Steady flow in the metering section of relatively narrow channel extruders can be regarded as a special case of curvilineal viscometric flow (as defined by Coleman, Markovitz, and Noll). It is shown that under these conditions the behavior of simple fluids can be adequately described from knowledge of a single shear stress function. To specify this function an extended form of the power‐law relationship is used, as it agrees well with viscometric measurements on polymer melts over a range of shear rates. The equations of motion and energy are solved taking into account the effect of transverse flow in the channel and of a superposed steady temperature profile. Results of the numerical computations are presented in terms of dimensionless parameters, and comparison is made with experimental results. The marked effect on non‐Newtonian fluid behavior on extruder output at low back pressures and on the shape of the screw characteristics is emphasized. The importance of temperature effects is measured by the magnitude of the (dimensionless) Brinkman number. Output versus pressure characteristics are calculated for a range of Brinkman numbers, and it is shown that the predicted temperature profiles in the channel approximate experimentally observed profiles. The results provide a rational basis for extruder design and for predicting the effect of geometrical parameters, operating conditions, and melt rheology on extruder performance.