**1. Introduction**

Generally, the investigation of the non-Newtonian fluid flow in two-dimensional problems is a hard task when multiple solutions are attempted to find because equations are related to high nonlinear terms. In spite of these challenges, the researchers are making efforts to tackle these problems for multiple solutions due to their wide range of applications in various science and industrial fields. The category of non-Newtonian fluids which pacts with the suspended micro-rotational particles, is known as the micropolar fluid. Eringen introduced the theory of micropolar [1,2]. He explained the impacts of couple stresses and local rotational inertia that cannot be described by the standard equations of Navier–Stokes. The micropolar equations are mathematically described for the theory of porous media and the theory of lubrication in the books by Lukaszewicz [3] and Eringen [4]. There are various applications of the micropolar fluids, for example, liquid crystals, particle suspension, animal blood, lubrication, turbulent shear flows, and paints. Lok et al., [5] considered the stagnation point flow of micropolar nanofluid and then succeeded to find the dual solution. It is also stated that only the

stable solution is the upper branch by performing an analysis of stability. Sheremet et al., [6] examined micropolar fluid with a convectional effect in a wavy triangular cavity. Bhattacharyya et al., [7] studied the micropolar fluid with thermal radiation effect and discovered that there exist dual solutions, while they did not perform the stability analysis. The dual solution has been discovered in the opposing flow case for the micropolar fluid [8], whereas stability analysis has not been performed in their examination of the multiple solutions. Ramzan et al., [9] investigated the nanofluid of micropolar non-Newtonian kind of fluid on the stretching sheet and claimed that velocity and angular velocity have an inverse relation with slip parameter. Turkyilmazoglu [10] analytically examined the Magnetohydrodynamic (MHD) flow of micropolar fluid. Shah et al., [11] examined micropolar nanofluid with the effect of the Casson parameter in the channel and stated that the thermal boundary layer becomes thicker for the higher values of the Brownian motion parameter. Some papers have recently shown some development on the micropolar fluid [12–16].

Nowadays, multiple solutions of fluid flow have received extensive consideration. Generally, many researchers stated in their studies that non-uniqueness of solutions, depending on the non-linearity in the fluid flow equations. Jawad et al. [17] stated that, due to the nonlinearity in fluid flow problems, there is more than one solution; however, seeking all possible solutions challenges researchers. Further, several of the approaches are ineffective because solutions are close together. Rohni [18] reported in her Ph.D. thesis that "the multiplicity of solutions in fluid dynamics and heat transfer is important to be able to be computed, since solutions arising from bifurcations often interact with one another producing otherwise inexplicable phenomena". Moreover, she claimed that, in order to notice all possible solutions of any fluid, the flow problem is still a challenge. Mishra and DebRoy [19] described that multiple solutions have many important applications when these are related to heat transfer, because the final qualities and structure of many products of material processing in the industries can be improved by the concept of multiple solutions. Lund et al., [20] considered nanofluid, with the base fluid as sodium alginate and solid particles of copper and silver, over the shrinking sheet and obtained dual solutions and reported that the existence of multiple solutions depends upon how the researchers set the ranges of different applied physical parameters. Furthermore, they specified that the dual solution depended on the range of the suction parameter. Khashi'ie et al. [21] highlighted the importance of the multiple solutions and the stability analysis during the examination of the mixed convection flow of the micropolar fluid. They stated "if the problem has non-unique solutions but the researchers manage to find one solution only, there is a probability that the solution is the lower branch solution (unstable/not real). This will lead to the misinterpretation of the flow and heat transfer characteristics". These papers include some significant studies on multiple solutions [22–24]. Here, it is worth highlighting that, when multiple solutions appeared in any fluid flow problem, the study of stability analysis should be considered. The first step in performing the stability analysis is, according to Rana et al. [25], to change the governing equation to unsteady form by adding a new time dependent variable for similarity. Lund et al., [26] found the dual solution of the Williamson fluid and claimed that there existed an infinite number of the values of the smallest eigenvalue by performing the stability of solutions. Waini et al., [27] reported that the stability of any solution dependent upon the sign of the value of the smallest eigenvalue. If the value of eigenvalue is positive, it indicated the stable solution, while the negative value of the smallest eigenvalue indicated the unstable solution [28].

The pointwise prime objectives of this study are, as follows:


The authors tried their best to find all possible solutions due to the importance of multiple solutions and lack of available literature in which triple solutions of any fluid flow problem were noticed. This paper is divided into six sections; Section 1 is for the brief introduction of micropolar fluid and multiple solutions. Mathematical formulation, derivation of stability analysis, and methodology are kept in Section 2, Section 3, and Section 4 respectively. Section 5 is for the result and discussion and Section 6 is for the conclusion.
