Polymer flows between rolls are treated as continua whose rheological behavior is described by a Maxwell model. Analytic expressions in quite simple and convenient form are obtained for the pressure gradient and the contact forces of friction. These are easily converted into relations for the specific pressure, thrust, and effective power for given starting parameters. The solutions obtained correctly reflect a number of qualitative characteristics of the flow of viscoelastic polymers. A comparison with the experimental data shows that the proposed method is suitable for use in engineering practice.