  Transient heat transfer in superfluid $^4$He (He II) is a complex process
that involves the interplay of the unique counterflow heat-transfer mode, the
emission of second-sound waves, and the creation of quantized vortices. Many
past researches focused on homogeneous heat transfer of He II in a uniform
channel driven by a planar heater. In this paper, we report our systematic
study of He II transient heat transfer in nonhomogeneous geometries that are
pertinent to emergent applications. By solving the He II two-fluid equation of
motion coupled with the Vinen's equation for vortex-density evolution, we
examine and compare the characteristics of transient heat transfer from planar,
cylindrical, and spherical heaters in He II. Our results show that as the
heater turns on, an outgoing second-sound pulse emerges, in which the vortex
density grows rapidly. These vortices attenuate the second sound and result in
a heated He II layer in front of the heater, i.e., the thermal layer. In the
planar case where the vortices are created throughout the space, the
second-sound pulse is continuously attenuated, leading to a strong thermal
layer that diffusively spreads following the heat pulse. On the contrary, in
the cylindrical and the spherical heater cases, vortices are created mainly in
a thin thermal layer near the heater surface. As the heat pulse ends, a
rarefaction tail develops following the second-sound pulse, in which the
temperature drops. This rarefaction tail can promptly suppress the thermal
layer and take away all the thermal energy deposited in it. The effects of the
heater size, heat flux, pulse duration, and temperature on the thermal-layer
dynamics are discussed. We also show how the peak heat flux for the onset of
boiling in He II can be studied in our model.
