A nonlinear inhomogeneous relaxation equation for the friction pressure is derived from the Boltzmann equation with the help of the moment method. Nonlinear terms emerge from both the flow term and the quadratic part of the Boltzmann collision operator. The importance of the latter contributions is characterized by coefficients which are expressed in terms of Chapman-Cowling collision integrals and can thus be evaluated for various collision models. Rheological consequences are analysed for stationary viscous flows in special geometries, viz. plane Couette flow, four-roller flow and uniaxial elongational flow. The resulting non-Newtonian viscosity and similar coefficients associated with normal pressure differences (viscometric functions) are discussed and displayed graphically as functions of the shear rate or the elongation rate.
