  We propose a novel device capable of measuring the relaxation time of
viscoelastic fluids as small as 1\,ms. In contrast to most rheometers, which by
their very nature are concerned with producing viscometric or
nearly-viscometric flows, here we make use of an elastic instability which
occurs in the flow of viscoelastic fluids with curved streamlines. To calibrate
the rheometer we combine simple scaling arguments with relaxation times
obtained from first normal-stress difference data measured in a classical shear
rheometer. As an additional check we also compare these relaxation times to
those obtained from Zimm theory and good agreement is observed. Once
calibrated, we show how the serpentine rheometer can be used to access smaller
polymer concentrations and lower solvent viscosities where classical
measurements become difficult or impossible to use due to inertial and/or
resolution limitations. In the absence of calibration the serpentine channel
can still be a very useful comparative or index device.
