where $G$, $M_B$ and $c$ are the gravitational constant, the mass of the black hole and the speed of light, respectively. In this unit system, distance, velocity and time are measured in units of $r_g=2GM_B/c^2$, $c$ and $t_g=2GM_B/c^3$ respectively, and the equations have been made dimensionless.

The Lagrangian formulation of the two-dimensional fluid dynamics equations for SPH [@m92] in cylindrical coordinate are given by [@lmc98] —\
The mass conservation equation, $$\frac{D \rho}{D t} = -\rho \nabla.{\bmath v},
\eqno(1)$$ where, $\frac{D}{Dt}$ denotes the co-moving time-derivative and $\rho$ is the density.

The radial momentum equation is given by, $$\frac{D v_r}{Dt} = -\frac{1}{\rho} \frac{\partial P}
{\partial r} + g_r + \frac{v^2_{\phi}}{r}.
\eqno(4a)$$

The vertical momentum equation is, $$\frac{Dv_z}{Dt} = -\frac{1}{\rho}\frac{\partial P}
{\partial z} + g_z.
\eqno(4b)$$

The azimuthal momentum equation is, $$\frac{D v_{\phi}}{Dt} = -\frac{v_{\phi} v_r}{r}
+\frac{1}{\rho} \left[ \frac{1}{r^2}\frac{\partial}
{\partial r} \left( r^2 \tau_{r\phi} \right) \right],
\eqno(4c)$$ where, $\tau_{r \phi}$ is the $r-\phi$ component of viscous stress tensor and is given by, $$\tau_{r\phi}=\eta r \frac{\partial \Omega}{\partial r},
\eqno(4d)$$ and the angular velocity is given by, $$\Omega = \frac{v_{\phi}}{r}.
\eqno(4e)$$

In Eqs. 4(a-b), $g_r$ and $g_z$ are the components of gravitational force [@pw80] and are given by, $$g_r = - \frac{1}{2(R-1)^2}\frac{r}{R}, \hskip 1.0cm {\rm 