  We demonstrate a close analogy between a viscoelastic medium and an
electrically conducting fluid containing a magnetic field. Specifically, the
dynamics of the Oldroyd-B fluid in the limit of large Deborah number
corresponds to that of a magnetohydrodynamic (MHD) fluid in the limit of large
magnetic Reynolds number. As a definite example of this analogy, we compare the
stability properties of differentially rotating viscoelastic and MHD flows. We
show that there is an instability of the Oldroyd-B fluid that is physically
distinct from both the inertial and elastic instabilities described previously
in the literature, but is directly equivalent to the magnetorotational
instability in MHD. It occurs even when the specific angular momentum increases
outwards, provided that the angular velocity decreases outwards; it derives
from the kinetic energy of the shear flow and does not depend on the curvature
of the streamlines. However, we argue that the elastic instability of
viscoelastic Couette flow has no direct equivalent in MHD.
