| dc.description.abstract | Laminar flow in pipes and channels with uniform and non-uniform cross-sections plays an important role in many engineering and physiological flow problems. Flow of blood through catheterised and stenosed (constricted) blood vessels are two such applications with intense biomedical and fluid dynamic interest. Understanding the flow structure in the presence of a catheter in a blood vessel enables one to minimise its influence on the flow and to improve the efficiency of pressure measurements. The study of flow in variable cross-sectional pipes provides deeper insight into the pathology of a proliferative disease of blood vessels called atherosclerosis or arteriosclerosis.
In recent years, the study of flow characteristics in pipes with elliptic cross-sections has attracted significant research interest due to their numerous applications. In high heat-load systems such as nuclear reactors, compact heat exchangers, etc., one has to study the flow in elliptic pipes. Blood vessels with elliptic cross-sections are also encountered in the cardiovascular system when the transmural pressure is low (e.g., pulmonary arteries) and in regions prior to bifurcations. Blood vessels may also become elliptic due to age or pathological conditions.
Motivated by these considerations, flows in elliptic pipes with uniform and non-uniform cross-sections are studied in this thesis to understand the effect of stenosis on blood flow. Flows in two eccentric circular cylinders and in two confocal elliptic cylinders are also considered to model the flow through a catheterised blood vessel. The thesis is divided into six chapters.
Chapter 1
This chapter provides a general introduction and presents the governing equations in appropriate orthogonal curvilinear coordinate systems for the problems studied in the rest of the thesis. Flow in eccentric circular cylinders is studied in bipolar coordinates, whereas flow in pipes with elliptic boundaries is studied using elliptic cylindrical coordinates. Chapters 2�deal with the flow of an incompressible viscous fluid, while chapters 5�consider the flow of an incompressible micropolar fluid. Each chapter begins with an introduction to the general aspects of the problem, including a survey of related work, followed by a brief description of the present investigation. Relevant original papers are listed at the end of the thesis.
Chapter 2
The steady unidirectional flow under a constant pressure gradient in two non-intersecting cylinders in relative motion, having eccentric circular and confocal elliptic cross-sections, is studied. An exact solution is obtained in each case in an appropriate coordinate system. Expressions for volume flux, wall shear stress, and friction factor are derived.
For eccentric cylinders in relative motion, the results corresponding to the limiting cases of concentric and fully eccentric (touching) cylinders are obtained directly by taking limits, without formulating separate limiting problems as previously done in literature. It is observed that, for flow due to the relative motion of the inner cylinder in the absence of a pressure gradient, the volume flux is maximum when the cylinders are concentric and minimum when they are touching.
Effects of relative motion, eccentricity, and radius ratio on velocity distribution, wall shear stress, and friction factors are analysed. The critical velocity of the inner cylinder for which the volume flux becomes zero is also obtained. For confocal elliptic cylinders, results for concentric circular cylinders are recovered by taking the limit in which the elliptic eccentricity approaches zero. The special case where the inner cylinder reduces to a strip joining the foci is also discussed.
Chapter 3
This chapter presents the steady flow of a homogeneous incompressible viscous fluid in an elliptic pipe of slowly varying cross-section and in two confocal elliptic cylinders with a non-uniform outer boundary. Governing equations of various orders are derived using the geometric parameter
??1,\varepsilon \ll 1,??1,
representing the ratio of the average undisturbed semi-major and semi-minor axes to the characteristic axial length. Closed-form solutions are obtained at successive orders of the asymptotic expansion in ?, for velocity and pressure gradient for a prescribed flow rate.
Pressure gradient, velocity distribution, flow resistance, and wall shear stress are analysed for various types of variable cross-sectional pipes. The effects of eccentricity, percentage constriction, and Reynolds number are studied. The influence of an inner cylinder on the flow is discussed. Non-linear terms become significant with increasing constriction percentage and Reynolds number.
Chapter 4
This chapter deals with pulsatile flow of an incompressible viscous fluid in an elliptic pipe of non-uniform cross-section and in two confocal elliptic cylinders of uniform cross-section. The same perturbation parameter
??1\varepsilon \ll 1??1
is used here.
The Womersley parameter
?=a??,\alpha = \frac{a \sqrt{\omega}}{\nu},?=?a???,
where ? is the oscillation frequency, a is the average of the undisturbed semi-axes, and ? is the kinematic viscosity, plays an important role.
Closed-form solutions for velocity components and pressure gradient up to first-order are obtained in terms of Mathieu and modified Mathieu functions. Their validity for low and high Womersley parameters is examined. A systematic method is developed for computing these special functions with real and complex arguments.
For large Womersley parameters, the local wall shear stress exhibits multiple extreme points in the region of maximum constriction.
An exact solution for oscillatory flow in confocal elliptic cylinders is obtained. The results for concentric cylinders are derived by taking the eccentricity to zero. Numerical results depict the behaviour of velocity and shear stress. Mean velocity decreases as the Womersley parameter increases.
Chapter 5
Steady and pulsatile flows of an incompressible micropolar fluid are studied in an elliptic pipe of variable cross-section and in a non-uniform channel. Using the geometric parameter representing slow variation, analytical expressions are derived for velocity, microrotation components, and pressure gradient. The flow structure resembles that of Newtonian flow with reduced pressure gradient and increased axial velocity.
The flow is governed by the Reynolds number and micropolar parameters
N=??+?,L1=??+?,N = \frac{\kappa}{\mu + \kappa}, \quad L_1 = \frac{\gamma}{\mu + \kappa},N=?+???,L1?=?+???,
where ?, ?, ? are material constants and l is a characteristic length. For pulsatile flow, the Womersley parameter ? and
N2=?j?N_2 = \frac{\rho j}{\mu}N2?=??j?
are additional parameters.
Pressure gradient decreases and centreline velocity increases with increasing N. Opposite trends occur with increasing L1L_1L1?. The influence of N2N_2N2? is similar to that of L1L_1L1?.
Chapter 6
This chapter addresses flow of two immiscible fluids in an elliptic pipe of uniform cross-section. Two cases are considered:
Newtonian fluid in the peripheral region and another Newtonian fluid of different viscosity in the core.
Newtonian fluid outside and a micropolar fluid in the core.
Exact solutions are obtained for both cases. Results for immiscible Newtonian fluids of different viscosities are recovered as limiting cases when micropolar parameters tend to zero.
The effects of viscosity ratio and eccentricity of the ellipse on flow are studied numerically. Relative viscosity increases with eccentricity, and velocity profiles become more blunted in the core region. | |
