  We study theoretically nonlinear dynamics induced by shear-flow instability
in segregated two-component Bose-Einstein condensates in terms of the Weber
number, defined by extending the past theory on the Kelvin-Helmholtz
instability in classical fluids. Numerical simulations of the Gross-Pitaevskii
equations demonstrate that dynamics of pattern formation is well characterized
by the Weber number $We$, clarifying the microscopic aspects unique to the
quantum fluid system. For $We \lesssim 1$, the Kelvin-Helmholtz instability
induces flutter-finger patterns of the interface and quantized vortices are
generated at the tip of the fingers. The associated nonlinear dynamics exhibits
a universal behavior with respect to $We$. When $We \gtrsim 1$ in which the
interface thickness is larger than the wavelength of the interface mode, the
nonlinear dynamics is effectively initiated by the counter-superflow
instability. In a strongly segregated regime and a large relative velocity, the
instability causes transient zipper pattern formation instead of generating
vortices due to the lack of enough circulation to form a quantized vortex per a
finger. While, in a weakly segregating regime and a small relative velocity,
the instability leads to sealskin pattern in the overlapping region, in which
the frictional relaxation of the superflow cannot be explained only by the
homogeneous counter-superflow instability. We discuss the details of the linear
and nonlinear characteristics of this dynamical crossover from small to large
Weber numbers, where microscopic properties of the interface become important
for the large Weber number.
