International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Index Copernicus Value (2013): 6.14 | Impact Factor (2013): 4.438 Volume 4 Issue 7, July 2015 www.ijsr.net Licensed Under Creative Commons Attribution CC BY A Mathematical Study of Pressure Drop and Shear Stress of Non Newtonian Blood Flow in an Artery with Bell Shaped Stenosis Lovely Jain 1 , S. P. Singh 2 Amity Institute of Applied Sciences, Amity University, Noida, Uttar Pradesh, India Dayalbagh Educational Institute, Dayalbagh University, Agra, India Abstract: In this paper, a mathematical model is proposed and analysed to study the effects of a bell shaped stenosis on pressure drop and shear stress, in an artery. Thebio rheological aspect is taken into account in the power law fluid representing the non-Newtonian fluid flow model. An analysis has been performed to estimate the effect of power law index and severity of the stenosis on various parameters such as pressure drop and wall shear stress. The analysis shows that as height and length of stenosis increases, not only pressure drop increases but shear stress also increases.The results are compared with the usual Newtonian fluid model. Keyword: Power law fluid model, shear stress, pressure drop, stenosisheight. 1. Introduction Stenosis in arteries of humans is a common occurrence and hemodynamic factors play a significant role in the formation and proliferation of this disease. It is a chronic disease in which thickening, hardening, and loss of elasticity of the arterial walls occurs, leading to impaired blood circulation. The thickening and hardening of the arteries are due to the build-up of calcium deposits in the lumen of the artery causing stenosis. It develops with aging, hypertension, diabetes, hyperlipidemia, and other diseased conditions in the artery. There have been a number of studies using mathematical model for the flow of blood in a stenosed artery,(Young, D. F., 1968; Shukla et. al., 1979; Shukla et. al., 1980;Chaturani and Sany, 1985; Pralhad and Schultz, 1988;Mishra and Chakravarty, 1986;Haldar, 1987;Moshkelani et.al., 2003)etc. An analysis on the effect of an axially symmetric, time dependent growth into the lumen of a tube of constant cross section through which a Newtonian fluid is steadily flowing is presented by Young, 1968. A theoretical solution of the unsteady-state momentum equation for the startup flow of a power law fluid in circular tubes is presented byPralhad and Schultz, 1988. Shukla et al.,1979, studied the effects of peripheral layer viscosity on physiological characteristics of blood flow through the artery with mild stenosis. It has been shown that the resistance to flow and the wall shear decrease as the peripheral layer viscosity decreases. A two-layered fluid model for blood flow through a stenosed tube has been developed by Pralhad and Schultz, 1988.The model consists of a core (suspension of RBC’s) and peripheral plasma layer. Resistance to flow and shear stress have been computed for different stenosis height. Effect of Stenosis on non- newtonian flow of the blood in an artery was studied by Shukla et. al., 1980. It has been shown that the resistance to flow and wall shear stress increases with the size of stenosis. Blood flow through a stenosed artery has been investigated by Chaturani and Sany, 1985. They represented blood flow by a non-Newtonian Herschel –Bulkley equation. It is observed that the wall shear stress and the flow resistance increase in Herschel Bulkley fluid in comparison with corresponding Newtonian fluid. The non-Newtonian fluid flow in a stenosed coronary bypass is investigated numerically by Chenand Wang, 2006using the carreau Yasuda model for shear thinning behavior of the blood. Results for the non-Newtonian flow, Newtonian flow and the rescaled Newtonian flow are presented.The effects of pulsatility, stenosis and non-Newtonian behavior of blood have been studied by Mandal et. al., 2007. They numerically solved the problem of non-Newtonian and nonlinear pulsatile flow through an irregularly stenosed arterial segment where the non-Newtonian rheology of the flowing blood is characterized by the generalized Power law model. The rate of flow, resistive impedance and the wall shear stress has been calculated.Unsteady response of non- Newtonian blood flow through a stenosed artery in magnetic field was studied by Ikbalet. al., 2008. Results are obtained for the flow velocity, flux and wall shear stress. 2. Assumptions The following assumptions are made in this chapter:  The blood is assumed to be homogeneous, incompressible and non-Newtonian fluid. It is assumed that motion of the flowing blood is steady and laminar.  The radial velocity can be neglected in the artery as the radial velocity in the artery is very small in comparison to the axial velocity (Young, 1968)  The stenosis developed in the artery is bell shaped axially symmetric and depends upon the axial distance z.  The maximum height of the stenosis is much less as compared to the length and unobstructed radius of the artery i.e. stenosis is mild.  There is no external force acting on the flowing blood. Paper ID: SUB156994 2650