The entropy generation is related to thermo-dynamic irreversibility; entropy generation is common in every category of heat transfer processes. The entropy generation technique is introduced by Bejan [1
]. The quantity of irreversibility equal to that of entropy can be measured during a process. Entropy analysis is an influential means to find the efficiency of a system or process. Limited studies on the entropy generation of turbulent nanofluid flow have been analyzed in the microchannel, stretching sheet and channels; the entropy generation in the microchannel and channel is studied more closely although the studies described can apply to nanofluid, fluid turbulent, or laminar flow [3
]. Some researchers have worked on entropy generation by considering different types of geometry. Ko and Cheng [4
] numerically studied the entropy generation in a channel wavy wall. According to the results of Ko and Cheng, amplitude of the wavy wall is increasing function of the entropy generation. Mahian et al. [5
] presented the entropy generation of water-TiO2
and ethylene glycol-Al2
nanofluids between two circular cylinders. The obtained results displayed that the entropy generation has inverse relation with the volume fraction of nanofluid. Tshehla and Makinde [6
] presented the entropy generation of steady flow between two concentric pipes. They demonstrated that entropy generation is directly relational to Brinkman number. Govindaraju et al. [7
] analyzed the impact of slip magnetohydrodynamic nanofluid on entropy generation in a stretching sheet. The authors found that the entropy generation reduces in the flow system in the presence of slip parameter and volume fraction. Govindaraju et al. [1
] presented the entropy generation study of an incompressible magnetohydrodynamic of viscous nanofluid flow over a stretching sheet by considering various categories of nanoparticles in the water-based fluid. They concluded that entropy generation has a direct relation with Hartmann number, magnetic parameter and dimensionless group parameter. Berrehal and Maougal [8
] presented entropy generation analysis for nanofluid flow over a wedge with convective boundary condition and thermal radiation. The authors found that the entropy generation can be decreased with deducing convection through boundary and with increase radiation parameter. Malvandi et al. [9
] discussed analytical solution of entropy generation for nanofluids over a flat plate. They demonstrated that the entropy generation depends on Prandtl number, Reynold number, Eckert number and volume fraction. Acharya et al. [10
] presented the entropy generation and heat transfer in a regenerative cooling channel of a rocket engine. Govindaraju et al. [2
] studied analytically the entropy generation of nanofluid flow over a stretching sheet with a uniform heat source-sink and inclined magnetic field. The results exposed that entropy generation has a direct relation with Ec and ϕ, while entropy generation decreases with the increase in A.
Magnetohydrodynamics attracted the attention of researchers due to natural phenomena, i.e., geophysics, astrophysics to several applications in engineering such as electromagnetic casting, liquid metal, plasma confinement and so on [11
]. Nadeem et al. [12
] discussed two-dimensional boundary layer flow over a stretching sheet with the effect of magnetohydrodynamics. The results exposed that when increasing the value of the Prandtl number and magnetic parameters, an opposite behavior is seen in the Sherwood number and Nusselt number. Rudraiah et al. [13
] numerically examined the impact of magnetic field on natural convection. The authors noted that heat transfer rate decreases with the effect of the magnetic field. Abbas et al. [14
] numerically discussed the effects of magnetohydrodynamics over a stretching continuous sheet in a rotating fluid.
The concept of a nanofluid was introduced by Choi [15
] who examined notable results with several possibilities of usage. Nanofluids are the latest class of nanotechnology concerning nanoparticles dispersed in base fluid. The heat transfer fluid became a significant interest in research due to various applications. The heat transfer rate of a nanofluid is greater due to greater thermal conductivities as compared to the base fluids [16
]. There are several papers on water-Ag nanofluid flow by many researchers. Atashafrooz [17
] studied the impact of water-Ag nanofluid flow over an inclined step by using numerical technique. Upreti et al. [18
] examined the water-Ag nanofluid over a flat porous plate with influences of injection/suction, heat absorption/generation and viscous-ohmic dissipation. Suleman et al. [19
] discussed the water-Ag nanofluid flow in a stretching cylinder with the effects of homogenous-heterogeneous reaction and Newtonian heating.
The geometry effect of nanoparticles is very important to change the thermal conductivity of the nanofluid [20
]. Several researchers have worked on Ag-water nanofluid flow. To the best of our knowledge, there is no such research on the shape effect of nanoparticles on magnetohydrodynamic Ag-water nanofluid flow over a stretching sheet with entropy generation. The motivation of the present paper is to investigate the shape effect of Ag nanoparticles on magnetohydrodynamic Ag-water nanofluid flow and heat transfer with entropy generation. Finally, graphs of velocity, temperature, Nusselt number and entropy generation are plotted and all their aspects are discussed.
This study is organized as follows: in the first section, we construct the mathematical formulation for the proposed model. The numerical solution of proposed model is obtained by homotopy analysis method in Section 3
. The entropy generation is discussed in Section 4
. The computational results are reported in Section 5
. Finally, conclusions are described in Section 6