  Patterned irradiation of cuprate superconductors with columnar defects allows
a new generation of experiments which can probe the properties of vortex
liquids by forcing them to flow in confined geometries. Such experiments can be
used to distinguish experimentally between continuous disorder-driven glass
transitions of vortex matter, such as the vortex glass or the Bose glass
transition, and nonequilibrium polymer-like glass transitions driven by
interaction and entanglement. For continuous glass transitions, an analysis of
such experiments that combines an inhomogeneous scaling theory with the
hydrodynamic description of viscous flow of vortex liquids can be used to infer
the critical behavior. After generalizing vortex hydrodynamics to incorporate
currents and field gradients both longitudinal and transverse to the applied
field, the critical exponents for all six vortex liquid viscosities are
obtained. In particular, the shear viscosity is predicted to diverge as
$|T-T_{BG}|^{-\nu z}$ at the Bose glass transition, with $\nu\simeq 1$ and
$z\simeq 4.6$ the dynamical critical exponent. The scaling behavior of the ac
resistivity is also derived. As concrete examples of flux flow in confined
geometries, flow in a channel and in the Corbino disk geometry are discussed in
detail. Finally, the implications of scaling for the hydrodynamic description
of transport in the dc flux transformer geometry are discussed.
