Using Laplace transformation technique, semi-analytical solutions are obtained for three basic viscoelastic fluid flow problems under the effect of the Jeffreys model. These semi-analytical solutions are not available in the literature. The present work investigates the effect of two types of driving forces on the flow behavior. These two types are the velocity-type and shear-type driving forces. The effect of the relaxation and retardation times on the flow behavior for these two types of driving forces may be viewed well using the obtained semi-analytical solutions. The three fundamental problems are transient Couette flow, transient wind-driven flow over finite domains and the transient Poiseuille flow in parallel-plates channels. It is shown that as the dimensionless relaxation time (lambda(1)) increases, the flow response to the imposed driving force becomes slower. This implies that the flow needs more time to feel the presence of the driving force and hence needs more time to attain steady-state behavior. On the other hand, the effect of the dimensionless retardation time (lambda(2)) depends on the type of the driving force imposed on the system. For a velocity-type driving force, the flow response becomes faster as the dimensionless retardation time (lambda(2)) increases and for a shear-type driving force the flow response becomes slower as the dimensionless retardation time (lambda(2)) increases. (c) 2005 Elsevier Ltd. All rights reserved.