azimuthal coordinate in three dimensions xii yj stream function u frequency u>~1 characteristic time Upper case Greek symbols T fluid boundary Ts traction boundary T u kinematic boundary T e finite element boundary finite element traction boundary T\u201e finite element kinematic boundary A increment (prefix) Q fluid domain fle finite element domain Lower case Roman symbols a pre-stenosis length, pre-orifice length a(y) steady part of pulsatile flow b post-stenosis length, post-orifice length b(y) magnitude of cosinusoidal part of pulsatile flow d nodal vector of unknowns f consistent load vector \/ length of stenosis, length of orifice IE inlet length n a axial cosine of unit outward normal to the boundary n 2 radial cosine of unit outward normal to the boundary p pressure &Padd additional pressure drop xiii r radial coordinate in three dimensions r(x) instantaneous radius t time u axial component of velocity UQ characteristic velocity v radial component of velocity vr radial component of velocity in three dimensions azimuthal component of velocity in three dimensions vz axial component of velocity in three dimensions x axial coordinate in two dimensions y radial coordinate in two dimensions z axial coordinate in three dimensions Upper case Roman symbols A steady streaming solution Ao(y) steady part of pulsatile approximation BQ(y) magnitude of cosinusoidal part of pulsatile approximation B(<) magnitude of cosinusoidal component Co(y) magnitude of sinusoidal part of pulsatile approximation C(t) magnitude of sinusoidal component C \u00b0 continuity of zeroth order derivatives C 1 continuity of first order derivatives Fr body force in radial direction body force in azimuthal direction Fz body force in axial direction J Jacobian matrix Jo Bessel's function of the first kind of order zero xiv K stiffness matrix M mass matrix Mj j\u2014th. bilinear isoparametric shape function Ni i\u2014th quadratic isoparametric shape function Q(t) instantaneous rate of flow R radius of the pipe Rw frequency Reynolds number R0 characteristic length Re Reynolds number 72,\/, error in approximating Poisson's equation 7\u00a3i error in approximating x\u2014momentum equation 7^ 2 error in approximating y\u2014momentum equation 72.3 error in approximating continuity equation Tin error in approximating x\u2014component of traction boundary 72-r2 error in approximating y\u2014component of traction boundary T tangent stiffness matrix U axial component of specified kinematic boundary Uo (maximum) entrance velocity V radial component of specified kinematic boundary X axial component of specified traction boundary Y radial component of specified traction boundary YQ Bessel's function of the second kind of order zero ZQ development length D ' . , , . . d d d \u2014 total material derivative = \u2014 + u\u2014\u2014V v\u2014 Dt at ox oy ' differentiation non-dimensional value differentiation with respect to time xv Acknowledgement My special thanks to my supervisor Dr. M.D. Olson, for his guidance and encour-agement throughout the course of my research work and in the preparation of this thesis. I would also like to thank Mr. Tom Nicol of the UBC Computing Center for many valuable suggestions in using the SPARSPAK package. Financial support in the form of a Research Assistantship from the Natural Sciences and Engineering Research Council of Canada and a Graduate Fellowship from the University of British Columbia is gratefully acknowledged. xvi To My Parents Indu and Madan Singh xvii C H A P T E R 1 Introduction and Literature Review 1.1 General Remarks F l u i d mechanics poses one of the most difficult problems for applied mathematics: the solution of the Navier-Stokes equations and simplifications thereof for flow conditions dictated by practical applications of the laws of fluid mechanics and technological de-velopments, rather than by academic questions. It was only after the advent of digital electronic computers, that the solutions for such problems could be developed with success. In the beginning of fluid mechanics research, l imit ing situations\u2014offering permissible simplifications of the governing equations to solvable ones\u2014were primar-i ly the goal of research. Thus came the Prandtl ' s lift line theory, the Prandtl-Meyer flow, or the Blasius series solution. Even though these problems were relatively quite simple, a l l of them already involved numerical work and a substantial amount of t ime i n which those solutions were derived. Prandtl 's lifting line theory yielded an integro-differential equation for the circulation, the Prandtl-Meyer flow a transcen-dental equation i n terms of M a c h number to be solved for the Prandtl-Meyer angle and the Blasius series solution lead to a nonlinear two point boundary value problem. T h e n came the desk calculator which enabled the computation of more complex flows such as supersonic nozzle flow by the method of characteristics. The first electronic computers immediately enabled the solution of the in i t ia l value two point boundary value problem for two-dimensional boundary layers and of the potential equation for flows wi th rather arbitrary boundary shapes, a substantial step forward. Krause [11] 1 Chapter 1: Introduction and Literature Review 2 has compiled a short history of early works in this field. W i t h the advent of high speed digital computers, researchers started developing numerical techniques for the solving hitherto unsolvable differential equations. The basic idea behind most of these techniques is to try to 'simulate' the physical phe-nomena in some approximate way and to improve the approximation in successive attempts. Computat ional fluid dynamics has always been at the forefront of the de-velopment of these numerical techniques [12]. Several reasons have contributed to this leadership among many disciplines some of which are: \u2022 the inherently nonlinear behavior of fluids (e.g., convection, turbulence) which must be accounted for, \u2022 the mixed hyperbolic-elliptic character of the governing part ial differential equa-tions, and \u2022 the need for engineers designing new airplanes to obtain much more accurate performance estimates and therefore more accurate results for the simulation than, say, their colleagues designing bridges. Another reason that follows from above is the size of typical problems. A problem in structural mechanics is considered 'large' if it exceeds 5 x 10 3 nodes, whereas large in fluid problems means more than 5 x 10 5 grid points [12]. For about 25 years the computational fluid dynamics grew around F in i te Dif-ference Methods, as these were relatively simple to understand and code, easy to vectorize and the structured grids typically associated wi th them described appro-priately the simple geometric complexity of the fields that were solved. However, as the computers became bigger and faster, attempts were made to simulate more and more complex flow domains and it soon became clear that structured grids were not flexible enough to describe these domains. It was at this point in time that unstruc-tured grids and Fini te Element Methods\u2014a natural way of discretizing operators on Chapter 1: Introduction and Literature Review 3 them\u2014entered the scene. Since then the finite element methods have become a de jure standard techniques, although some researchers s t i l l prefer the finite difference methods. Since the in i t ia l development efforts for the finite element methods for fluid me-chanics, much attention has been devoted to the treatment of the primit ive variable (velocities and pressure) formulations. The dominant approach used in the finite el-ement methods is the mixed order interpolation scheme which attempts to eliminate the spurious pressure modes [31], i.e., the tendency to produce 'checkerboard' pres-sure distributions [24]. This scheme is analogous to the 'staggered gr id ' used in the finite difference methods. However, mixed order interpolation is not total ly effective at el iminating the spurious pressure modes for simple elements, i.e., bilinear triangle or bilinear quadrilaterals where constant element pressures are used [24]. De Bre-maecker [5] has suggested use of penalty functions to avoid this problem; Rice and Schnipke [24] have presented an equal order velocity pressure interpolation that does not show the spurious pressure modes at a l l ; Zinser and B e n i m [37] have suggested a segregated velocity pressure formulation of Navier-Stokes equations which results in an approximate pressure equation and allows uncoupling of pressure and velocity solutions. Numerous other suggestions have been made. For the present investiga-t ion, we have used a mixed order interpolation wi th quadratic bilinear element where pressure is not constant over the element; along the element boundaries the variation is linear whereas in the domain the variation is linear plus one quadratic cross term. This eliminates the above mentioned 'checkerboard' pressure distributions. 1.2 Stenosis and Pulsatile Flow The world \"stenosis\" is a generic medical term which means a narrowing of any body passage, tube, or orifice. Thus, an arterial stenosis refers to a narrowed or con-stricted segment of an artery and may be caused by impingement of extravascular Chapter 1: Introduction and Literature Review 4 masses or due to intravascular atherosclerotic plaques which develop at the walls of the artery. In the arterial systems of humans, it is quite common to find narrowings, or stenoses, some of which are, at least approximately, axisymmetric or 'collar-like.' Pa r t i a l occlusion of blood vessels due to stenosis is one of the most frequently oc-curring abnormalities in the cardiovascular system of humans; it is a well established fact that about 75% of al l deaths are caused due to circulatory diseases [2]. Of these, atherosclerosis is the most frequent. Considerable studies related to the circulatory flow in blood vessels were given major attention towards the beginning of the present century. Quite a few analytical and experimental investigations related to blood flow w i t h different perspectives have already been carried out. Interest in studies of this particular domain of biomechanics has increased wi th the discovery that many car-diovascular diseases are associated wi th the flow conditions in the blood vessels; one major type of flow disorder results from stenosis. Diagnosis of arterial diseases is usually carried out by the method of X-ray angiog-raphy which involve considerable risk of morbidity [3]. So any attempts to develop non-invasive techniques for detecting arterial disease like stenosis are very worthwhile. As a result, numerical modeling of arterial blood flow has long been of interest, but development of realistic models has occurred only in recent times [22]. Causes of the slow progress include nonlinearities in the model equations, unsteady flow in the vas-cular system, and elastic properties of arteries. Most of the studies have been carried out assuming that blood is a Newtonian fluid. However it has now been accepted that blood behaves more like a non-Newtonian fluid under certain conditions, in particular at low shear rate. Srivastava [28] has presented a study of the effects of stenosis on the flow of b lood when blood is represented by a couple stress fluid, a special type of non-Newtonian fluid which takes particle size into account. This , however, is beyond the scope of the current investigation. Young [35] has presented a survey of some of the early works done in this field. Chapter 1: Introduction and Literature Review 5 Over the years, Daly [4], Deshpande [6,7], Yoganathan et al. [34], Phi lpot et al. [21], Porenta, Young and Rogge [22], Forrester and Young [8,9] and others have shed considerable light on the complex problem of blood flow through an artery in the presence of stenoses. Daly studied the effects of pulsatile flow through stenosed canine femoral arteries for constrictions i n the range 0-61% and presented a comparison of in-vivo and in-vitro results. Local flow reversal was observed in the wake of the stenosis during systole and during diastolic flow reversal. He also concluded that the rate of development of the local reverse flow is very sensitive to the height of the stenosis, a result supported by Forrester as well. Deshpande obtained numerical solutions for steady flow through axisymmetric, contoured constrictions in a rigid tube uti l izing full Navier-Stokes equations i n cyl in-drical coordinates. Laminar boundary conditions, normally applicable only at x = \u00b1 o o , were applied at finite values of x by using a transformation of the type r\\ = tanh(Kx) , where is K is a constant chosen to group the grid points i n an efficient manner. The stenosis geometry was modeled as a full cosine wave. Yoganathan et al. obtained steady flow velocity measurements in a pulmonary artery model with varying degrees of pulmonic stenosis. Phi lpot et al. used a 7-mW He-Ne laser as the light source for flow visualization in a Pulmonary artery model. They photographed the setup over the entire systolic period, thus observing the development and dissi-pation of the flow. Porenta et al. developed a one dimensional analysis which included taper, branches and obstructions and used Kantovich-like numerical technique and yielded stable nu-merical solution after 3 cycles, independent of in i t i a l values. The arterial cross-section was of the form a \u2022 exp(\u2014ax), where a is a small positive number. Low frequency behavior was approximated wi th seven cosine and seven sine components of the gen-eralized Fourier series. Propagation of a pressure wave was evident from his results. Forrester modeled the stenosis as a full cosine wave and used the Karman-Pohlhausen Chapter 1: Introduction and Literature Review 6 method as described by Schlichting [26] to model the flow using a five parameter fourth order polynomial and obtained an analytical solution. He has also reported some experimental results which supported his solution scheme. In particular, his experiments clearly reveal the development of separation downstream of the stenosis and the reattachement points as well as variation of incipient separation point and crit ical Reynolds number wi th height of stenosis. 1.3 Analysis in Time Domain Pulsati le flow in the arteries is highly variable wi th time and to understand al l the effects of stenosis, one has to solve the full Navier-Stokes equations for any time t. In general, this can be done by obtaining the finite element solution for the steady part and then carrying out simultaneous numerical integration, choosing a sufficiently small step so as not to allow divergence of the results. This , however, is extremely costly in terms of C P U time and storage since, at each step, we must solve for the effects of the quadratic, nonlinear, convection terms. McLach lan [14] and Nayfeh and Mook [17] have pointed out that the solution of the differential equations wi th such nonlinearities has a steady part as well as an oscillating part. The techniques available for determining the steady state behavior and the slowly varying periodic behavior of forced nonlinear systems can be divided into two groups: the first group includes the method of averaging and multiple scales. W i t h this group one determines the equa-tions describing phases and amplitudes which are to be solved simultaneously. The second group includes the Lindstedt-Poincare technique and the method of harmonic balance. Details of these methods are presented in [14,17]. Pat tan i [19,20] has modified the method of averaging and shown that it can be used successfully for this k ind of problem. Using the Floquet theory, he has also shown the solution to be stable. This method is used here. Chapter 1: Introduction and Literature Review 7 1.4 Scope of Present Investigation This thesis describes the development work on extending the finite element method to cover the effects of stenoses on pulsatile flow in arteries. The problem configuration of interest is that of an axisymmetric, or 'collar-like' artery. The pulsatile flow is approximated as a combination of steady, sinusoidal and cosinusoidal velocities. In Chapter 2, the fundamentals of fluid flow i n cyl indrical coordinates are pre-sented. The chapter begins by stating the governing differential equations for the fluid, derived in [26] based-on the principle of conservation of mass and momen-t u m , and choosing non-dimensional parameters to effectively characterize the flow through a pipe. Kinemat ic and traction boundary conditions are also presented and non-dimensionalized. In the last section, the Poisson's equation for stream function is derived. Chapter 3 is about discretization of the various differential equations and the boundary conditions using a family of locally defined shape functions. Residuals are formed and Galerkin's procedure is used to minimize those residuals. The modified method of averaging is used to solve for the steady streaming part and slowly varying periodic parts. A Newton-Raphson iteration scheme is outlined for the solution of the resulting set of nonlinear, algebraic equations. Numerical results are described in Chapter 4. Besides the pulsatile flow through stenoses, several other l imit ing problems are also solved to check the code. These results compare favorably wi th published analytical and experimental results in the literature. Streaklines and streamlines are plotted for different values of the non-dimensional parameters Re and Rw. In the last section, conclusions and suggestions for further investigations are presented. C H A P T E R 2 Derivation of Governing Equations In this chapter, the governing equations for axisymmetric fluid flow are presented. The velocity-pressure primitive variable form of the Navier-Stokes equations for two dimensional incompressible viscous flow is used. The equations of motion are non-dimensionalized to effectively characterize the flow situation. Final ly , an approximate representation for pulsatile flow is presented. A l l the analysis is based on incompress-ib i l i ty assumptions which hold well for liquids. 2.1 Conservation Equations and Boundary Conditions The basic equations governing viscous, incompressible fluid flow are the conservation equations: conservation of mass and conservation of momentum. These equations along w i t h the specified boundary and ini t ia l conditions, in general, need to be solved over a domain f i bounded by a contour T which is composed of two distinct parts\u2014 kinematic boundary and traction boundary\u2014denoted as Tu and T s , respectively, as shown in figure (2.1) If r,