  Recent observations show cool, oscillating prominence threads fading when
observed in cool spectral lines and appearing in warm spectral lines. A
proposed mechanism to explain this evolution is that the threads were heated by
turbulence driven by the Kelvin--Helmholtz instability that developed as a
result of wave-driven shear flows on the surface of the thread. As the
Kelvin--Helmholtz instability is an instability that works to mix the fluids,
in the solar corona it can be expected to work by mixing the cool prominence
material with that of the hot corona to form a warm boundary layer. In this
paper we develop a simple phenomenological model of nonlinear Kelvin--Helmholtz
mixing, using it to determine the characteristic density and temperature of the
mixing layer, which for the case under study with constant pressure across the
two fluids are $\rho_{\rm mixed}=\sqrt{\rho_1\rho_2}$ and $T_{\rm
mixed}=\sqrt{T_1T_2}$. One result from the model is that it provides an
accurate, as determined by comparison with simulation results, determination of
the kinetic energy in the mean velocity field. A consequence of this is that
the magnitude of turbulence, and with it the energy that can be dissipated on
fast time-scales, as driven by this instability can be determined. For the
prominence-corona system, the mean temperature rise possible from turbulent
heating is estimated to be less than 1\% of the characteristic temperature
(which is found to be $10^5$\,K). These results highlight that mixing, and not
heating, are likely to be the cause of the observed transition between cool to
warm material in Okamoto et. al (2015). One consequence of this result is that
the mixing creates a region with higher radiative loss rates on average than
either of the original fluids, meaning that this instability could contribute a
net loss of thermal energy from the corona, i.e. coronal cooling.
