  Context. Evolution of massive stars is affected by a significant loss of mass
either via (nearly) spherically symmetric stellar winds or by aspherical
mass-loss mechanisms, namely the outflowing equatorial disks. However, the
scenario that leads to the formation of a disk or rings of gas and dust around
massive stars is still under debate. Aims. We study the hydrodynamic and
thermal structure of optically thick, dense parts of outflowing circumstellar
disks that may be formed around various types of critically rotating massive
stars, for example, Be stars, B[e] supergiant (sgB[e]) stars or Pop III stars.
Methods. We specify the optical depth of the disk along the line-of-sight from
stellar poles. Within the optically thick dense region we calculate the
vertical disk thermal structure using the diffusion approximation while for the
optically thin outer layers we assume a local thermodynamic equilibrium with
the impinging stellar irradiation. We use two of our own types of hydrodynamic
codes: two-dimensional operator-split numerical code and unsplit code based on
the Roe's method. Results. Our models show the geometric distribution and
contribution of viscous heating that begins to dominate in the central part of
the disk. In the models of dense viscous disks the viscosity increases the
central temperature up to several tens of thousands of Kelvins. The high
mass-loss rates and high viscosity lead to instabilities with significant waves
or bumps in density and temperature in the very inner disk region. Conclusions.
The two-dimensional radial-vertical models of dense outflowing disks including
the full Navier-Stokes viscosity terms show very high temperatures that are
however limited to only the central disk cores inside the optically thick area,
while near the edge of the optically thick region the temperature may be low
enough for the existence of neutral hydrogen.
