We provide and analyse a model for the growth of bacterial biofilms based on
the concept of extra-cellular polymeric substance (EPS) as a polymer solution,
whose viscoelastic rheology is described by the classical Flory-Huggins theory.
We show that one-dimensional solutions exist, which take the form at large
times of travelling waves, and we characterise their form and speed in terms
of the describing parameters of the problem. Numerical solutions of the time-
dependent problem converge to the travelling wave solutions.