The 2D disk structure with advective transonic inflow-outflow solutions around black holes

We solved analytically viscous two-dimensional (2D) fluid equations for accretion and outflows in spherical polar coordinates ($r, \theta, \phi$) and obtained explicitly flow variables in $r-$ and $\theta -$directions around black holes (BHs). We investigated global transonic advection-dominated accretion flow (ADAF) solutions in $r-$direction on an equatorial plane with using Paczy\'nski-Wiita potential. We used radial flow variables of ADAFs with symmetric conditions on the equatorial plane, as initial values for integration in $\theta-$direction. In the study of 2D disk structure, we used two-azimuthal components of viscous stress tensors namely, $\tau_{\rm r\phi}$ and $\tau_{\rm\theta\phi}$. Interestingly, we found that the whole advective disk is not participating in outflow generation and the outflows form close to the BHs. Normally, outflow strength increased with increasing viscosity parameter ($\alpha_1$), mass-loss parameter ($s$) and decreasing gas pressure ratio ($\beta$). Outflow region increased with increasing $s$, $\alpha_1$ for $\tau_{\rm r\phi}$ and decreasing $\alpha_2$ for $\tau_{\rm\theta\phi}$. The $\tau_{\rm\theta\phi}$ is effective in angular momentum transportation at high latitude and outflows collimation along an axis of symmetry since it changes polar velocity ($v_{\rm\theta}$) of the flow. The outflow emission is also affected by the ADAF size and decreased with decreasing it. Transonic surfaces formed for both inflows ($v_{\rm r}<0$, very close to BH) and outflows ($v_{\rm r}>0$). We also explored no outflows, outflows and failed outflows regions, which mainly depend on the viscosity parameters.