scholarly journals Time dependent periodic Navier-Stokes flows on a two-dimensional torus

1996 ◽  
Vol 179 (3) ◽  
pp. 577-597 ◽  
Author(s):  
Zhi-Min Chen ◽  
W. G. Price
Keyword(s):  
Time Dependent ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Stokes Flows

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

2018 ◽  
Vol 613 ◽  
pp. A75 ◽  
Author(s):  
P. Kurfürst ◽  
A. Feldmeier ◽  
J. Krtička
Keyword(s):  
Mass Loss ◽  
Optical Depth ◽  
Massive Stars ◽  
Thermal Structure ◽  
Time Dependent ◽  
Navier Stokes ◽  
Viscous Heating ◽  
Two Dimensional ◽  
Dense Region ◽  
Loss Rates

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. It is also unclear how various forming physical mechanisms of the circumstellar environment affect its shape and density, as well as its kinematic and thermal structure. 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. We calculate self-consistent time-dependent models of temperature and density structure in the disk’s inner dense region that is strongly affected by irradiation from a rotationally oblate central star and by viscous heating. Methods. Using the method of short characteristics, we specify the optical depth of the disk along the line-of-sight from stellar poles. Within the optically thick dense region with an optical depth of τ > 2∕3 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. For time-dependent hydrodynamic modeling, we use two of our own types of hydrodynamic codes: two-dimensional operator-split numerical code based on an explicit Eulerian finite volume scheme on a staggered grid, and unsplit code based on the Roe’s method, both including full second-order Navier-Stokes shear viscosity. Results. Our models show the geometric distribution and contribution of viscous heating that begins to dominate in the central part of the disk for mass-loss rates higher than Ṁ ≳ 10−10 M⊙ yr−1. In the models of dense viscous disks with Ṁ > 10−8 M⊙ yr−1, the viscosity increases the central temperature up to several tens of thousands of Kelvins, however the temperature rapidly drops with radius and with distance from the disk midplane. 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, for example.

Stability of Navier–Stokes Equations

2006 ◽  
Author(s):  
Jean-Yves Chemin ◽  
Benoit Desjardins ◽  
Isabelle Gallagher ◽  
Emmanuel Grenier
Keyword(s):  
Stokes System ◽  
Stokes Equations ◽  
Time Dependent ◽  
Navier Stokes ◽  
Well Posedness ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
The Stability ◽  
Three Space

In this chapter we intend to investigate the stability of the Leray solutions constructed in the previous chapter. It is useful to start by analyzing the linearized version of the Navier–Stokes equations, so the first section of the chapter is devoted to the proof of the well-posedness of the time-dependent Stokes system. The study will be applied in Section 3.2 to the two-dimensional Navier–Stokes equations, and the more delicate case of three space dimensions will be dealt with in Sections 3.3–3.5.

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