Unsteady plane and axisymmetric stagnation flow of an incompressible viscous fluid on the body that moves along the oncoming flow with a time-dependent velocity is studied in this work. Similarity solutions for the full Navier-Stokes equations are obtained and the results of the flow velocity, shear stress and stream lines are reported for both plane two dimensional flow and axisymmetric flow. The results show that the features of the similar boundary flow highly depends on a characteristic parameter beta. There exists a critical value beta(c) below which no similarity solution to the flow is found. When beta(c) < beta < 0, two solution branches exist and different flow patterns appear for each branch. Flow with monophonically growing velocity, reversed flow and flow with S-shaped velocity are obtained for various values of beta. The boundary layer thickness of the plane and axisymmetric flows is tabulated, the streamlines of the flow are demonstrated, and the shear stress over the boundary layer is also discussed. (C) 2011 Elsevier Ltd. All rights reserved.