Shocks in relativistic viscous accretion flows around Kerr black holes

We study the relativistic viscous accretion flows around the Kerr black holes. We present the governing equations that describe the steady state flow motion in full general relativity and solve them in 1.5D to obtain the complete set of global transonic solutions in terms of the flow parameters, namely specific energy (${\cal E}$), specific angular momentum (${\cal L}$) and viscosity ($\alpha$). We obtain a new type of accretion solution which was not reported earlier. Further, we show for the first time to the best of our knowledge that viscous accretion solutions may contain shock waves particularly when flow simultaneously passes through both inner critical point ($r_{\rm in}$) and outer critical point ($r_{\rm out}$) before entering into the Kerr black holes. We examine the shock properties, namely shock location ($r_s$) and compression ratio ($R$, the measure of density compression across the shock front) and show that shock can form for a large region of parameter space in ${\cal L}-{\cal E}$ plane. We study the effect of viscous dissipation on the shock parameter space and find that parameter space shrinks as $\alpha$ is increased. We also calculate the critical viscosity parameter ($\alpha^{\rm cri}$) beyond which standing shock solutions disappear and examine the correlation between the black hole spin ($a_k$) and $\alpha^{\rm cri}$. Finally, the relevance of our work is conferred where, using $r_s$ and $R$, we empirically estimate the oscillation frequency of the shock front ($\nu_{QPO}$) when it exhibits Quasi-periodic (QP) variations. The obtained results indicate that the present formalism seems to be potentially viable to account for the QPO frequency in the range starting from milli-Hz to kilo-Hz as $0.386~{\rm Hz}\le \nu_{QPO} \left(\frac{10M_\odot}{M_{BH}} \right) \le 1312$ Hz for $a_k=0.99$, where $M_{BH}$ stands for the black hole mass.