We consider a number of expansion flows for different non-Newtonian liquids. In the first example, the liquid is an aqueous polymer solution and the expansion flows of interest may be considered to be either two-dimensional or three-dimensional. The large vortex activity which is created by fluid inertia is damped by fluid elasticity. In the second group of experiments, a fibre suspension in a Newtonian matrix is investigated in both an axisymmetric contraction how and an axisymmetric expansion flow. The vortex enhancement in the contraction confirms earlier work by other workers. The strong vortex activity in the expansion (at relatively low Reynolds numbers) is unexpected and provocative. The second part of this paper contains an attempt to simulate the observed contraction and expansion hows using a versatile finite-element technique which is able to encompass viscoelastic liquids and fibre suspensions. It is also able to handle three-dimensional hows. The agreement between experiment and numerical simulation in all cases is very satisfactory.