Flow instability or transition from laminar to turbulent for the various fluids is a subject of research for the fluid dynamics. Various approaches have been devised to look into the various aspects of the transition from laminar to turbulent flow. The present investigation deals with the application of Energy gradient method to study the effect of the eccentricity and presence of the secondary solid phases in the flow instability for the fully developed annular flow. For the annulus of different radius ratio, the different eccentricity ratio has been considered for the Newtonian fluid. Also for the concentric case, the effect of the solid secondary phase has been studied with the possibility of the solid deposition at the bottom of the annulus. These cases have practical relevance with the annular flow in the well bore in case of horizontal drilling. The derivations have been carried out for all these cases for flow instability analysis. (C) 2016 Elsevier B.V. All rights reserved.