  We explore the stability of the interface between two phase-separated Bose
gases in relative motion on a lattice. Gross-Pitaevskii-Bogoliubov theory and
the Gutzwiller ansatz are employed to study the short- and long-time stability
properties. The underlying lattice introduces effects of discreteness, broken
spatial symmetry, and strong correlations, all three of which are seen to have
considerable qualitative effects on the Kelvin-Helmholtz instability.
Discreteness is found to stabilize low flow velocities, because of the finite
energy associated with displacing the interface. Broken spatial symmetry
introduces a dependence not only on the relative flow velocity, but on the
absolute velocities. Strong correlations close to a Mott transition will stop
the Kelvin-Helmholtz instability from affecting the bulk density and creating
turbulence; instead, the instability will excite vortices with Mott-insulator
filled cores.
