A mathematical model is developed to describe ice accretion and water flow on a cold substrate of arbitrary shape. It is shown how the model may be applied to practical substrate shapes, such as flat surface, cylinder and aerofoil. A numerical scheme to solve the governing equations is then described. Results are presented for an aerofoil under conditions appropriate to in-flight icing and for a cylinder in conditions for atmospheric icing. (C) 2004 Elsevier Ltd. All rights reserved.