Theoretical Study of Flow Phenomena in the Constricted Tubes

Abstract

The major health issue in the modern world is cardiovascular disease. Cardiovascular diseases are actually group diseases associated with the abnormal functioning of the blood vessels. These diseases have a close relationship with the nature of blood movement. In the view of this, the study of blood flow through various types of arteries is of considerable importance in many cardiovascular diseases. One of such disease is Stenosis, is progressive of cardiovascular disease. Stenosis can happen in all arteries. It is one of the most frequent abnormalities in blood circularity. Stenosis is defined as a partial occlusion of the arteries due to the accumulation of fatty substances, cholesterol, calcium, cellular waste products and other substances on inner walls of the artery. Cardiovascular system can cause circulatory disorders by reducing or occluding the blood supply which may result in serious consequences like chest pain, shortness of breath, heart attack and strokes. The hemodynamic behavior of the blood flow is influenced by the presence of the arterial stenosis. If the stenosis is present in an artery, normal blood flow is disturbed. The actual causes of stenosis are not well known but its effects on the cardiovascular system can be understood by studying the blood flow through stenosed arteries. The main aim of this book is to analyze the flow problems in the stenosed arteries, because the flow problems play vital role in the arterial network system, which will furnish a better assessment of physiology on human body. The objective of this book is to examine the different physiological situations and develops mathematical models by contemplating different non-Newtonian fluids with mixture of nanoparticles in constricted tubes under various situations. The main fluid flow characteristics namely, resistance to the flow, pressure drop, wall shear stress, temperature profile and nanoparticle phenomena are discussed in detail. This can play a significant role in the development and progression of the pathological conditions. Hence a detailed knowledge of the flow field in a stenosed artery may help in proper understanding and prevention of arterial diseases. This book divided into six chapters. The chapter wise summary of the work done is given as follows. CHAPTER-I This chapter deals with brief introduction of arterial stenosis, basic equations non-Newtonians fluids and their applications are given. CHAPTER-II This deals an extensive survey of literature, which will show the significance of the problems considered. CHAPTER-III In this chapter, the effects of post stenotic dilatations and stenosis on different non-Newtonian fluids are examined. Section-3.1: This section describes the flow of steady incompressible couple stress fluid in a circular tube with stenosis and dilatations have been investigated. The stenosis was assumed to be axially symmetric and mild. The flow equations have been linearized and the expressions for the resistance to the flow, velocity, pressure drop, wall shear stress have been derived. The effects of various parameters on these flow variables have been investigated. It has been observed that, pressure drop \(\Delta p\) increases with volumetric flow rate \(q\) and height of the stenosis \(\delta 1\). But decreases with couple stress fluid parameters, \(\eta\) and dilatation 02 . The résistance to the flow increases with height of the stenosis \(\delta 1\) but decreases with couple stress fluid parameters, \(n\) and post stenotic dilatation \(\delta 2\). Wall shear stress h increases with height of the stenosis \(\delta 1\) and couple stress fluid parameter. But decreases with dilatation 2. The velocity u increases with couple stress fluid parameters, \(\eta\) and dilatation \(\delta 2\). But decreases with stenosis height \(\delta 1\). Section-3.2: An attempt has been made to study the effects stenosis and post- stenotic dilatation of a couple-stress fluid with nanoparticles has been investigated under the assumption of mild stenosis. The equations are solved analytically by using Homotopy perturbation method. Expressions for resistance to the flow and wall shear stress have been calculated and the effects of various relevant parameters on these flow variables have been studied through the graphs. It is found that, the resistance to the flow increases with the heights of the stenosis 1 and Brownian motion parameter Nb but decreases with dilatation \(\delta2\), couple stress fluid parameters, \(\eta\) and thermophoresis parameter Nt. Wall shear stress h increases with the height of the stenosis \(\delta1\), thermophoresis parameter Nt, local temperature Grashof number Gr and local nanoparticle Grashof number Br. But decreases with dilatation \(\delta2\), couple stress fluid parameters, \(\eta\) and Brownian motion parameter Nb. Section-3.3: A mathematical model for the steady flow of an incompressible micropolar fluid with nanoparticles through a tube with stenosis and post stenotic dilation has been analyzed by assuming stenosis is to be mild. The analytical solutions of the governing equations are obtained by using Homotopy perturbation method. It is observed that, the resistance to the flow increases with the heights of the stenosis \(\delta1\), micropolar parameter m, thermophoresis parameter Nt, local temperature Grashof number Gr and local nanoparticle Grashof number Br. But decreases with the coupling number N, dilatation \(\delta2\) and Brownian motion parameter Nb. It is seen that, the wall shear stress h increases with the heights of the stenosis \(\delta1\) coupling number N and Brownian motion parameter Nb. But decreases micropolar parameter m, thermophoresis parameter Nt, local temperature Grashof number Gr, dilatation 2 and local nanoparticle Grashof number Br. It can be seen that, the temperature profile \(\theta\)t and nanoparticle phenomena \(\sigma\) increases with the increase of thermophoresis parameter Nt and decreases with the increase of Brownian motion parameter Nb. CHAPTER-IV In this chapter deals with blood flow through an artery in the presence of overlapping stenosis by assuming blood as non-Newtonian fluid. Section-4.1: Couple stress fluid model has been applied to analyze the effects on blood flow characteristics due to presence of an overlapping stenosis in arteries. The governing equations are solved and the analytical solutions are obtained in terms of modified Bessel functions. The effects of various physical parameters on these flow variables have been studied through the graphs. The flow characteristics i.e. resistance to the flow, pressure drop \(\Delta p\) and wall shear stress \(\tau\)h increases with height of the stenosis \(\delta\) and length of the stenosis L0. The resistance to the flow and pressure drop decreases with couple stress fluid parameters, \(\eta\). But the wall shear stress increases with couple stress fluid parameters, \(\eta\). Section-4.2: A mathematical model for the steady flow of an incompressible micropolar fluid with nanoparticles through a tube having overlapping stenosis has been studied by assuming stenosis is to be mild. The analytical solutions of the governing equations are obtained by using Homotopy perturbation method. Effect of different physical parameters on pressure drop, resistance to the flow, wall shear stress, temperature profile and nanoparticle phenomenon of the fluid are studied. It is observed that both pressure drop \(\Delta p\) and wall shear stress \(\tau\)h increases with the heights of the stenosis \(\delta\) length of the stenosis L0, coupling number N and Brownian motion parameter Nb. But decreases with micropolar parameter m, thermophoresis parameter Nt, local temperature Grashof number Gr and local nanoparticle Grashof number Br. Resistance to the flow increases with the heights of the stenosis \(\delta\) , length of the stenosis L0, micropolar parameter m, thermophoresis parameter Nt, local temperature Grashof number Gr and local nanoparticle Grashof number Br. But decreases with coupling number N and Brownian motion parameter Nb. CHAPTER-V In this chapter a mathematical model is developed to analyze the flow phenomenon through a non-uniform tube having multiple stenoses has been investigated. Section-5.1: The flow of steady and an incompressible couple stress fluid with nanoparticles through a non-uniform channel and having two stenoses has been considered. Solutions have been obtained for mild stenoses. The flow equations have been linearised and the expressions for pressure drop, resistance to the flow, wall shear stress have been derived. Homotopy perturbation method is used to solve the coupled equations of temperature profile and nanoparticle phenomena and analytical methods have been applied to the present study to find the other variables. The resistance to the flow increases with the heights of the stenosis (\(\delta\)1& \(\delta\)2), couple stress fluid parameters, \(\eta\) and thermophoresis parameter Nt but decreases with local temperature Grashof number Gr, local nanoparticle Grashof number Br, inclination \(\alpha\) and Brownian motion parameter Nb. Wall shear stress \(\tau\)h increases with the heights of the stenosis (\(\delta\)1& \(\delta\)2), Brownian motion parameter Nb, local temperature Grashof number Gr, local nanoparticle Grashof number Br and inclination. But decreases with couple stress fluid parameters, \(\eta\) and thermophoresis parameter Nt. Section-5.2: The steady flow of Nanofluid through a circular tube of non-uniform cross section and is having two stenoses. Cylindrical polar coordinate system r, \(\theta\), z is taken so that z- axis coincides with the centre line of the tube. It is assumed that the tube is inclined at an angle ‘’ to the horizontal axis. The stenoses are supposed to be mild and develop in an axially symmetric manner. The effects of different physical parameters on these flow variables have been investigated through the graphs. It is observed that, the resistance to the flow increases with the heights of the stenosis (1 and \(\delta\)2), thermophoresis parameter Nt, local temperature Grashof number Gr and local nanoparticle Grashof number Br. But decreases with Brownian motion parameter Nb. It is found that Pressure drop increases with the heights of the stenosis (\(\delta\)1& \(\delta\)2) and Brownian motion parameter Nb but decreases with thermophoresis parameter Nt, local temperature Grashof number Gr, local nanoparticle Grashof number Br and inclination \(\alpha\). It is also observed that the wall shear stress \(\tau\)h is increasing with the heights of the stenoses (\(\delta\)1& \(\delta\)2) and Brownian motion parameter Nb but decreases with thermophoresis parameter Nt. It is observed that, nanoparticle phenomena \(\sigma\) increases with the increase of Brownian motion parameter Nb and thermophoresis parameter Nt. It can be seen that, temperature profilet increases with the increase of thermophoresis parameter Nt and decreases with the increase of Brownian motion parameter Nb.