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Context. Evolution of massive stars is affected by a significant loss of masseither via (nearly) spherically symmetric stellar winds or by asphericalmass-loss mechanisms, namely the outflowing equatorial disks. However, thescenario that leads to the formation of a disk or rings of gas and dust aroundmassive stars is still under debate. Aims. We study the hydrodynamic andthermal structure of optically thick, dense parts of outflowing circumstellardisks that may be formed around various types of critically rotating massivestars, 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 fromstellar poles. Within the optically thick dense region we calculate thevertical disk thermal structure using the diffusion approximation while for theoptically thin outer layers we assume a local thermodynamic equilibrium withthe impinging stellar irradiation. We use two of our own types of hydrodynamiccodes: two-dimensional operator-split numerical code and unsplit code based onthe Roe's method. Results. Our models show the geometric distribution andcontribution of viscous heating that begins to dominate in the central part ofthe disk. In the models of dense viscous disks the viscosity increases thecentral temperature up to several tens of thousands of Kelvins. The highmass-loss rates and high viscosity lead to instabilities with significant wavesor bumps in density and temperature in the very inner disk region. Conclusions.The two-dimensional radial-vertical models of dense outflowing disks includingthe full Navier-Stokes viscosity terms show very high temperatures that arehowever 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 lowenough for the existence of neutral hydrogen.

Two-dimensional modeling of density and thermal structure of dense circumstellar outflowing disks