Study of blood flow using power law and Harschel-Bulkley non-Newtonian fluid model through elastic artery

To study the hydrodynamics of the arterial system a knowledge of the elastic constants of the arterial wall, from which one can predict the ‘natural’ velocity, is of fundamental importance (Mc Donald, 1974).The aim of the present study is to analyze the effect of non-Newtonian nature of blood through an Elastic artery. The artery is considered elastic and blood is considered under power-law model and Harschel-Bulkley model. The effect of elastic nature of arterial wall on longitudinal velocity of blood is analyzed for both the models of blood under consideration. In the proposed study we found that in the elastic artery the velocity profiles for power law model is advanced compare to the Herchel-Bulkley model for fixed value of power index n and other parameter and with the increase of power index the velocity of fluid decreases with the increase of power index for both the fluid. Also we observe that for linear transmular pressure the velocity of the blood decreases with the increase of elasticity upto unit value of z which is transition point, away from this point the result turned reverse. The velocity of fluid decreases with the down stream for each value of elasticity of the vessel but in case of small value of elasticity the downstream velocity from the transition point decreases enormously compare to higher elastic value of the vessel. Keywords: Artery, Elastic tube, Pulsatile flow, Non-Newtonian fluid. I. INTRODUCTION Artery is an elastic tube whose diameter will vary with a pulsating pressure; in addition it will propagate pressure and flow waves created by the ejection of blood by the heart, at a certain velocity which is largely determined by the elastic properties of the wall. To study the hydrodynamics of the arterial system knowledge of the elastic constants of the arterial wall, from which one can predict the ‘natural’ velocity, is of fundamental importance (McDonald, 1974). The behavior of an elastic tube containing a viscous fluid submitted to pulsatile flow has been the object of many studies concerning the analysis of blood flow. The non-Newtonian fluids flow through elastic tubes impart many information to understand bio-fluid dynamics proximal to the real physiological conditions. (Mehrotra et al., 1985) studied the pulsatile blood flow in a stenosed artery. ( Tsangaris and Drikakis, 1989) studied the pulsating blood flow in an initially stressed, anisotropic elastic tube. They also discussed the propagation of pressure waves. A model of pulsatile flow in a uniform deformable vessel is discussed by (Johnson et al., 1992). (Gupta and Agarwal, 1993) studied the laminar nonNewtonian power-law fluid flow development in a pipe. Casson fluid flow in a pipe filled with a homogeneous porous medium is studied by (Dash and, Mehta 1996) and examined the effect of permeability factor and yield stress of the fluid on shear stress distribution, wall shear stress, plug flow radius, flow rates and frictional resistance. (Pedrizzetti et al., 2002) studied the pulsatile flow inside moderately elastic artery, and found influence of wall elasticity on the flow and on the unsteady wall shear stress. ( Ahmad and Misra 2003) studied the Casson’s fluid flow in an artery like elastic tube to see the dependence of velocity field and shear stress on elastic character of the tube. They obtained the velocity distribution by solving generalized equation coupled with Casson constitutive equation. (Stephanis et al., 2003) proposed a new coefficient of elasticity related to the elastic state of the blood vessels. ( Johnston et al.,(2004) studied the non-Newtonian blood flow in human right cornary arteries. They concluded that, while the Newtonian model of blood viscosity is a good approximation in regions of mid-range to high shear, it is advisable to use the generalized power law model in order to achieve better approximation of wall shear stress at low shear. (Mandal, 2005) discussed non-Newtonian and non-linear blood flow through a stenosed artery. The nonNewtonian rheology of the flowing blood is characterized by the generalized power law model. (Sankar and Hemalatha 2007) studied the pulsatile flow of blood through catheterized artery by modeling blood as Harschel Bulkley fluid and the catheter and artery as rigid coaxial circular cylinders. They observed that the velocity and flow rate decreases, whereas wall shear stress and longitudinal impedance increase for increasing value of yield stress with other parameters held fixed. (Benam et al., 2008) designed an experimental set up for study of characteristics of pulsatile flow in elastic tubes, aiming to simulate arterial blood flow. The pulsatile flow of blood through mild stenosed artery treating the blood as Herschel–Bulkley fluid is studied by (Sankar and Lee 2009) . They observed that the plug core radius , pressure drop and wall shear stress increase with the increase of yield stress or the stenosis height. (Nadeem and Akabar 2011) analyzed the Power law fluid model for blood flow through a tapered artery with a stenosis, ( Mallik et al., 2013) studied a non-Newtonian fluid model for blood flow using power law through an atherosclerotic arterial segment having slip velocity. (Kumar and Diwakar 2013) Obtained A mathematical model of power law fluid with an application of blood flow through an artery with stenosis. The aim of the present study is to analyze the effect of non-Newtonion nature of blood through an artery. The artery is considered elastic and blood is considered under power-law model and Harschel-Bulkley model. The effect of elastic nature of arterial wall on longitudinal velocity of blood is discussed for both the model of blood under consideration. II. METHODS A. Formulation & solution of the problem In the present problem non-Newtonian blood flow through an elastic artery is considered. The vessel is considered field with blood at rest and surrounded by fluid. The axis of the vessel is Proceedings of ICFM 2015 International Conference on Frontiers in Mathematics 2015 March 26-28, 2015, Gauhati University, Guwahati, Assam, India Available online at http://www.gauhati.ac.in/ICFMGU ISBN: 978-81-928118-9-5 230 taken along the axis of z. For mathematical convenience, artery considered as a long cylindrical tube so that entrance and end effect can be avoided. The following assumption are taken into account (i) All physical properties are constants (ii) The flow is study and fully developed (iii)The flow is axisymmetric and laminar (iv) The arterial wall is elastic Physical model of the problem Considering the transmural pressure difference of the vessel as described in (Mazumdar, 1992) Th = r p z ( )− p0 # % (1) where h be the wall thickness of the tube, r the radius of the tube, p0 the exterior pressure and p the interior Blood pressure, p− p0 is the transmural pressure difference, T is the tension per unit length and per unit thickness of the tube. According to Hooke’s law the tension T is defined by T = E (r − r0) r0 (2) where r = r0 is equilibrium position when tension T is zero, E is Young’s modules of the tube wall. Using the relation (1) in (2), we have r = ro 1− ro ( p(z)− po ) / Eh (3) B. The Governing eq. of Motion Governing eq. of motion of a Newtonian fluid in cylindrical co-­‐ordinate system (r , θ , z) where * r and * z denote the radial and axial coordinates respectively and θ* is the azimuthal angle is μ 1 r d dr r du * dr !