and}
\hskip 1.0cm
g_z = - \frac{1}{2(R-1)^2}\frac{z}{R},
\eqno(4f)$$ where $R = \sqrt{r^2 + z^2} $.

The form of dynamic viscosity parameter is given by [@ss73], $$\eta = \nu \rho = \alpha \rho a h$$ where, $\nu$ is the kinematic viscosity, $\alpha$ is the viscosity parameter, $a~(=\sqrt{{\gamma P}/{\rho}})$ is the sound speed, $\rho$ is the mass density, $h~[= \sqrt{\frac{2}{\gamma}}ar^{1/2}(r-1)]$ is the disc half height estimated using hydrostatic equilibrium [@cd04] and $v=\sqrt{v^2_r+v^2_z}$.

The conservation of energy is given by, $$\frac D{Dt}\left( \epsilon +\frac 12{\bf v}^2\right) =-\frac P\rho \nabla
{\bf v+v\cdot }\left( \frac{D{\bf v}}{Dt}\right) +\frac 1\rho \nabla \left(
\bf \vec{ {\bf \tau } } {\bf :v} \right),
\eqno(4g)$$ with $$\left( \frac{D{\bf v}}{Dt}\right) =-\frac 1\rho \nabla P+{\bf g},$$

where, $P=\left( \gamma -1\right) \rho \epsilon$ is the equation of state of ideal gas, g is the gravitational acceleration and $\epsilon$ is the internal energy, respectively. ${\bf \vec{{\bf \tau }}:v}$ is the vector resulting from the contraction of the stress tensor with the velocity vector. We include only $\tau _{r\phi }$ (namely, the $r-\phi$ component) since it is the dominant contributor to the viscous stress. A complete steady solution requires the equations of energy, angular momentum and mass conservation supplied by transonic conditions at the critical 