**Triple Local Similarity Solutions of Darcy-Forchheimer Magnetohydrodynamic (MHD) Flow of Micropolar Nanofluid Over an Exponential Shrinking Surface: Stability Analysis**

**Liaquat Ali Lund 1,2, Dennis Ling Chuan Ching 3, Zurni Omar 1, Ilyas Khan 4,\* and Kottakkaran Sooppy Nisar <sup>5</sup>**


Received: 26 April 2019; Accepted: 28 May 2019; Published: 20 August 2019

**Abstract:** In this paper, the MHD flow of a micropolar nanofluid on an exponential sheet in an Extended-Darcy-Forchheimer porous medium have been considered. Buongiorno's model is considered in order to formulate a mathematical model with different boundary conditions. The governing partial differential equations (PDEs) of the nanofluid flow are changed into a third order non-linear quasi-ordinary differential equation (ODE), using the pseudo-similarity variable. The resultant ODEs of the boundary value problems (BVPs) are renewed into initial value problems (IVPs) using a shooting method, and then the IVPs are solved by a fourth order Runge-Kutta (RK) method. The effects of various physical parameters on the profiles of velocity, temperature, microrotation velocity, concentration, skin friction, couple stress coefficients, heat, and concentration transfer are demonstrated graphically. The results reveal that triple solutions appear when *S* ≥ 2.0337 for *K* = 0.1 and *S* ≥ 2.7148 for *K* = 0.2. A stability analysis has been performed to show the stability of the solutions; only the first solution is stable and physically possible, whereas the remaining two solutions are not stable.

**Keywords:** pseudo-similarity variable; micropolar nanofluid; darcy forchheimer model; MHD flow; triple solution; stability analysis
