In circumstellar disks, the size of dust particles varies from submicron to several centimeters, while planetesimals have sizes of hundreds of kilometers. Therefore, various regimes for the aerodynamic drag between solid bodies and gas can be realized in these disks, depending on the grain sizes and velocities: Epstein, Stokes, and Newton, as well as transitional regimes between them. This means that simulations of the dynamics of gas–dust disks require the use of a drag coefficient that is applicable for a wide range for sizes and velocities for the bodies. Furthermore, the need to compute the dynamics of bodies of different sizes in the same way imposes high demands on the numerical method used to find the solution. For example, in the Epstein and Stokes regimes, the force of friction depends linearly on the relative velocity between the gas and bodies, while this dependence is non-linear in the transitional and Newton regimes. On the other hand, for small bodies moving in the Epstein regime, the time required to establish the constant relative velocity between the gas and bodies can be much less than the dynamical time scale for the problem—the time for the rotation of the disk about the central body. In addition, the dust may be concentrated in individual regions of the disk, making it necessary to take into account the transfer of momentum between the dust and gas. It is shown that, for a system of equations for gas and monodisperse dust, a semi-implicit first-order approximation scheme in time in which the interphase interaction is calculated implicitly, while other forces, such as the pressure gradient and gravity are calculated explicitly, is suitable for stiff problems with intense interphase interactions and for computations of the drag in non-linear regimes. The piecewise drag coefficient widely used in astrophysical simulations has a discontinuity at some values of the Mach and Knudsen numbers that are realized in a circumstellar disk. A continuous drag coefficient is presented, which corresponds to experimental dependences obtained for various drag regimes.
- SMOOTHED PARTICLE HYDRODYNAMICS
- HYPERBOLIC CONSERVATION-LAWS
- STIFF RELAXATION
- PROTOPLANETARY DISKS
- 2-FLUID DUST