The impression test of a Newtonian fluid under a constant load and for a small Reynolds number flow was simulated by using the finite element method. It is found that the penetration velocity is a constant if the surface tension/viscosity ratio is less than 0.1 cm/s, the product of surface tension and indenter radius is less than 0.05 of the applied load, and the penetration depth is less than1/3of the indenter radius. Such constant penetration velocity is proportional to the applied load and inversely proportional to the viscosity and the indenter radius. The stick or slip boundary condition at the interface between the indenter and the fluid has no effect on the penetration velocity under these conditions. ©1997 American Institute of Physics.