  Numerical methods for solving the Navier-Stokes equations for classical (or
normal) viscous fluids are well established. This is also the case for the
Gross-Pitaevskii equation, governing quantum inviscid flows (or superfluids) in
the zero temperature limit. In quantum flows, like liquid helium II at
intermediate temperatures between zero and 2.17 K, a normal fluid and a
superfluid coexist with independent velocity fields. The most advanced existing
models for such systems use the Navier-Stokes equations for the normal fluid
and a simplified description of the superfluid, based on the dynamics of
quantized vortex filaments, with ad hoc reconnection rules. There was a single
attempt (C. Coste, The European Physical Journal B - Condensed Matter and
Complex Systems, 1998) to couple Navier-Stokes and Gross-Pitaevskii equations
in a global model intended to describe the compressible two-fluid liquid helium
II. We present in this contribution a new numerical model to couple a
Navier-Stokes incompressible fluid with a Gross-Pitaevskii superfluid. Coupling
terms in the global system of equations involve new definitions of the
following concepts: the regularized superfluid vorticity and velocity fields,
the friction force exerted by quantized vortices to the normal fluid, the
covariant gradient operator in the Gross-Pitaevskii based on a slip velocity
respecting the dynamics of vortex lines in the normal fluid. A numerical
algorithm based on pseudo-spectral Fourier methods is presented for solving the
coupled system of equations.Finally, we numerically test and validate the new
numerical system against well-known benchmarks for the evolution in a normal
fluid of different types or arrangements of quantized vortices (vortex crystal,
vortex dipole and vortex rings). The new coupling model opens new possibilities
to revisit and enrich existing numerical results for complex quantum fluids.
