**1. Introduction**

Micropolar fluid is a polar fluid which contains rigid randomly oriented or spherical particles. It can be defined as a fluid with micro structures and belongs to the nonsymmetric stress tensor [1]. Furthermore, this fluid model is employed to analyze the behavior of liquid crystals and exotic polymeric fluid or lubricant colloidal suspensions. Ariman et al. [2,3], Eringen [4–6], and Lukaszewicz [7] discussed the properties and applications of the micropolar fluid in details. The concept of the electrically conducting fluids motion in the presence of a magnetic field is called magnetohydrodynamics, or MHD for short. The word MHD is the combination of the words magneto, hydro, and dynamics, which mean magnetic, fluid and motion, respectively. MHD is also known as magnetofluid dynamics and hydromagnetic, which can be defined as the study of the dynamics of the electromagnetic field and the electrically conducting fluids. Recently, Kumar et al. [8] examined the MHD flow of micropolar fluid with a porous medium. Micropolar fluid with an MHD effect on a shrinking sheet along a weak

concentration has been considered by Gupta et al. [9]. Turkyilmazoglu [10] found the exact solution of micropolar fluid within the existence of the MHD effect. The MHD flow of micropolar fluids with a porous medium had been considered by many researchers, such as Sheikh et al. [11], Akhter et al. [12], Siddiq et al. [13], Dero et al. [14], Hayat et al. [15,16], Ahmed et al. [17], and Waqas et al. [18].

In the last couple of years, the use of nanofluid as a convectional fluid, in order to increase the heat transfer rate, has pulled in extensive consideration among researchers. Research demonstrated that dissolving different sorts of nanoparticles, such as nonmetal, polymeric and metal mixed in the base fluids, provides good thermal properties [19,20]. The term nanofluid, which was introduced by Choi and Eastman in 1995 [21], can be defined as a fluid that is a mixture of regular (base) fluids with nano-meter sized particles (less than 100 nm). These particles may contain oxides, carbon nanotubes, and metals. On the other hand, oil, ethylene glycol, and water are generally considered to be the base fluids. These fluids have different physical and chemical properties from regular fluids [22]. There are two approaches to study nanofluids, namely the experimental and numerical one. Many researchers considered the numerical approach to understand the behavior of nanofluids and introduced new concepts to understand them. Khanafer et al. [23] built up a model to contemplate the heat transfer improvement of Cu-water nanofluid in a two-dimensional enclosure. Meanwhile, Buongiorno [24] constructed a new non-homogeneous model in which velocity of base fluids and nanoparticles are not equal to zero. This model consists of seven slip parameters, which are Brownian diffusion, diffusiophoresis, gravity settling, fluid drainage, inertia, thermophoresis, and the Magnus effect. The references of the development of nanofluids can be found in the book by Nield and Bejan [25] and also in the published review articles on nanofluid, such as Mahian et al. [26–28] and Wong and Leon [29]. Recently, a few researchers have considered nanoparticles with non-Newtonian base fluid in the presence of MHD effects, such as Mahdy [30], Rehman et al. [31], Hamid et al. [32], Eid et al. [33], and Prabhakar et al. [34].

It can be observed from previously published literature that not much work has been done on the Extended-Darcy-Forchheimer porous medium, due to the fact that the governing equations cannot be reduced to self-similarity solutions through the use of a similarity transformation, particularly when using exponential similarity variables. Similarly, the MHD flow of micropolar nanofluid over an exponential shrinking surface has also not been considered because the equation of the angular velocity cannot be transformed into a self-similarity solution. Keeping in view these drawbacks, we attempt to employ a new approach which is a pseudo-similarity variable in the governing equations of fluid flow in order to obtain a local similar solution, as adopted by a few researchers in their studies [35–38]. The key objective of the present work is to consider the MHD flow of micropolar nanofluid over an exponential shrinking surface in an Extended-Darcy-Forchheimer porous medium. The resultant equations, after performing the pseudo-similarity variable in the form of a third-order non-linear quasi-ordinary differential equation, have been solved using the shooting method with the RK-method; we found triple solutions. When multiple solutions exist in any problem, it is necessary to conduct a stability analysis in order to determine the stable solutions. Consequently, this analysis is also taken into account in this research.
