# Entropy Optimization in MHD Nanofluid Flow over an Exponential Stretching Sheet
## 1 Introduction  
Nanofluids are advanced heat transfer fluids with applications in a large variety of industrial processes. Over the last few years, the advancements in nanofluids have generated considerable research interest on account of their novel features that make them conceivably beneficial in a number of industrial processes, such as in glass blowing, cancer therapy, plastic and polymer extrusion, micro-forming, and air conditioning. The potential benefits and challenges of using nanofluids with controlled particle characteristics for various heat transfer applications have been investigated by many researchers. In particular, the ion-slip effects on MHD flow, which find applications in nuclear power reactors, power generation, and in several areas of astrophysics and geophysics, have been studied by [1, 2]. The MHD free-forced convective nanofluid fluid flow over a permeable medium was studied by [PERSON] et al. [3]. The Casson nanofluid flow with entropy generation was investigated by [PERSON] et al. [4]. Considering mixed convection on fluid flow is very significant for industrial applications, as free or forced convection may not be enough independently to disperse adequate thermal energy. [PERSON] et al. [5] examined the mixed convection in a water-based nanofluid in an MHD stagnation point flow. In physical flows, a magnetic field may have a significant impact on the flow of nanofluids. Fluid flow in electrical and magnetic fields may be used, for example, to control the cooling rate. Such an implementation has been analyzed by [PERSON] et al. [6]. A source of heat (or sink) in a flowing fluid is increasingly becoming important to researchers. These are important in manufacturing processes and other applications, which include nuclear waste disposal, storage of food, and exothermic processes in reactors. The effect of activation energy and an exponentially fluctuating temperature-dependent source in MHD heat transfer was investigated in [7, 8, 9, 10]. A review of previous research shows that, when estimating the performance of thermal systems, heat irreversibility is unavoidable and remains at the core of these processes. The second law of thermodynamics is used in quantifying irreversibility in optimal thermal system design. The generation of entropy is a criterion for the non-optimal operation of a thermal system. [PERSON] et al. [11] studied the generation of entropy in the flow of a convective nanofluid through a vertically expanding surface. Their research found that when the Brinkman number increased, so did the level of chaos in the system. [PERSON] et al. [12,13] presented the formulation for entropy generation in forced convection flow of radiative viscous nanofluids. [PERSON] et al. [14] calculated the entropy optimization for a magnetized flow of a nanofluid with heat radiation. [PERSON] et al. [15] studied the volumetric entropy production in a non-Newtonian fluid passing a stretched surface with linear thermal radiation. More current relevant research addressing the optimization of entropy in the direction of a stretched surface with varied flow parameters may be found in [PERSON] et al. [16] and the references therein. The aim of the current study was to investigate entropy generation and the effect of the Bejan number on 2-dimensional MHD viscoelastic incompressible nanofluid flow through an exponentially expanding sheet. The influences of dominant factors such as fluid buoyancy, viscous dissipation, heat absorption, and heat generation were studied by [PERSON] et al. [17]. The flow over an impermeable surface in the presence of thermal radiation and viscous dissipation was analyzed by [PERSON] et al. [18]. There are currently many demands for heating and cooling processes using fluids containing metallic nanoparticles. To control the rate of heat transfer in the vicinity of the expanded lamina, both entropy and Bejan number play a key role.