## 1 Introduction  
Investigations of the momentum and heat transfer specifications in an obstructed duct with multiple obstacles has many thermal applications including compact heat exchangers, flow around arrays of nuclear fuel rods, cooling of electronic devices, oil or gas flows in reservoirs, chimney stacks, power generators, etc. [1]. The heat transfer improvement in all mentioned applications is an essential need. Nanofluids are recognized as high heat transfer performance fluids, which can be used in many cooling systems including cooling of electronic components, oil coolers, inter coolers, and coolant in microchannel heat sink [2,3]. The particulate fouling as a destructive phenomenon in nanofluids should be taken into consideration as it can affect the favorable improved thermal properties of these fluids. Accurate understanding of the particle motion is essential to supress the destructive effects of this phenomenon. In past years, researchers used different models for simulating nanofluid flow for different problems. These models are single-phase approach, Eulerian model, volume of fluid model, mixture model, and Eulerian-Lagrangian approach (Discrete phase model). [PERSON] et al. [PERSON] et al. (2017) reviewed these models. They concluded that the Eulerian-Lagrangian approach is more precise and reliable for simulating the nanofluid because it takes into account the interactions between two phases by considering the Brownian, drag, gravity, and thermophoresis forces between them. Moreover, this model has ability to predict particle distribution and calculate concentration of nanoparticles in domain. [PERSON] et al. [PERSON] et al. (2016) performed a comparison between single-phase and Lagrangian trajectory approaches for simulating the nanofluid flow in a tube. They found that the Lagrangian trajectory approach predicts a higher heat transfer coefficient in comparison with the single-phase one as the Lagrangian trajectory approach considers the dynamic of particles and the interactions between liquid and particulate phases. [PERSON] et al. [PERSON] et al. (2017) used the Eulerian-Lagrangian approach to investigate the nanofluid flow in a microchannel. They reported the same findings about the greater prediction of heat transfer coefficient by this model in comparison to the single-phase one. In another study, [PERSON] et al. [PERSON] et al. (2017) studied the nanofluid turbulent flow in helically-coiled tubes. They used the Eulerian-Lagrangian approach. Besides the numerical study, they performed an experimental work on this problem to benchmark the accuracy of the numerical model. They observed that the Eulerian-Lagrangian model presents the results with higher accuracy in comparison to the single-phase one. Some researchers studied the flow and heat transfer in an obstructed channel. [PERSON] et al. [PERSON] et al. (2017) studied the convective heat transfer across an obstructed duct with a built-in heated square obstacle. Their results indicated that the Strouhal number increases with an enhance in the Richardson number. [PERSON] et al. [PERSON] et al. (2017) simulated the heat exchange in a duct with wall-mounted square obstacles. They reported that the flow accelerates near faces with a decrease in the distance between obstacles. This leads to enhance in the heat transfer rate from obstacles. [PERSON] and [PERSON] [PERSON] and [PERSON] (2017) investigated the influences of nanofluid on heat transfer improvement in an obstructed duct. They applied the single-phase approach for modelling the nanofluid for this problem. They observed about 60% enhancement for heat transfer in the duct by using the nanoparticles and the blocks. Readers are referred to the most significant studies on nanoparticles in [PERSON] et al.