Self-similar Evolutionary Solutions for Accreting Magneto-fluid around a Compact Object with Finite Electrical Conductivity

In this paper, we investigate the time evolution an accreting magneto-fluid with finite conductivity. For the case of a thin disk, the fluid equations along with Maxwell equations are derived in a simplified, one-dimensional model that neglects the latitudinal dependence of the flow. The finite electrical conductivity is taken into account for the plasma through Ohm law; however, the shear viscous stress is neglected, as well as the self-gravity of the disk. In order to solve the integrated equations that govern the dynamical behaviour of the magneto-fluid, we have used a self-similar solution. We introduce two dimensionless variables, $S_0$ and $ε_ρ$, which show the magnitude of electrical conductivity and the density behaviour with time, respectively. The effect of each of these on the structure of the disk is studied. While the pressure is obtained simply by solving an ordinary differential equation, the density, the magnetic field, the radial velocity and the rotational velocity are presented analytically. The solutions show that the $S_0$ and $ε_ρ$ parameters affect the radial thickness of the disk. Also, the radial velocity and gas pressure are more sensitive to electrical conductivity in the inner regions of disk. Moreover, the $ε_ρ$ parameter has a more significant effect on physical quantities in small radii.

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