which relates the angular velocity @W to the observed torque, M, at the inner cylinder, through the rheological parameters defining the fluid. Subsequent measurement of @W and M allows these parameters to be determined for a given fluid model. Where possible, the findings of the perturbation analysis are compared directly with experimental measurements involving a model of such a recirculating rheometer.