  Differentially rotating flows of unmagnetized, highly conducting plasmas have
been created in the Plasma Couette Experiment. Previously, hot-cathodes have
been used to control plasma rotation by a stirring technique [C. Collins et
al., Phys. Rev. Lett. 108, 115001(2012)] on the outer cylindrical
boundary---these plasmas were nearly rigid rotors, modified only by the
presence of a neutral particle drag. Experiments have now been extended to
include stirring from an inner boundary, allowing for generalized circular
Couette flow and opening a path for both hydrodynamic and magnetohydrodynamic
experiments, as well as fundamental studies of plasma viscosity. Plasma is
confined in a cylindrical, axisymmetric, multicusp magnetic field, with $T_e<
10$ eV, $T_i<1$ eV, and $n_e<10^{11}$ cm$^{-3}$. Azimuthal flows (up to 12
km/s, $M=V/c_s\sim 0.7$) are driven by edge ${\bf J \times B}$ torques in
helium, neon, argon, and xenon plasmas, and the experiment has already achieved
$Rm\sim 65$ and $Pm\sim 0.2 - 12$. We present measurements of a
self-consistent, rotation-induced, species-dependent radial electric field,
which acts together with pressure gradient to provide the centripetal
acceleration for the ions. The maximum flow speeds scale with the Alfv\'{e}n
critical ionization velocity, which occurs in partially ionized plasma. A
hydrodynamic stability analysis in the context of the experimental geometry and
achievable parameters is also explored.
