A theory of flowing granular materials with incompressible grains is presented which is similar to one proposed by Goodman and Cowin in 1972. Using a nonlinear version of the theory, boundary‐value problems corresponding to gravity flow in a closed vertical channel and in a channel with a free surface inclined at various angles from the vertical are considered in detail. In analyzing these problems, the dependence of the free energy and viscosity on the volume fraction are specialized to agree with those proposed as a result of recent experimental work. Solutions of these problems demonstrate many of the characteristics normally assumed in other treatments of granular materials. In particular, for vertical flow, when the volume fraction is greater than (less than) the critical volume fraction, a compressive (tensile) force must be exerted on the walls to maintain isochoric flow. Pluglike regions in which the velocity is almost constant develop, and there is a relatively thin shear region near the wall. For the flow with a free surface, as grain size or angle of inclination increases, the solutions tend to two rigid motions separated by a thin shear layer.