ner_tags: O O O O O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O B-Formula I-Formula I-Formula I-Formula O O O B-Formula I-Formula I-Formula I-Formula O O O B-Formula I-Formula I-Formula I-Formula O O O B-Formula I-Formula I-Formula I-Formula O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O
tokens: did for the Schwarzschild case (Chattopadhyay Kumar 2016 ), but now for Kerr metric. And then using these values, we look for transonic jet solution. We iterate c φ and n c for the same values of Z φ and r b and by estimating the value of from the disc. This goes on until the entropy of the transonic jet is in between the PSD and pre-shock disc or . Once the transonic solution is found, then from Section 2.4 , we compute the , which is fed to equation ( 35 ) to recalculate the r sh . Once the new r sh is obtained, steps (iii)–(v) are repeated but with the new value of , till r sh converges to a value. The resulting jet is the self-consistent jet solution driven by disc properties. While the jet solution is being obtained, shock transition in jets is also studied. 4 RESULTS We are connecting two flows, namely accretion and jet, and since both the flows are quite complicated, we would first present all possible advective accretion solutions, then all possible jet solutions and then self-consistent accretion-jet solutions. 4.1 Accretion In this section, we present only accretion solution, i.e. we follow steps (i)–(iii) of Section 3 . BH accretion disc is necessarily transonic, i.e. the solutions falling on to a BH pass through at least one sonic point. Therefore, a lot of insight can be gained by studying the critical point conditions. For a given value of and λ, the critical points can be uniquely determined. Let us denote and . Therefore, using the critical point conditions (equations 19 and 20 ) in equation ( 16 ), we obtain energy as a function of r c for given values of λ, a s and ξ (equation 21 ). We plot with r c for a s = −0.99 (Fig. 1 a), a s = −0.5 (Fig. 1 b), a s = 0.0 (Fig. 1 c), a s = 0.5 (Fig. 1 d), a s = 0.8 (Fig. 1 e) and a s = 0.99 (Fig. 1 f). Each of the curves plotted from top to bottom is for λ = 1.5–4.0 with interval dλ = 0.5 and with same composition parameter, ξ = 1.0. Critical points are formed because gravity increases the infall velocity, but because accretion is an example of convergent flow, the temperature of the flow also increases during accretion (compressional heating). As temperature increases, sound speed also increases, but in accretion the rise of sound speed is less than the bulk velocity. Therefore, the flow that is subsonic at large distances becomes supersonic within a certain point (critical point). However, if the infalling matter is rotating, then the centrifugal term competes with the gravitation term in a way that MCP may form. As a s increases, frame dragging enhances the centrifugal term significantly, therefore, even matter with low angular momentum exhibits MCP, for example the flow with λ = 2 [dotted in Figs 1 (a)–(f)] exhibits MCP for a s = 0.99 but not for a s ≤ 0.8. On the other hand, if the flow is retrograde, then MCP occur only for higher angular momentum flow. Fig. 1 (c) is similar to fig. 3a of Chattopadhyay Chakrabarti ( 2011 ). This phenomenon is better exhibited with ‘kite-tail’ plot.