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The radiative heat transfer performance with viscous dissipation on entropy generation in the MHD flow of variable thermal conductivity viscous Cu–water nanofluid through a porous parallel channel is investigated in this paper. The governing non-linear differential equations are solved using power series for small values of thermal conductivity variation parameter, which are then analysed by Hermite- Padé approximation method. The effects of the physical governing flow parameters on velocity, temperature and entropy generation are discussed extensively both numerically and graphically. A stability analysis has been performed for the local rate of heat transfer which signifies that the lower solution branch is stable and physically acceptable. The entropy generation of the system increases at the two porous plates and also the fluid friction irreversibility is dominant there.