Although the flow behavior of polymer melts and solutions can generally be adequately described by differential and integral constitutive equations, it is often difficult to obtain solutions to the equations of motion using such constitutive models. Consequently, simpler theological models are sometimes used to gain some physical insight and at least some qualitative information about the nature of non-Newtonian flows. The generalized Newtonian fluid can be used to study nonlinear viscous effects, and the second-order fluid can be used to study elastic effects in steady flows. Here, we illustrate how a first-order fluid model can be used to study elastic effects in unsteady flows. Solutions for velocity-driven and pressure-driven flows illustrate that the first-order fluid model does describe some of the effects caused by the elasticity of the fluid.