  We study the electron temperature profiles for an inhomogeneous electron flow
in the hydrodynamic regime. We assume that the inhomogeneity is due to a weakly
non-uniform distribution of the momentum relaxation time within a spherically
constricted area. We show that the temperature profile dramatically depends on
the drive strength and the viscosity of the electron liquid. In the absence of
viscosity, a Landauer-dipole-like temperature distribution, asymmetrically
deformed along the current by the inelastic electron-phonon scattering, emerges
around the inhomogeneity. We find that both the Landauer-dipole temperature
profile and its asymmetry in the direction of the driving electric field exist
in all dimensionalities and are, therefore, universal features of inhomogeneous
hydrodynamic electron flow. We further demonstrate that the electron viscosity
suppresses the thermal Landauer dipole and leads to the appearance of a "hot
spot" exactly at the center of the constriction. We also calculate the phonon
temperature distribution, which can be directly measured in experiments on
thermal nanoimaging.
