MA Ye, NG Chiu-On, CHANG Yin-Yee

Department of Mechanical Engineering, The University of Hong Kong, Hong Kong, China,

WAVE INDUCED OSCILLATORY AND STEADY FLOWS IN THE ANNULUS OF A CATHETERIZED VISCOELASTIC TUBE*

MA Ye, NG Chiu-On, CHANG Yin-Yee

Department of Mechanical Engineering, The University of Hong Kong, Hong Kong, China,

E-mail: cong@hku.hk

(Received March 30, 2010, Revised August 2, 2010)

A perturbation analysis based on equations of motion in Lagrangian form is presented for the oscillatory and time-mean viscous flows induced by a propagating wave of small amplitude in an annulus with a viscoelastic outer wall. Owing to the steady streaming effect, the existence of a catheter in a blood vessel brings in an additional steady pressure gradient, a correction to that predicted by the linear theory, and an additional steady shear stress, which may increase the possibility of hemolysis of red blood cells.

annulus flow, steady streaming, catheterized artery

1. Introduction

The study of oscillatory flow of a viscous fluid contained in a flexible tube is of importance particularly to biomechanics. In clinical situations such as balloon angioplasty, a pressure transducer with a long fine catheter is inserted in a peripheral artery to measure the pressure gradient over a large part of the arterial tree during angioplasty procedures[1]. The insertion of catheters in blood vessels will, however, alter the pressure distribution, thereby inducing errors to the pressure gradient measurement[2]. Also, hemolysis of blood samples is a common clinical problem encountered in emergency operations. Excessively large shear stress in the annulus flow due to catheterization may destroy the red blood cells, leading to inaccuracy in assay results and often to the need for repeated blood draws[3].

With an objective to understand blood flow in an annular region through a stenotic artery, several studies have been performed to estimate the correction to the measured values of blood pressure gradient[1,2,4-6]. Due to the nonlinear phenomenon known as steady streaming, which amounts to a time-independent component of flow[7], there exists a higher-order non-zero time-mean pressure gradient in addition to the oscillatory component found at the leading order. Sarkar and Jayaraman[1]derived a correction to the mean pressure drop as predicted by the linear theory in pulsatile flow through a catheterized stenosed rigid artery. The effects of the catheter size and oscillation frequency on the time-mean quantities such as velocity, impedance and wall shear stress were studied. A nonlinear analysis of the annulus flow in an elastic tube was further carried out by Sarkar and Jayaraman[2]. They also showed that the mean pressure gradient would change with catheter size at all frequencies. The geometry as well as the elasticity of the wall could play an important role in the dynamics of the flow even for small catheter radius. More complex effects, such as artery clot, artery curvature and non-Newtonian fluid, were investigated by Jayaraman and Sarkar[4], Jayaraman and Dash[5]and Sankar[6], respectively. The associated mass transport problem in oscillatory flow through a catheterized artery was also studied by, e.g., Sarkar and Jayaraman[8].

The above-mentioned works are based on the Eulerian formulation, in which the Taylor expansiontheorem is employed to approximate the boundary conditions on a moving wall by those on the mean position of the wall[9]. Such an approach would, however, limit the validity of the second-order results to waves of extremely small amplitude when the oscillation frequency is high[10]. To circumvent this problem, we adopt in this work the Lagrangian description, by which the conditions on a moving boundary are prescribed exactly by referring to its undisturbed position[11]. The Lagrangian description is intrinsically particle specific, and hence its analysis will yield results that can be interpreted as the conditions experienced by individual particles, e.g., blood cells, in the flow. The Lagrangian steady streaming velocity, which shows the time-mean movement of particles, is found directly by this approach. Ma and Ng[10]recently developed a Lagrangian model to investigate the nonlinear flow induced by oscillatory pressure forcing through a thick-walled flexible tube without catheter insertion. Their mathematical formulation is largely followed in this work. In particular, their solutions for oscillatory as well as time-mean motions with zero initial wall stresses will be used as particular cases for comparison in this article.

Sharp and Mohammad[3]found that the probability of hemolysis would increase as a result of increasing pressure difference and catheter size. They defined a threshold for hemolysis of red blood cells as a function of exposure time and shear stress. Below the threshold, hemolysis is much less probable to occur. In this regard, it is essential to determine a catheter size that will not cause the blood cells to be subject to excessively large shear stress during the process of catheterization. This has motivated the present study.

Specifically, the present work is to study the oscillatory and time-mean flows, as induced by purely oscillatory pressure forcing, in a catheterized artery with viscoelastic outer wall. The Lagrangian coordinates are employed. The catheter, or the inner tube, is assumed to be rigid, and the condition outside the outer tube can be either stress-free (i.e., a free tube without constraint by the surrounding tissues), or zero-displacement axially (i.e., a tethered tube constrained by the surrounding tissues). Effects of non-Newtonian fluid, stenosis, curvature, gravity and initial stresses of the outer wall are not considered in this work. A perturbation analysis is performed by introducing a small ratio of wave amplitude to tube radius. The wavenumber and wave attenuation are found of the first order and the steady fluid dynamic parameters like the axial mass transport velocity, pressure gradient, and shear stress are found of the second order. Results are generated to illustrate the effects of oscillation frequency and catheter radius on the first-order oscillatory flow and the second-order steady quantities. Finally, based on the results, a critical catheter radius is suggested.

2. Mathematical formulation

Figure 1 shows a schematic diagram of the annular geometry and the cylindrical coordinate system. The radius of the rigid catheter, and the initial inner and outer radii of the outer wall are represented by d, a and b, respectively. By the Lagrangian description, the instantaneous radial and axial coordinates of a particle of fixed identity, ( r, z), and the pressure, p, are functions of the initial coordinates of the particle, ( R, Z ), and time, t. Axisymmetry is assumed, and hence any dependence on the azimuthal positionθ is eliminated. As a result of an oscillatory pressure gradient of angular frequency σ, a progressive wave of wavenumber k is induced in the system, where k and σ are related by a dispersion equation, as will be deduced later.

Fig.1 Schematic diagram of the problem, where description is based on Lagrangian axial and radial coordinates, Z and R. Viscous fluid contained in the annulus between a viscoelastic tube and a rigid catheter is subjected to time-periodic oscillatory pressure forcing at Z=0. A fluid element initially centered at (R, Z), upon undergoing deformation, moves to a new center ( r, z) at timet

will be used as an ordering parameter for the perturbation analysis below.

Following Ma and Ng[10], we introduce the following normalized variables, which are distinguished by a caret:

where the subindex l is used to distinguish between the fluid and outer wall domains

where ρ represents the density, τijare the deviatoric stress components, T and N are, respectively, the tangential and normal stress components on a material surface as seen in an ( R, Z) plane.

The motions of the fluid and the wall are governed by the continuity and momentum equations in the Lagrangian system as follows. The continuity equation is

The dimensionless stress components are given below:

More details about the deduction of the Lagrangian equations of motions in the cylindrical system can be found in Ma and Ng[10]. In the equations above, three dimensionless parameters are introduced:

where νl=μl/ρlis the kinematic viscosity. These parameters have the following physical meanings: α is a frequency parameter known as the Womersley number, β represents the significance of the tube elasticity, γ is a ratio of the tube viscosity to the fluid viscosity.

The normalized boundary conditions read as follows:

Substitution of the expansions above into the governing Eqs.(5) - (9), the stress components (10) -(15) and the boundary conditions (17) - (19), and collecting terms of equal power of ε, we may obtain the first- and second-order problems as detailed in the following sections.

3. First-order problem

At O()ε, the governing equations for the fluid are

The system will have a non-trivial solution if and only if the determinant of the coefficient matrixA vanishes:

This condition yields the characteristic equation governing the eigenvalue, which is the complex wavenumberEquation (56) admits four complex solutions forwhere the real partdenotes the wavenumber, and the imaginary partis the attenuation constant. Our focus is on the forward traveling waves, so we only consider the two solutions with＞ 0 and＜0. Each of these two solutions corresponds to a distinct wave mode. The wave with a slower phase velocity C1= σ/kr1is called Young wave representing a pressure wave propagating in the fluid, while the one with a higher phase velocity C2= σ/kr2is called Lamb wave representing a wave traveling largely along the wall. Most of our discussions below will be on the Young wave mode, which is more important in the present study.

The constants B1-B7are related to the pressure amplitude poby the following relations: where Δklis the cofactor obtained by eliminating the kth row and the lth column of the matrix A.

In Case 2 for a tethered tube, B4=0 and the dispersion equation becomes much simpler, which can be expressed as

The Lamb wave does not exist this time and only the Young wave can survive in the tethering case.

Fig.2 Phase velocity C1(m/s) and imaginary wavenumberof the Young wave as functions of the Womersley numberα

4. Second-order problem

The overbar denotes time average over one wave period, and the asterisk denotes the complex conjugate.

According to the forcing terms on the right-hand side of the time-averaged equations, the second-order solutions of the steady motion in both the fluid and the wall can be expressed as follows:

where UsR, UsLand psfare functions ofonly. The termindicates that all the second-order steady solutions decay along the axial direction. The strength of the axial decay is represented by the magnitude of imaginary part wavenumber, which is always negative as illustrated in Fig.2.

The boundary conditions for the steady streaming of the fluid can be written as

Substitution of Eq.(66) into Eqs. (61) - (63) and the boundary conditions (67) - (68) readily yields the time-mean pressure and axial steady streaming velocity at=0.

where the integration constant M1and M2can be determined from the boundary conditions (67) and (68).

The steady Lagrangian drift, which is largely in the axial direction, is now determined. This steady current is affected through the parameter B1by the oscillation frequency and catheter radius, as given in Eq.(57). Our second-order analysis and the resulting expression for the Lagrangian drift in annulus flow appear not to have been reported in the existing literature. By virtue of the Lagrangian approach, the boundary conditions are prescribed exactly on the interfaces, and hence the expressions deduced above are good irrespective of the displacement amplitude of the boundaries.

Fig.3 Cross-sectional profiles of steady axial velocity at=0

Fig.5 Cross-sectional profiles of steady shear stress

Numerical results, which are obtained with the computational package Mathcad Version 14, are presented below to help us look into the effects of the catheter radius on the second-order steady quantities. Figure 3 shows the steady axial velocity profiles at the steady pressure gradient as has been illustrated in Fig.4. As a matter of fact, a catheter of radius d? =0. 3 will take up only 9% of the cross-sectional area of the tube. An order-of-magnitude analysis will show that, with a blockage ratio equal to 10%, the increase in the section-mean velocity is also of the order 10%. Such a limited increase in velocity will also lead to a limited increase in the magnitude of the dynamics quantities. The results here suggest that in practice a catheter radius no greater than d?=0. 3 should be used in order to avoid too large an increase in pressure or stress to be induced in the fluid.

5. Concluding remarks

In this article, we have examined the oscillatory as well as the steady Lagrangian flows of a fluid as induced by a wave propagating in the annulus of a catheterized viscoelastic tube. The problem is entirely Lagrangian in formulation and analysis. The steady axial velocity has been solved as an analytical function of the wall and fluid properties, the Womersley number, and the catheter radius, as given in Eq.(71) in the second-order problem. Because of wave damping, all the time-mean quantities contain an exponential factor for axial decay along the tube. The influences of the oscillation frequency and the catheter radius on steady streaming velocity, steady pressure gradient and steady shear stress have been studied. A catheter radius that is 30% of the radius of the inner wall surface is a possibly critical radius, below which the steady pressure gradient and the steady shear stress in most part of the fluid may not be appreciably affected by the presence of the catheter.

Extension of the present model is possible on taking into account additional effects due to non-Newtonian fluid, large amplitude wave, stenosis and curvature of the artery wall. For a problem with complex geometry or strong nonlinearity, numerical efforts should be employed[21,22]. Our analytical results presented here can be used as a benchmark to test the accuracy of a computational scheme for solving the Lagrangian problem numerically.

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10.1016/S1001-6058(09)60094-9

* Project supported by the Research Grants Council of the Hong Kong Special Administrative Region, China (Grant No. HKU 715609E).

Biograsphy: MA Ye (1982-), Male, Ph. D., Engineer