In this example we will consider the stability of a viscoelastic filament subject to exponential stretching. This is a model for a filament stretching rheometer. In order to determine the true extensional properties of a viscoelastic fluid the deformation in the device must approach ideal uniaxial elongation. Consider a cylindrical viscoelastic sample placed between two coaxial plates. We will assume that the bottom plate is fixed and that the top plate is pulled at an exponential rate. Such a device is being used by a number of research groups [8,9,10]. Of interest is to determine whether the deformation in the central portion of such a device approaches ideal uniaxial elongation.