tokens: open circles joined with dotted line are for shock free accretion solutions. See the text for details. Until now, we have studied the various properties of accretion shock waves around black holes. These shocks are non-dissipative in nature as the specific energy remains conserved across the shock front (Chakrabarti 1989 ). Indeed, shocks of this kind are radiatively inefficient as well. Apparently, the realistic shock waves are likely to be dissipative, where a part of accreting energy is escaped from the shock location through the disc surfaces. This essentially causes the reduction of the overall specific energy profile in the PSC region (Singh Chakrabarti 2011 ). The plausible mechanism that perhaps regulates the energy dissipation at the shock is the thermal Comptonization process (Chakrabarti Titarchuk 1995 ) that eventually reduces the thermal energy in the PSC. Based on the above consideration, we assume that the loss of energy across the shock front is proportional to the temperature difference between the immediate pre-shock and post-shock flow and therefore, the energy loss across the shock front is calculated as (Das, Chakrabarti Mondal 2010 ) where a and a denote the immediate pre-shock and post-shock sound speed and f represents the fraction of the available thermal energy lost at shock. In this analysis, we treat f as a free parameter and chose its value as f 0.998 for representation. Evidently, this is the measure of energy dissipation across the shock (Das etal. 2010 ; Singh Chakrabarti 2011 ; Sarkar Das 2013 ; Kumar Chattopadhyay 2013 ). We then compute the maximum energy dissipation () by freely varying all the flow parameters and plot it as function of in for various values of accretion rates in Fig. 12 (a). Here, solid, dotted and dashed curves represent the results corresponding to , 0.075 and 0.1, respectively. We observe that for a given accretion rate (), initially shows gradual increment with in , which eventually approaches towards a saturation value in the gas pressure dominated domain. Meanwhile, it was shown by numerical simulation (Machida etal. 2006 ) that in the magnetically dominated state, accretion disc becomes cool as compared to a gas pressure dominated disc. This clearly indicates that the gas pressure dominated flow is expected to possess higher thermal energy content and accordingly, we obtain enhanced when in is larger. Moreover, as is increased, cooling becomes more effective in the flow resulting the lowering of thermal energy content. Therefore, for a given in , remains smaller for flows having higher . When in is large, the effect of synchrotron cooling due to the increase of accretion rate becomes practically insignificant. Hence, when in 10 4 , asymptotes to a saturation value irrespective to the flow accretion rate. Figure 12. Variation of maximum energy dissipation at shock () as a function of at inner sonic point ( in ) for three different values of accretion rates, , 0.075 and 0.1 denoted by
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