Blood stenosis is considered one of the most serious risks which face humanity nowadays. In addition, it is also one of the most apparent symptoms of COVID (19) (Corona Virus). Consequently, this research is shedding light on studying the blood flow in case of having blood clots and artery elasticity in the presence of stenosis during studying the flow. Hematopoiesis requires a model of the yield stress fluid, and among the available yield stress fluid models for blood flow, the Herschel-Bulkley model is preferred (because Bingham, Power-law and Newtonian models are its special cases). Navier stokes equation is used to simulate this subject in a mathematical way. The elasticity on the stenosis arterial walls is simulated by Rubinow & Keller model [24] and Mazumdar model [25]. The results reveal exciting behaviors that, in turn, require adequate study of non-Newtonian fluid flow phenomena, especially the results showed that the increase in the parameters related to the elasticity of the walls facilitating the flow of blood through the stenosis area. In addition, a comparison between two elasticity models (Rubinow & Keller model and Mazumdar model) is considered. Further, for normal artery without stenosis, our results are the same as those obtained by Vajravelu et.al [22].