Dilute aqueous solutions of certain polymers have transport properties which differ considerably from those of pure water. It is shown how engineering estimates of regions where polymer solution elasticity will be important can be made by comparing the shear wave velocity and the relevant kinematical velocity. The transition from laminar shearing flow in plane Poiseuille and rotational Couette geometries is related to solution rheological properties using linear and non-linear stability theory, and verified experimentally for the Couette flow. Torque reduction in the Coutette geometry is explained using a non-linear analysis. Rheological properties in dilute solutions can be estimated from the onset and torque of the secondary flow in the rotational Couette geometry. (Author)