X. Ma G. C. Lee State University of New York at Buffalo, Buffalo, NY 14260
S. G. Wu Beijing Polytechnic University, Beijing, China
Numerical Simulation for the Propagation of Nonlinear Pulsatile Waves in Arteries The behavior of nonlinear pulsatile flow of incompressible blood contained in an elastic tube is examined. The theory takes into account the nonlinear convective terms of the Navier-Stokes equations. The motion of the arterial wall is characterized by a set of linearized differential equations. The region bounded by the flexible arterial wall is mapped into a fixed area in which numerical discretization takes place. The finite element method (Galerkin weighted residual approach) is used for the solution of this nonlinear system. The results obtained are pressure distribution, velocity profile, flow rate and wall displacements along the elastic tube (20 cm long).
1
Introduction Because many cardiovascular disorders are closely associated with the flow conditions in the blood vessel [1], the characteristics of blood flow in arteries have received much attention. Earlier studies have concentrated on linearized Navier-Stokes equation and small elastic deformation [3, 14, 15]. These simplifying assumptions made it possible to obtain analytic solutions which provided much insight into the mechanics of the blood flow. However, they are inadequate in describing the velocity field, especially in large arteries (Ling and Atabek, 1972). Therefore analysis based on the nonlinear equations for flow in large blood vessel has been given emphasis in recent years [11], For pulsatile flow in arteries, the effects of nonlinearity mainly come from the following: the nonlinear acceleration terms in the flow equation; the finite strain and nonlinear elasticity (or nonlinear viscoelasticity) of the arterial wall and the nonlinear motion of the elastic tube. Among them the most important nonlinear effect in describing the flow field lies in the convective acceleration terms of the Navier-Stokes equations. In the past decades, little progress has been made in this area of study. As early as 1958, Lambert established a onedimensional model and obtained the solution by the method of characteristics. Later, this model was extended by several authors [8, 9], Because the model is one-dimensional, the velocity profile could not be determined. Therefore this model is less informative than the linearized theories which can provide velocity distributions. In 1972, Ling and Atabek developed a nonlinear model called "local flow theory". They employed an approximate numerical method to obtain the flow profile through the locally measured pressure-radius function, namely R = R(p). This
theory worked well in describing the velocity profile at a given fixed location. However, it does not provide a continuous solution over a segment of the elastic tube. Recently, Hung et al. [12] made a comparative study of nonlinear pulsatile flow in rigid and deformable stenotic vessels. In this work, the exponential type of coordinate transformation is employed. The motion of the wall is characterized by an algebraic relation. In this paper, the nonlinear Navier-Stokes equations are used to characterize the flow field. For the equations of motion of the arterial wall, the linearized version of a model based on our previous study is employed. The finite element method is used for the solutions of this coupled system over a finite segment of the tube (20 cm). 2
Equations of Motion of the Blood The physical model and coordinate system are shown in Fig. 1. The equations which govern axial-symmetric pulsatile flow are the Navier-Stokes and continuity equations. These equations are: d2u ldp (\d_ du du du du (1) + -2 p dz \r dr drTdz dt dz dr dv dv dv I dp I1 d dv\ d2v 3 (2) dt dz dr p dr \r dr drr&~
midle surface in steady state
s>-ds dz
1 a
dv
v
{R2idi l ? aJ + az 2 R\2
(29)
where dR
dR
RC^Tr^Tz The continuity equation is expressed in E q . (27).
6
The Finite Element Model 6.1 FEM Model for the Flow. The finite element solution process is performed in t h e transformed (£, z) space which is discretized into triangular elements (revolutionary). T h e pressure varies linearly within the element a n d is defined at t h e corner points (Nf). T h e velocity field is approximated b y the quadratic polynomial a n d the velocity components are defined Journal of Biomechanical Engineering
unNi^+v:Ni^\R2^dz
+
J_dp ~pR 3£
+v
K3
2
«
In the above, (a) and (b) are simply the results of long wave assumption, in which iai?/3zl « 1. The cross derivative of flow velocity is typically a troublesome term. Since this term comes from the diffusion in the z direction (small compared to diffusion in the r direction), we can neglect this term. By considering these simplifications, the Navier-Stokes equations are then reduced to: du
0
0
where the super d o t represents the time derivative and the mass coefficients: M,
dR a_R 2J
S. G. Wu Beijing Polytechnic University, Beijing, China
Numerical Simulation for the Propagation of Nonlinear Pulsatile Waves in Arteries The behavior of nonlinear pulsatile flow of incompressible blood contained in an elastic tube is examined. The theory takes into account the nonlinear convective terms of the Navier-Stokes equations. The motion of the arterial wall is characterized by a set of linearized differential equations. The region bounded by the flexible arterial wall is mapped into a fixed area in which numerical discretization takes place. The finite element method (Galerkin weighted residual approach) is used for the solution of this nonlinear system. The results obtained are pressure distribution, velocity profile, flow rate and wall displacements along the elastic tube (20 cm long).
1
Introduction Because many cardiovascular disorders are closely associated with the flow conditions in the blood vessel [1], the characteristics of blood flow in arteries have received much attention. Earlier studies have concentrated on linearized Navier-Stokes equation and small elastic deformation [3, 14, 15]. These simplifying assumptions made it possible to obtain analytic solutions which provided much insight into the mechanics of the blood flow. However, they are inadequate in describing the velocity field, especially in large arteries (Ling and Atabek, 1972). Therefore analysis based on the nonlinear equations for flow in large blood vessel has been given emphasis in recent years [11], For pulsatile flow in arteries, the effects of nonlinearity mainly come from the following: the nonlinear acceleration terms in the flow equation; the finite strain and nonlinear elasticity (or nonlinear viscoelasticity) of the arterial wall and the nonlinear motion of the elastic tube. Among them the most important nonlinear effect in describing the flow field lies in the convective acceleration terms of the Navier-Stokes equations. In the past decades, little progress has been made in this area of study. As early as 1958, Lambert established a onedimensional model and obtained the solution by the method of characteristics. Later, this model was extended by several authors [8, 9], Because the model is one-dimensional, the velocity profile could not be determined. Therefore this model is less informative than the linearized theories which can provide velocity distributions. In 1972, Ling and Atabek developed a nonlinear model called "local flow theory". They employed an approximate numerical method to obtain the flow profile through the locally measured pressure-radius function, namely R = R(p). This
theory worked well in describing the velocity profile at a given fixed location. However, it does not provide a continuous solution over a segment of the elastic tube. Recently, Hung et al. [12] made a comparative study of nonlinear pulsatile flow in rigid and deformable stenotic vessels. In this work, the exponential type of coordinate transformation is employed. The motion of the wall is characterized by an algebraic relation. In this paper, the nonlinear Navier-Stokes equations are used to characterize the flow field. For the equations of motion of the arterial wall, the linearized version of a model based on our previous study is employed. The finite element method is used for the solutions of this coupled system over a finite segment of the tube (20 cm). 2
Equations of Motion of the Blood The physical model and coordinate system are shown in Fig. 1. The equations which govern axial-symmetric pulsatile flow are the Navier-Stokes and continuity equations. These equations are: d2u ldp (\d_ du du du du (1) + -2 p dz \r dr drTdz dt dz dr dv dv dv I dp I1 d dv\ d2v 3 (2) dt dz dr p dr \r dr drr&~
midle surface in steady state
s>-ds dz
1 a
dv
v
{R2idi l ? aJ + az 2 R\2
(29)
where dR
dR
RC^Tr^Tz The continuity equation is expressed in E q . (27).
6
The Finite Element Model 6.1 FEM Model for the Flow. The finite element solution process is performed in t h e transformed (£, z) space which is discretized into triangular elements (revolutionary). T h e pressure varies linearly within the element a n d is defined at t h e corner points (Nf). T h e velocity field is approximated b y the quadratic polynomial a n d the velocity components are defined Journal of Biomechanical Engineering
unNi^+v:Ni^\R2^dz
+
J_dp ~pR 3£
+v
K3
2
«
In the above, (a) and (b) are simply the results of long wave assumption, in which iai?/3zl « 1. The cross derivative of flow velocity is typically a troublesome term. Since this term comes from the diffusion in the z direction (small compared to diffusion in the r direction), we can neglect this term. By considering these simplifications, the Navier-Stokes equations are then reduced to: du
0
0
where the super d o t represents the time derivative and the mass coefficients: M,
dR a_R 2J
