## 2. Statement of the problemThe considerations given in the previous section allow us to
perform the analysis on the basis of the numerical solution of the
Euler gas dynamic equations. Suppose the flow pattern to be
axisymmetric with the gravitating object placed at the origin of the
coordinate system and with the rotation axis aligned with the
where is the internal energy per unit mass and is the polytropic index. The system of governing equations in the Cartesian coordinate system shown in Fig. 1 reads: The superscript T means transposing. In the above system is the total energy per unit volume. We normalized the quantities of density, velocity, and pressure by , , , where is the reference length defined by the formula approximating the solid portion of the contracted magnetosphere of the star (Elsner & Lamb 1976) In the first quadrant one has .
The angle is counted off the
We assume the inflow at to be
supersonic. Thus, we must specify all quantities on the outer
boundary. Taking into account this fact and inspecting the structure
of the system (7)-(11), we arrive at the following set of
dimensionless parameters of the problem:
, ,
, and where is the -component of the velocity vector. Neglecting for the moment the velocity components in the angular directions, we can calculate the entropy function and the total enthalpy as We now choose the dimensionless to be 100 and introduce the following distribution of the angular velocity along the outer boundary: The first interval here corresponds to the constant angular
momentum and the second to the constant angular velocity distribution,
respectively. The To modify the quantities of and at , we assume and and calculate the new values from the formulas Because of the symmetry of the problem, it suffices to seek solution only in the first quadrant. The conditions on the inner boundary are the following: nonpenetration is adopted for , and absorbing boundary conditions are used for . The essence of these absorbing boundary conditions is described in Pogorelov & Semenov (1997). We use the quantity extrapolation at the supersonic exit and apply the relations in the rarefaction fan to accelerate the outflow velocity to the sonic value if . These conditions give results nearly identical to those obtained with the "vacuum" boundary conditions by Hunt (1971) if the latter are used in the framework of the Godunov-type schemes (Toro 1997). The initial conditions are believed to be not very important if we seek a steady-state solution. On the other hand, they are of crucial importance if we study transient phenomena. © European Southern Observatory (ESO) 2000 Online publication: January 29, 2001 |