  The Hydrodynamics of Superfluid Turbulence (HST) describes the flows (or
counterflows) of HeII in the presence of a chaotic set of vortex filaments. The
HST equations govern both a slow variation of the hydrodynamic variables due to
dissipation related to the vortex tangle and fast processes of the first and
second sound propagation. This circumstance prevents effective numerical
simulations of the problems of unsteady heat transfer in HeII. By virtue of a
pertinent multi-scale perturbation analysis we show how one can eliminate the
fast processes to derive the evolution equation for the slow processes only. We
then demonstrate that the long-term evolution of a transient heat load of
moderate intensity obeys the nonlinear heat conductivity equation. The second
example of the methods developed is investigation of unsteady processes in the
second sound resonator. The latter is frequently used for study of
nonstationary behavior of vortex tangle, just by monitoring of the quality
factor behavior. This procedure however is wrong when characteristic times of
processes are comparable (or smaller) than the time constant of resonator. We
show how to extract the correct information on the vortex line density (VLD)
dynamics with use of procedure we developed.
  PACS numbers: 47.32.Cc, 47.37.+q, 67.40.Vs., 05.10.Gg.
