This article considers transient flows for planar contractions and Oldroyd-B fluids, with increasing flow-rate boundary conditions. We employ a novel hybrid finite volume/element time-dependent algorithm. This scheme is shown to be second-order accurate. The hybrid scheme consists of a Taylor-Galerkin finite element discretisation, and a cell-vertex fluctuation-distribution finite volume approach. These two approaches are coupled at each time-step to solve the parabolic/hyperbolic system of partial differential equations. The finite element section is applied to the mass and momentum conservation equations, whilst the hyperbolic constitutive equation is treated via finite volume discretisation. The application of this time-accurate scheme to complex flows reveals some novel features, in contrast to time-independent (constant flow-rate) driving boundary conditions. In particular, we highlight dynamic flow structure evolution on the field and in stress. (C) 2004 Elsevier B.V. All rights reserved.