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This paper explores the mathematical model for couple stress fluid flow through an annular region. The above model is used for studying the blood flow be-tween the clogged (stenotic) artery and the catheter. The asymmetric nature of the stenosis is considered. The closed form expressions for the physiological parameters such as impedance and shear stress at the wall are obtained. The effects of various geomet-ric parameters and the parameters arising out of the fluid considered are discussed by considering the slip velocity and tapering angle. The study of the above model is very significant as it has direct applications in the treatment of cardiovascular diseases.