  The superfluid/normal-fluid interface of liquid 4He is investigated in
gravity on earth where a small heat current Q flows vertically upward or
downward. We present a local space- and time-dependent renormalization-group
(RG) calculation based on model F which describes the dynamic critical effects
for temperatures T near the superfluid transition T_lambda. The model-F
equations are rewritten in a dimensionless renormalized form and solved
numerically as partial differential equations. Perturbative corrections are
included for the spatially inhomogeneous system within a self-consistent
one-loop approximation. The RG flow parameter is determined locally as a
function of space and time by a constraint equation which is solved by a Newton
iteration. As a result we obtain the temperature profile of the interface.
Furthermore we calculate the average order parameter <psi>, the correlation
length xi, the specific heat C_Q and the thermal resistivity rho_T where we
observe a rounding of the critical singularity by the gravity and the heat
current. We compare the thermal resistivity with an experiment and find good
qualitative agreement. Moreover we discuss our previous approach for larger
heat currents and the self-organized critical state and show that our theory
agrees with recent experiments in this latter regime.
