Time evolution of a thick disc with finite conductivity around a nonrotating compact object is presented. Along with the Maxwell equations and the Ohm's law, the Newtonian limit of the relativistic fluid equations governing the motion of a finitely conducting plasma is derived. The magnetofluid is considered to possess only the poloidal components of the electromagnetic field. Moreover, the shear viscous stress is neglected, as well as the self-gravity of the disc. In order to solve the equations, we have used a self-similar solution. The main features of this solution are as follows. The azimuthal velocity is somewhat increased from the Keplerian value in the equator plane to the super-Keplerian values at the surface of disc. Moreover, the radial velocity is obtained proportional to the meridional velocity. Magnetofluid does not have any nonzero component of the current density. Subsequently, the electromagnetic force is vanished and does not play any role in the force balance. While the pressure gradient maintains the disc structure in latitudinal direction, magnetofluid has no accretion on the central compact object. Analogously to the parameter α in the standard model, our calculations contain one parameter η0 which specifies the size of the electrical resistivity.

In this paper, we investigate the time evolution of an accreting magneto-fluid with finite conductivity. For the case of a thin disk, the fluid equations along with Maxwell's equations are derived in a simplified, one-dimensional model that neglects the latitudinal dependence of the flow. The finite electrical conductivity of the plasma is taken into account by Ohm's 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, S0 and ρ, which represent the size of the 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 S0 and ρ parameters affect the radial thickness of the disk. Also, radial velocity and gas pressure are more sensitive to the electrical conductivity in the inner regions of disk. Moreover, the parameter ρ has a more significant effect on the physical quantities for small radii.

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In this paper, we explore the self-similarity time evolution of a hot accretion flow around a compact object in the presence of a toroidal magnetic field. We focus on a simplified model which is axisymmetric, rotating, unsteady viscous-resistive under an advection-dominated stage. In this work, we suppose that both the kinematic viscosity and the magnetic diffusivity to be a result of turbulence in the accretion flow. To describe such a flow, we apply magneto-hydrodynamics equations in spherical coordinates, [Formula: see text] and adopt unsteady self-similar solutions. By neglecting the latitudinal dependence of the flow, we obtain a set of one-dimensional differential equations governing the accretion system. In this research, we encounter two parameters related to the magnetic field; one of them is, [Formula: see text], defined as the ratio of the magnetic pressure to the gas pressure and the other one, [Formula: see text] applied in the magnetic diffusivity definition. Our results show that [Formula: see text] is a function of position, and increases towards outer layers. On the other hand, we examine different strength of magnetic field by choosing different value of [Formula: see text] which is the value of [Formula: see text] at the inner edge of disc. We see that both [Formula: see text] and [Formula: see text] have positive effect on growing the radial infall velocity but density and gas pressure decrease by larger values of these parameters. Moreover, the rotational velocity and temperature of accreting material reduce considerably under the influence of a stronger magnetic field. We also focus on the behavior of the mass accretion rate appearing as a descending function of position. Finally, our solutions confirm that the radial trend of the physical quantities in a dynamical accretion flow is different from the ones in a steady flow. However, the effect of various parameters on the physical quantities in our model is qualitatively consistent with similar steady models.

The effect of toroidal magnetic fields on a supercritical accretion disc (slim disc) around a rotating object is examined. In this research, it is supposed that angular momentum transport is due to viscous turbulence and the α-prescription is used for the kinematic coefficient of viscosity. Moreover, the general relativistic effects are neglected. The degree of advection which demonstrates the fraction of energy that accretes by matter on-to the central object is considered by f parameter. For the steady-state structure of such accretion flows, a set of self similar solution is presented. Our solutions will include two important non-dimensional parameters β and a. β is the ratio of the magnetic pressure to the gas pressure, the so-called friction of magnetic pressure, which shows the magnetic field strength. The ratio of the angular velocities of the central body and the accretion flow is indicated by the rotating parameter a. The possible combined effects of magnetic field, spin of central object and degree of advection is investigated. We also show the effect of rotating parameter a on the physical quantities of disc is different for co and counter rotating flows. Moreover, by increasing the degree of advection and strength of magnetic field, the behavior of the radial and angular velocities becomes reversed with respect to a. The model implies that the surface temperature, thickness and luminosity of disc strongly depends on rotation parameter and strength of magnetic field.