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tokens: n s is the number density of the s th species of the fluid. The mass density is given by Chattopadhyay ( 2008 ) and Chattopadhyay Ryu ( 2009 ), where , is the composition parameter and . The electron and proton masses are and , respectively. For single-temperature flow, the isotropic pressure is given by The EoS for multi-species flow is (Chattopadhyay 2008 ; Chattopadhyay Ryu 2009 ) The non-dimensional temperature is defined with respect to the electron rest mass energy, . Using equations ( 19 ) and ( 20 ), the expression of the energy density in equation ( 21 ) simplifies to where The expressions of the polytropic index, the adiabatic index and the sound speed are given as Integration of first law of thermodynamics (equation 13 ) by assuming adiabatic flow ( Q + = 0) and using the EoS (equation 22 ) gives us the adiabatic relation of multi-species relativistic flow (Chattopadhyay Kumar 2013 ; Kumar et al. 2013 ), where k 1 = 3(2 − ξ)/4, k 2 = 3ξ/4 and and is the constant of entropy. Equation ( 24 ) is the generalized version of . Combining equations ( 24 ) and ( 14 ), we get the expression of entropy accretion rate, Re-arranging equations ( 10 )–( 14 ) with the help of equations ( 9 ), ( 7 ), ( 19 ), ( 20 ) and ( 22 ) in geometric units, we present the spatial derivative of flow variables v , l and Θ, where Here, contains an extra term l 2 v /( r 2 γ 2 ) compared to the inviscid case (Chattopadhyay Chakrabarti 2011 ). There is γ φ term in the expression of disc height (equation 12 ). The radial derivative of equation ( 14 ) implies that the radial derivative of the specific angular momentum will be non-zero, which causes the extra term to appear. There are many height prescriptions (Lasota 1994 ; Riffert Herold 1995 ; Peitz Appl 1997 ), and choice of any one of them apart from the one used will not affect the result qualitatively. Then, Moreover, These differential equations are integrated by using fourth-order Runge–Kutta numerical method with the help of using critical point conditions and l′Hospital rule at critical point. 2.2.1 Sonic point equations Mathematical form of critical point equation is , which gives two equations as and Here, the subscript ‘c’ denotes the same physical quantities described in equations ( 26 )–( 28 ), but evaluated at the location of the critical point. The velocity gradient on the sonic point, i.e. (d v /d r ) c , is obtained by employing l′Hospital rule. 2.2.2 Relativistic shocks for viscous flow The relativistic shock conditions were first obtained by Taub ( 1948 ), which for viscous flow in the presence of mass-loss are where . We have solved four shock conditions ( 31 )–( 34 ) simultaneously, where viscous shear tensor () is continuous across the shock and we obtained the relation between pre-shock (suffix ‘−’) and post-shock (suffix ‘+’) flow variables, where . Here, , . From equation ( 11 ), viscous shear tensor can be written as . 2.3 Outflow equations The jet being tenuous, we idealize it to be